Description
Self-organization is a process where some form of global order or coordination arises out of the local interactions between the components of an initially disordered system.
ABSTRACT
Title of dissertation: DATA VISUALIZATION OF ASYMMETRIC DATA
USING SAMMON MAPPING AND APPLICATIONS OF
SELF-ORGANIZING MAPS
Haiyan Li, Doctor of Philosophy, 2005
Dissertation directed by: Professor Bruce L. Golden
Department of Decision and Information Technologies
Data visualization can be used to detect hidden structures and patterns in data sets
that are found in data mining applications. However, although efficient data visualization
algorithms to handle data sets with asymmetric proximities have been proposed, we
develop an improved algorithm in this dissertation.
In the first part of the proposal, we develop a modified Sammon mapping
approach that uses the upper triangular part and the lower triangular part of an
asymmetric distance matrix simultaneously. Our proposed approach is applied to two
asymmetric data sets: an American college selection data set, and a Canadian college
selection data set which contains rank information. When compared to other approaches
that are used in practice, our modified approach generates visual maps that have smaller
distance errors and provide more reasonable representations of the data sets.
In data visualization, self-organizing maps (SOM) have been used to cluster
points. In the second part of the proposal, we assess the performance of several software
implementations of SOM-based methods. Viscovery SOMine is found to be helpful in
determining the number of clusters and recovering the cluster structure of data sets. A
genocide and politicide data set is analyzed using Viscovery SOMine, followed by
another analysis on the public and private college data sets with the goal to find out
schools with best values.
DATA VISUALIZATION OF ASYMMETRIC DATA USING SAMMON MAPPING
AND APPLICATIONS OF SELF-ORGANIZING MAPS
by
Haiyan Li
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2005
Advisory Committee:
Professor Bruce L. Golden, Chair
Professor Edward A. Wasil
Assistant Professor Paul F. Zantek
Assistant Professor Steven Gabriel
Professor Ali Haghani
©Copyright by
Haiyan Li
2005
ii
ACKNOWLEDGEMENTS
The work with this dissertation has been extensive and trying, but in the first
place exciting, instructive, and fun. Without help, support, and encouragement from
several persons, I would never have been able to finish this work.
First of all, I would like to thank my advisor, Dr. Bruce Golden, for his generous
time and commitment. It is not often that one has an advisor that always contributes the
time for listening to the little problems and roadblocks that unavoidably crop up in the
course of performing research. Throughout my doctoral work he encouraged me to
develop independent thinking and research skills. He continually stimulated my
analytical thinking. His technical and editorial advices were essential to the completion
of this dissertation proposal and have taught me innumerable lessons and insights on the
workings of academic research in general.
I am also appreciative and thankful for the support and patience of Dr. Edward
Wasil, who has read through my draft copies, listened to my complaining and encouraged
me every step of the way. His insightful comments were crucial for editing the drafts
into the dissertation proposal. His timeliness in returning the drafts did not go unnoticed.
I am indebted to Dr. Paul Zantek who has been a source of support throughout my
graduate studies. His data mining class has been instrumental in shaping my
understanding of clustering and data visualization. I am very grateful to Dr. Steven
iii
Gabriel for the many valuable comments that helped shape this work. I would like to
express my appreciation to Dr. Ali Haghani for gladly serving on my dissertation
committee.
Thanks also to all my colleagues at the Department of Decision and Information
Technologies for providing a good working atmosphere, especially Stacy Calo for
assisting me with collecting survey and data entry during my late months of pregnancy.
At last, to my family thanks for supporting me with your love and understanding.
iv
TABLE OF CONTENTS
LIST OF TABLES………………………………………………………………………..vi
LIST OF FIGURES ......................................................................................................... viii
Chapter 1 Introduction................................................................................................. 1
1.1 Motivation and the Problem of Interest ...................................................... 1
1.2 Summary of Objectives............................................................................... 6
1.3 Dissertation Organization ........................................................................... 7
Chapter 2 Literature Review........................................................................................ 9
2.1 Data Visualization..................................................................................... 10
2.2 Multidimensional Scaling......................................................................... 13
2.2.1 Overview................................................................................................... 13
2.2.2 Problem definition and stress function ..................................................... 15
2.3 Sammon Mapping..................................................................................... 17
2.3.1 Overview................................................................................................... 17
2.3.2 SM algorithm............................................................................................ 18
2.3.3 Implementation ......................................................................................... 19
2.4 Self-organizing Maps................................................................................ 22
2.4.1 Overview................................................................................................... 22
2.4.2 SOM algorithm......................................................................................... 23
2.4.3 SOM software packages ........................................................................... 24
2.5 Asymmetric Proximity.............................................................................. 27
2.5.1 Overview................................................................................................... 27
2.5.2 Mathematical modeling for asymmetric proximity data........................... 28
2.5.3 Other approaches of analyzing asymmetric proximity data ..................... 31
Chapter 3 Constructing Sammon Maps from Asymmetric Data............................... 36
3.1 Modification of Sammon Mapping Method ............................................. 37
v
3.2 Implementation of the Modified SM Method........................................... 42
3.3 Performance Measurements of Sammon Maps ........................................ 48
Chapter 4 Visualizing American College Selection Data ......................................... 51
4.1 Description of the Data Set ....................................................................... 51
4.2 Experimental Design................................................................................. 57
4.3 Discussion of the Results.......................................................................... 60
Chapter 5 Visualizing Canadian Ranked College Data............................................. 81
5.1 Description of Canadian Ranked College Data ........................................ 82
5.2 Modeling Steps ......................................................................................... 85
5.3 Discussion of the Results.......................................................................... 87
Chapter 6 Self-Organizing Maps: State Sponsored Murder Data Set ....................... 95
6.1 Introduction............................................................................................... 95
6.2 Evaluating SOM-based Methods .............................................................. 96
6.2.1 Constructing data sets ............................................................................... 97
6.2.2 Measuring performance ............................................................................ 99
6.2.3 Comparison results and conclusions....................................................... 100
6.3 Self-Organizing Maps: the State Sponsored Murder Data Set ............... 104
Chapter 7 Self-Organizing Maps: Best Values in Colleges .................................... 122
7.1 Data Description and Preprocessing....................................................... 123
7.2 Discussion of Results.............................................................................. 133
Chapter 8 Future Work............................................................................................ 152
References …………………………………………………………………………..154
vi
LIST OF TABLES
Table 1.1 Asymmetric proximity matrix of schools A, B, and C. ....................... 4
Table 1.2 Example of a 3 × 3 asymmetric distance matrix.................................. 6
Table 2.1 Matrix of distances among 10 buildings............................................ 14
Table 2.2 Stress guidelines suggested by Kruskal (1964a). .............................. 17
Table 2.3 An example of car switching data. .................................................... 28
Table 2.4 Asymmetric distance matrix taken from the American college
selection data ..................................................................................... 34
Table 3.1 Asymmetric distance matrix of three data points. ............................. 41
Table 3.2 Results in our proposed approach with random starts 1. ................... 41
Table 3.3 Results in our proposed approach with random starts 2. ................... 41
Table 3.4 Results in the common approach with random starts 1. .................... 41
Table 3.5 Results in the common approach with random starts 2. .................... 42
Table 3.6 Asymmetric distance matrix of 30 American colleges. ..................... 44
Table 3.7 Symmetrized distance matrix of 30 American colleges. ................... 45
Table 3.8 Asymmetric distance matrix of four data points................................ 49
Table 3.9 Order relationships............................................................................. 50
Table 4.1 Adjacency matrix for six schools. ..................................................... 53
Table 4.2 One hundred schools selected from The Fiske Guide for analysis. ... 54
Table 4.3 Average error measures obtained from the modified SM method. ... 58
Table 4.4 Average order preservation coefficients obtained from the modified
SM method......................................................................................... 58
Table 4.5 Order preservation coefficients for the six data sets (standard vs.
modified)............................................................................................ 77
Table 4.6 Order preservation coefficients for the six data sets (Merino’s vs.
modified)............................................................................................ 77
Table 4.7 Error measures for the six data sets (standard vs. modified). ............ 78
Table 4.8 Error measures for the six data sets (Merino’s vs. modified)............ 78
Table 5.1 44 Canadian universities collected from surveys. ............................. 83
Table 5.2 Competitors of 44 Canadian universities........................................... 84
Table 5.3 Order preservation measures of different gap values of Canadian
ranked................................................................................................ 91
Table 5.4 Order preservation measures of different gap values of Canadian
ranked................................................................................................ 91
Table 5.5 Error measures of different gap values of Canadian ranked college
data.................................................................................................... 91
Table 5.6 Error measures of different gap values of Canadian ranked college
data (Merino’s vs. modified)............................................................. 92
vii
Table 6.1 Pairwise classification notation. ...................................................... 100
Table 6.2 Cluster recovery rates (in %). .......................................................... 101
Table 6.3 Cluster recovery rates (in %) by level of dispersion........................ 102
Table 6.4 Values of the Rand statistics............................................................ 102
Table 6.5 Cluster recovery rates (in %) for Viscovery. ……………………..104
Table 6.6 Genocide and politicide data set from Harff (2003). ....................... 106
Table 6.7 Transformed genocide and politicide data....................................... 108
Table 6.8 Notation for the transformed genocide and politicide data set. ....... 108
Table 6.9 Cluster profiles of the genocide and politicide data set. .................. 110
Table 6.10 Modified genocide and politicide data set 1 (M1)........................... 112
Table 6.11 Modified genocide and politicide data set 2 (M2)........................... 113
Table 6.12 Notation of the transformed genocide and politicide data with
added variables……………………………………………………..114
Table 6.13 Transformed modified genocide and politicide data set 1 (M1)...... 115
Table 6.14 Transformed modified genocide and politicide data set 2 (M2)...... 116
Table 6.15 Cluster profiles of the first modified data set (M1). ........................ 118
Table 6.16 Cluster profiles of the second modified data set (M2). ................... 119
Table 7.1 Seven common variables in Kiplinger’s public and private data
sets .………………………………………………………………..124
Table 7.2 Four additional variables in Kiplinger’s public and private data
sets. .................................................................................................. 124
Table 7.3 Kiplinger’s (2003) public college data. ........................................... 126
Table 7.4 Kiplingers’ (2004) private college data. .......................................... 130
Table 7.5 Summary of clusters of the public colleges. .................................... 134
Table 7.6 Cluster profiles of the public colleges. ............................................ 135
Table 7.7 Summary of clusters of the public colleges. .................................... 144
Table 7.8 Cluster profiles of the private college data. ..................................... 145
viii
LIST OF FIGURES
Figure 2.1 Example using principle components analysis. ................................. 11
Figure 2.2 Illustration of a principal curve.......................................................... 13
Figure 2.3 Example of multidimensional scaling. .............................................. 14
Figure 2.4 Sammon map of the 10 building........................................................ 18
Figure 2.5 Illustration of a self-organizing map.................................................. 23
Figure 2.6 Sammon map produced by SOM_Pak............................................... 25
Figure 2.7 Screenshot of Viscovery. Three groups (As, Bs, and Cs) are
shown in the....................................................................................... 26
Figure 2.8 Sammon map of the upper triangular matrix. .................................... 35
Figure 2.9 Sammon map of the lower triangular matrix. .................................... 35
Figure 3.1 Sammon map of the asymmetric distance matrix for data set 30A
generated by the modified SM method. ............................................ 46
Figure 3.2 Sammon map of the symmetrized distance matrix for data set
30A generated by the standard SM method. ..................................... 46
Figure 4.1 Directed graph generated from the adjacency matrix........................ 53
Figure 4.2 Map of 100 schools generated by the modified SM method. ............ 61
Figure 4.3 Map of 100 schools generated by the standard SM method. ............. 61
Figure 4.4 Map of 100 schools generated by Merino’s method. ........................ 62
Figure 4.5 Map of group A generated by the modified SM method................... 65
Figure 4.6 Map of group A generated by the standard SM method.................... 65
Figure 4.7 Map of group A generated by the Merino’s method. ........................ 66
Figure 4.8 Map of 30 schools from group A generated by the modified SM
method. .............................................................................................. 69
Figure 4.9 Map of 30 schools from group A generated by the standard SM
method. .............................................................................................. 69
Figure 4.10 Map of 30 schools from group A generated by Merino’s method. ... 70
Figure 4.11 Map of group B generated by the modified SM method. .................. 70
Figure 4.12 Map of group B generated by the standard SM method. ................... 71
Figure 4.13 Map of group B generated by Merino’s method. .............................. 71
Figure 4.14 Map of group C generated by the modified SM method. .................. 73
Figure 4.15 Map of group C generated by the standard SM method. ................... 73
Figure 4.16 Map of group C generated by Merino’s method. .............................. 74
Figure 4.17 Map of group D generated by the modified SM method................... 74
Figure 4.18 Map of group D generated by the standard SM method.................... 75
Figure 4.19 Map of group D generated by Merino’s method. .............................. 75
Figure 5.1 Sammon map of the Canadian ranked college generated by the
modified method................................................................................ 88
Figure 5.2 Sammon map of the Canadian ranked college generated by the
standard method................................................................................. 89
ix
Figure 5.3 Sammon map of the Canadian ranked college generated by Merino’s
method. .............................................................................................. 89
Figure 5.4 Sammon map of the Canadian ranked college generated by the
modified method. ............................................................................... 92
Figure 5.5 Sammon map of the Canadian ranked college generated by the
modified method. ............................................................................... 93
Figure 6.1 Example of a four-cluster data set. .................................................... 98
Figure 6.2 Resulting SOM map of the genocide and politicide data set. .......... 109
Figure 6.3 Resulting SOM map of the first modified data set (M1). ................ 117
Figure 6.4 Resulting SOM amp of the second modified data set (M2). ........... 117
Figure 7.1 Map for the public college data set. ................................................. 134
Figure 7.2 Map for the private college data set................................................. 144
1
Chapter 1
Introduction
In this chapter, we provide background information on data visualization and
describe the motivation and objectives of our research.
1.1 Motivation and the Problem of Interest
Data visualization is the process of “representing data as a visual image” (Latham,
1995) in which an image is created using a combination of points, lines, coordinate
systems, numbers, symbols, words, shadings, and colors to represent different measured
quantities (Tufte, 1983).
Data visualization is often used to make apparent any pattern in a data set that is
large in size or dimensionality. For example, analyzing increasing amounts of data, such
as customer data, to discover hidden patterns is a major problem facing businesses and
organizations today. Visualization, together with other data mining techniques such as
clustering and classification, can be employed to generate a data map that serves as a
guide and provides the user with insights, i.e., detecting customer purchase patterns. The
ability to show patterns attracts decision makers to use data visualization as a tool to get a
2
better understanding of the data set and then make better decisions. For example,
consider the problem facing each high-school senior: selecting an undergraduate
American university or college to pursue a bachelor’s degree. Students can consult
rankings in popular publications and reference books such as The Fiske Guide, which
provides information on 300 universities and colleges. The information contained in
these publications and books is not easy to assimilate. Condon et al. (2002) built a model
of American universities that enables a student to visualize the data. Information that
cannot be revealed in lists and tables, such as similarities between the universities, can be
directly quantified according to the distances between universities on visual maps. These
maps can assist students in identifying similar universities to consider.
As data visualization receives more and more attention, a variety of methods for
visualization have been proposed including self-organizing maps (SOM), multi-
dimensional scaling (MDS), and Sammon mapping (SM). These methods have been
widely used in data visualization, as they are easy to implement and have modest
computational requirements. Our proposed work is closely related to the Sammon
mapping method, which we will describe in later chapters. SOM, MDS, and SM usually
deal with the problems that have symmetric structures, for example, symmetric distance
matrices. These methods take the table format of data sets as input, where rows are
observations and columns are attributes. MDS and SM also accept a distance matrix as
input but they require a symmetric distance matrix.
3
A proximity matrix or a similarity matrix contains the pairwise distances or
similarities between all pairs of data items in a data set. In a proximity matrix, the
distances are usually assumed to be symmetric. However, in practice, there are many
interesting problems in which asymmetric proximities arise, especially in marketing or
human behavior surveys. Asymmetric proximity is one type of proximity in which the
pairwise distances are not symmetric. For example, in terms of teaching quality, the
president of university A thinks that university B is the most competitive rival, while the
president of university B thinks that the closest competitor is university C and not
university A, and the president of university C thinks that university A is the closest
competitor. The corresponding proximity matrix is shown in Table 1.1. If school j is a
competitor of school i, then d
ij
= 1, otherwise, d
ij
= 0. Clearly, the matrix is asymmetric,
i.e. entry (i, j) = entry (j, i) for some i and j, i = j.
Since visualization methods normally work on symmetric cases, a question that
needs to be answered is how to visualize asymmetric cases such as the matrix in Table
1.1. A visualization method that can handle asymmetry may need to be developed, or
some modification may need to be made to an existing visualization method. A simple
modification could be made by averaging off-diagonal entries to create a symmetric
matrix. However, replacing asymmetric distances with an averaged distance alters the
structure of the data set in a way that may result in a less accurate representation.
In order to maintain the asymmetric information of the data set, Merino and
Munoz (2001) developed asymmetry coefficients. Mathematically, they defined an
asymmetry coefficient of a data observation as a summation of the standardized
similarities of the data observation to all of the other data observations (details such as
4
Competitor
School
A B C
A -- 1 0
B 0 -- 1
C 1 0 --
Table 1.1 Asymmetric proximity matrix of schools A, B, and C.
transforming distances to similarities are described in the next chapter). Merino and
Munoz incorporated asymmetry coefficients into the MDS and SM objective functions
and corresponding search procedures. Their approach dealt with the symmetrized
distance matrix of data observations. Based on their computational results, Merino and
Munoz observed that data observations with large asymmetry coefficients were more
influential in determining the structure of a map. However, data observations that are
similar to many other data observations are usually dominant in forming the structure of a
visual map, regardless of whether asymmetry coefficients are introduced into the MDS or
SM methods. When most of the data observations have similar asymmetry coefficients,
there is little or no impact the coefficients have on influencing the structure of a map.
In order to visualize asymmetric cases, we can examine the upper triangular part
and the lower triangular part of the matrix simultaneously rather than study the
symmetrized distance matrix derived by averaging corresponding upper and lower entries
in the asymmetric distance matrix, or arithmetically calculate asymmetric coefficients.
We expect that the visual maps generated by taking into account the upper triangular part
and the lower triangular part of the matrix simultaneously are a better representation of
the asymmetric data, and hopefully produce more insight into the data sets. For example,
5
we apply our approach to a small 3 × 3 asymmetric distance matrix given in Table 1.2.
We use the GRG Solver, an optimization software developed by Frontline Systems
(www.solver. com), to solve the small problem in our proposed approach and in one of
the traditional approaches (i.e., the standard Sammon mapping approach which takes the
average values of asymmetric parts as input) for comparison. Our approach seeks to
optimize a Sammon mapping objective function directly on the whole asymmetric
distance matrix. The common approach aims to solve the same optimization problem on
the symmetrized distance matrix of the asymmetric data. The results show that our
proposed approach is better than the one in the common approach (detailed discussions
are given in Chapter 3) even in this tiny data set with only three data points. In addition,
an asymmetric case with ranking information will be analyzed to extend our research on
visualization of asymmetric problems.
Once an algorithm is implemented, it is necessary to check that if the algorithm
works effectively as compared to other algorithms. Meanwhile, it is helpful to see if the
algorithm is reliable on different data sets. We will apply our algorithm to twodata sets:
American college selection data and Canadian college data. The proximity matrix of
each data set is asymmetric. The American college selection data are gathered from the
Fiske Guide (1999). Canadian college data are collected from a survey that we
conducted. Detailed information about each data set will be given in later chapters.
Another part of our study focuses on applications of data visualization.
Clustering is such an application of data visualization that is widely used to detect hidden
patterns
6
Asymm A B C
A 0 1 3
B 2 0 2
C 3 4 0
Table 1.2 Example of a 3 × 3 asymmetric distance matrix.
that cannot be observed directly from enormous amounts of data. Several SOM-based
software implementations for clustering are available either commercially or free of
charge, such as Som_Pak (1997) and Viscovery (2002). However, there is little
information indicating which implementation works better in practice. In our work, we
will evaluate the performance of four software implementations of SOM-based clustering
methods. Based on our evaluation, we will analyze several applications, such as
clustering a state murder data set (Harff, 2003) and finding out colleges with best values
(2003, 2004).
1.2 Summary of Objectives
A summary of the research work that has been done is as follows.
Firstly, we developed a new visualization method that uses data with asymmetric
proximities.
Second, we implemented our visualization method using a gradient descent
algorithm.
7
Third, we applied our method to twodata sets: American college selection and
Canadian college selection and compared our results with the results generated by the
standard method and the Merino’s method.
Fourthly, we assessed four SOM-based clustering methods and analyzed
clustering applications: state-sponsored data and college data with the best values.
1.3 Dissertation Organization
In Chapter 2, an overview of the literature on data visualization, including
multidimensional scaling, Sammon mapping, and self-organizing maps, is provided. We
define asymmetric distances and discuss several techniques that are used to handle an
asymmetric distance matrix for visualization.
In Chapter 3, our modified Sammon mapping algorithm and its implementation
are described and evaluated.
In Chapter 4, our modified Sammon mapping method is applied to American
college selection data. We give the construction procedures of the data set. We discuss
the results and compare our results to results generated by the commonly used standard
SM method and Merino’s method.
In Chapter 5, problems with ranking information are introduced. An example is
given to show the process of constructing a distance matrix that incorporates ranking
information. Canadian college selection data is analyzed by using our modified SM
method. We discuss our preliminary results and provide insights gained from our work.
8
In Chapter 6, the performances of four SOM-based clustering software
implementations are evaluated. We analyze an application (i.e., state-sponsored data)
using Viscovery.
In Chapter 7, we apply Viscovery to public and private college data to find out
colleges with best values. The visual maps of the public college data and the private
college data are given and the results are discussed.
In Chapter 8, we summarize our research and point out our future work.
9
Chapter 2
Literature Review
Data visualization is one technique that can help researchers and business
decision makers discover patterns in a data set. Data visualization has received lots of
attention in the literature. Many methods (e.g., multidimensional scaling and Sammon
mapping) and systems have been proposed and implemented in research and business
areas like biomedical science, marketing, and financial services.
In this chapter, we provide an overview of the relevant literature in data
visualization. First, we survey the papers that pertain to data visualization. Second, we
examine papers on multidimensional scaling, Sammon mapping, and self-organizing
maps. Third, we discuss several approaches dealing with asymmetric proximity data.
10
2.1 Data Visualization
There are many visualization methods that have been proposed for illustrating
structures and multivariate relationships among data items. These methods can be
classified into two categories: linear visualization methods and nonlinear visualization
methods. In this section, we will discuss some of these methods.
Principle component analysis (PCA) (Hotelling, 1933) is a standard method in
data analysis. Principle components are a set of variables that define a projection that
encapsulates the maximum amount of variation in a dataset and is orthogonal (and
therefore uncorrelated) to the previous principle component of the same (see Figure 2.1).
Projection pursuit (Friedman, 1987) tries to show the best visual representation that
reveals as much of the non-normally distributed structure of the data set as possible. A
neural network implementation of this method is given by Fyfe and Baddeley (1995).
PCA cannot take into account nonlinear structures and projection pursuit cannot
project the nonlinear structures onto a low-dimensional display if the data set has many
dimensions and is highly nonlinear. Several approaches have been proposed to project
nonlinear, high-dimensional structures onto a low-dimensional space. The most common
methods allocate each individual data point onto a lower dimensional display and then
optimize the display so that the distances between the points are as close as possible to
the original distances. These methods differ in the selection of the objective function and
the optimization approach.
Multidimensional scaling (MDS) refers to a group of methods that use proximities
among data points to produce a representation of the data set (Kruskal, 1964a; Shepard,
1962). The representation consists of a geometric configuration of the points on a map
11
Figure 2.1 Example using principle components analysis.
where each point corresponds to one of the data items. MDS is widely used in
behavioral, economic, and social sciences to analyze the pairwise proximities of the data
points (e.g., similarity of brands in a market survey). MDS is discussed in greater detail
in a later section.
Another nonlinear visualization method is Sammon mapping (Sammon, 1969).
Sammon mapping (SM) is closely related to MDS. It tries to optimize an objective
function in order to preserve the relative pairwise distance between data points. Details
on MDS and Sammon mapping are provided in Sections 2.2 and 2.3.
Principal curves are a nonlinear generalization of PCA that projects a data set
onto a nonlinear manifold after a linear manifold of the data set has been generated using
PCA. Here, a manifold (or surface) refers to a topological space on which every point
12
has a neighborhood sharing some essential features of the data set (i.e., a sphere is a
manifold). Principal curves were first defined as self-consistent smooth curves (Hastie
and Stuetzle, 1989) that pass through the middle of a d-dimensional probability
distribution or data cloud (see Figure 2.2). Extensions of principal curves use
multidimensional base functions to construct a nonlinear manifold, such as adaptive
principal surfaces (LeBlanc and Tibshirani, 1994). Another popular approach is to use
variants of the self-organizing map (SOM) algorithm (e.g., apply self-organizing maps to
extract principal curves and extended principal curves from data (Der et al., 1998)). The
self-organizing map is an efficient tool for the visualization of high-dimensional data sets
(Vesanto, 1999).
Nonlinear visualization methods are computationally very intensive for large data
sets. The triangulation method (Lee et al., 1977) can be used to reduce the computational
complexity. An important property of the triangulation method is that the distances to its
nearest two neighbors can be preserved exactly when inserting a new data item. Using
the triangulation method, data items will be projected onto a map one by one and the
nearest neighbor distances can always be preserved, that is, the generated map is based on
a subset of distances in the original space. The projection process is, thus, substantially
faster than nonlinear visualization methods. However, the generated map from the
triangulation method may not be as accurate as the map from nonlinear visualization
methods since the projection preserves only a part of distances.
13
Figure 2.2 Illustration of a principal curve.
2.2 Multidimensional Scaling
2.2.1 Overview
As we mentioned in Section 2.1, multidimensional scaling is a collection of
visualization methods that project proximity data onto lower dimensional space. In
general, the goal of multidimensional scaling is to provide a visual representation of
proximities in a set of investigated objects. Proximity data are always represented as
distances. For example, in Table 2.1, each entry represents the pairwise distance between
two buildings.
To better illustrate the idea of MDS, consider an example that visualizes the
locations of 10 buildings. Given the symmetric matrix of distances among 10 buildings
(see Table 2.1), MDS produces the map given in Figure 2.3.
14
A B C D E F G H I J
A 0 14.5 14 1 12.5 17.5 12 11.5 16.5 12
B 14.5 0 1 14.5 7 9.5 5.5 5 9.5 11.5
C 14 1 0 14 6 8.5 4.5 4 8.5 11.5
D 1 14.5 14 0 12.5 17.5 12 11.5 16.5 12
E 12.5 7 6 12.5 0 9.5 3 2 9.5 10.5
F 17.5 9.5 8.5 17.5 9.5 0 7.5 7.5 1.5 11
G 12 5.5 4.5 12 3 7.5 0 1 7.5 11
H 11.5 5 4 11.5 2 7.5 1 0 7.5 11
I 16.5 9.5 8.5 16.5 9.5 1.5 7.5 7.5 0 10.5
J 12 11.5 11.5 12 10.5 11 11 11 10.5 0
Table 2.1 Matrix of distances among 10 buildings.
A
B C
D
E
F
G
H
I
J
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
-8 -6 -4 -2 0 2 4 6 8 10
Figure 2.3 Example of multidimensional scaling.
15
In this example, the relationship between the original distances among data points
and resulting distances shown on the map is positive, that is, the smaller the original
distance, the closer the resulting distance between points, and vice versa. If the original
proximity data had been represented as similarities, the relationship would have been
negative which means the smaller the similarity between two data items, the farther apart
in the map they would be. In our study, proximity data are represented as distances
among data items.
2.2.2 Problem definition and stress function
From a slightly more technical point of view, for a set of observed distances
between every pair of N items, multidimensional scaling methods aim to find a visual
representation of the items in lower dimensional space such that the resulting distances
among items match the original distances as closely as possible. The metric version of
MDS aims to find configurations for data items where the resulting distances are as close
as possible to the original distances of data items. Nonmetric MDS tries to keep the rank
orders of the distances among data items as close as possible to the original rank orders.
We consider only nonmetric MDS in this dissertation.
The mathematical definition of MDS now follows. Given N items and a
corresponding distance matrix where entry d
ij
is the pairwise distance between data items
i and j, MDS seeks to find vector configurations ] ,..., 1 : [
*
p k x x
ik i
= =
-
and
] ,..., 1 : [
*
p k x x
jk j
= =
-
for data items i and j (i, j = 1,…,N, i j) in a p-dimensional space
(p N – 1), such that the Euclidian distance between
-
i
x and
-
j
x
16
¿
=
÷ = ÷ =
p
k
jk ik j i ij
x x x x d
1
2 * * * * *
) ( ,
ij ij j i
d d ~ ¬
<
*
.
approximates the corresponding distance
ij
d , for all pairs of data items i and j.
Since the proximity matrix is assumed to be symmetric, it is sufficient to take into
account each pair of data items i and j just once. However, it may not be possible to
perfectly represent the original distances (Johnson and Wichern, 1998) in a given lower
dimensional space. Therefore, a numerical measure is needed to indicate the closeness
and the measure is called a stress function.
Kruskal’s stress (Kruskal, 1964a), known as Stress formula 1 or Stress 1 for short,
measures the extent to which a representation deviates from a perfect match and is
defined as
¿
¿
<
<
÷
j i
ij
j i
ij ij
d
d d
2
2 *
) (
.
If the stress is zero, then the resulting pairwise distances are exactly the same as
the pairwise distances in the original distance matrix. However, in order to be useful, it
is not necessary to require a zero stress value as long as a certain amount of distortion is
tolerable. Kruskal (1964a) provides guidelines for the amount of stress to tolerate (see
Table 2.2).
Multidimensional scaling has been applied in many areas--the literature is vast
and dispersed over many periodicals and books. We will not attempt to give an overview
of the developments to date, and refer the reader elsewhere for details (Borg and
Groenen, 1997; Cox and Cox, 1994).
17
Stress Goodness of fit
0.2 Poor
0.1 Fair
0.05 Good
0.025 Excellent
0 Perfect
Table 2.2 Stress guidelines suggested by Kruskal (1964a).
2.3 Sammon Mapping
2.3.1 Overview
Frequently, a Sammon map is used for data exploration. A practical area of
Sammon mapping is in the visualization of protein structures based on measures of
similarity between molecules. For example, Sammon maps have been used to analyze
protein sequence relationships (Agrafiotis, 1997).
Like other MDS visualization methods, SM deals with proximity data. We are
given distances between every pair of data items (we possibly have no direct access to
any high-dimensional data but we do have access to a measure of distances between
every two data items). SM tries to reconstruct the original data based solely on the given
distance matrix. For example, given the distances between two buildings, SM can be
used to construct a map for the coordinates of the building themselves. A demonstration
of SM is presented in Figure 2.4 using the distance matrix shown in Table 2.1.
18
Figure 2.4 Sammon map of the 10 building.
2.3.2 SM algorithm
SM is an unsupervised nonlinear method that tries to preserve relative distances
(Lerner et al., 1998). Here “unsupervised” means no targeted information or outcome to
predict. To preserve the inherent structure, the algorithm that generates a Sammon map
employs a nonlinear transformation of the observed distances among data items when
mapping data items from a high-dimensional space onto a low-dimensional space.
If we denote the distance (usually Euclidean distance) between two data items i
and j, i j, in the original dimensional space by
ij
d and the distance in the required
projected space by
'
ij
d , the error function of SM is given by
19
¿ ¿
¿ ¿
= + =
= + =
÷
=
n
i
n
i j ij
ij ij
n
i
n
i j
ij
d
d d
d
E
1 1
2 '
1 1
) (
1
.
In the error function E, the smaller the error value, the better the map we get.
However, in practice, we are often unlikely to obtain perfect maps especially when the
data set is large and in high-dimensional space. Therefore, approximate preservation is
the likely result when we project high-dimensional data onto a low-dimensional plane.
The error function of SM is similar to Kruskal’s S stress function (see Section
2.2.2) except that each squared difference of distances is divided by the corresponding
input distance rather than the squared input distance. In other words, the only difference
between SM and MDS is that the errors in distance preservation are normalized with the
distance in the original space. Because of the normalization, SM places greater emphasis
on smaller distances rather than on larger distances; this is different from Kruskal’s MDS
which treats small and large distances roughly the same.
2.3.3 Implementation
SM can also be viewed as an optimization problem and its error function can be
minimized using several available techniques. Sammon solved the minimization problem
using steepest gradient descent that is also referred to as pseudo-Newton minimization
(Becker and Le Cun, 1989). This optimization procedure can be achieved by iteratively
using the following rule
2 '
2
'
' '
) (
) (
) (
) (
) ( ) 1 (
t x
t E
t x
t E
t x t x
ik
ik
ik ik
c
c
c
c
÷ = + o .
20
Note that
'
ik
x is the k
th
coordinate of the position of data item i in the required
projected space, and is called “magic factor” which controls the step size of updating
coordinates.
We point out that Sammon (1969) suggested a value of 0.3 to 0.4 for . However,
since is experimentally determined, the suggested value may not be optimal for all
problems. Therefore, multiple experiments are necessary in order to find an appropriate
value of to minimize E.
Because a second derivative is used in the denominator, the update rule can be
unstable at some points where the second derivative is very small. To avoid the
instability, an alternative minimization technique, called normal gradient descent, has
been used, where
) (
) (
) ( ) 1 (
'
' '
t x
t E
t x t x
ik
ik ik
c
c
÷ = + o .
When employing the gradient descent procedure to search for a minimum error
value, a local minimum in the error surface could be obtained. Therefore, several
experiments with different random initializations may be necessary. Another problem is
the computational requirement of SM is O(n
2
). The pairwise distances and the
derivatives have to be computed each iteration. Therefore, as the number of data items
increases, the computational time increases dramatically. To lower the computational
overhead, the Hamming metric has been used as a distance measure instead of the
Euclidean metric (White, 1972). This showed some improvement in the computational
efficiency, but the resulting maps could be distorted when the input space is the
Euclidean space (Chien, 1978) and the interpretation becomes more complex.
21
So far, all of the problems that we have discussed are assumed to have symmetric
proximity data, that is, the distance between data items i and j denoted by d
ij
is exactly
same as the distance between data items j and i denoted by d
ji
. However, in practice,
there are lots of interesting problems that have asymmetric proximity data, which means
that the distance d
ij
may be different from the distance d
ji
.
To our knowledge, there are few papers that discuss the visualization of
asymmetric proximity data sets. One of our major research goals is to develop a method
that visualizes asymmetric proximity data sets. Our objective function and search
procedure for implementing SM with asymmetric data are discussed in Chapter 3.
22
2.4 Self-organizing Maps
2.4.1 Overview
The self-organizing map (SOM) invented by Kohonen in the early 1980s is a type
of neural network based on the idea of self-organized or unsupervised learning (Kohonen,
1995). This means that the algorithm has no targeted information or outcome to predict.
Consequently, SOMs are ideal for clustering where no requirement of output fields is
defined. However, SOM can also be employed to visualize high-dimensional data items
(Flexer, 1999).
Being a stable and flexible method for clustering and visualization, SOM has been
used for a wide range of purposes, ranging from controlling industry processes to
analyzing gene data (Kaski et al., 2001). Many applications of SOM are given in the
survey by Oja et al. (2003).
An illustration of the mechanism of SOM is shown in Figure 2.5. The 3 3× map
consists of two layers: the input layer, and the output layer, which is often referred to as
the output map. All the input neurons are connected to all the output neurons, and
weights are assigned to each connection (not shown).
23
Figure 2.5 Illustration of a self-organizing map.
2.4.2 SOM algorithm
The self-organizing map trains by locating data items, one by one, to the output
map. When an input data item is presented to an output neuron, its characteristics are
compared with those of all output neurons, which are given initial weights. The neuron
with the characteristics that are most similar to that of the input data item is chosen to
represent the input data item; at the same time, the surrounding neurons of this chosen
neuron are adjusted to be more similar to the chosen neuron in order to attract input data
items similar to the mapped data item. This chosen neuron has a better chance, as
compared to other neurons, of representing input data items that have similar patterns,
and its neighbors are gradually able to represent similar input data items.
Each input data item is attracted to one and only one neuron, while each neuron
can attract one or more data items. Each neuron i has a reference vector m
i
= [m
i1
, …,
m
in
], which is used to represent an input data item, where n is the dimensionality of the
Input layer
Output layer
24
input space. When locating an input data item x on the output map, the neuron w is the
winner selected based on the minimum Euclidean distance, that is,
{ }
i
i
m x w ÷ = min , where .. is Euclidean distance.
During the training process, the winning neuron, also called the best matching
unit, and its neighbors are allowed to modify their reference vectors as close to the
current input data item as possible. The general form of the modification is given by
)] ( ) ( )[ ( ) ( ) ( ) 1 ( t m t x t h t t m t m
i wi i i
÷ + = + o ,
where ) (t o is the learning rate that controls the training speed and ) (t h
wi
is the
neighborhood function centered on the winning neuron w (this function indicates the
radius of neighborhood set) and ) (t x is the input at time t. Initially, the neighborhood
function is set to a large value; this value decreases monotonically with time, as does the
learning rate.
2.4.3 SOM software packages
In this section, we focus on two SOM software packages that are publicly
available: SOM_Pak and Viscovery SOMine.
SOM_Pak was developed at the Helsinki University of Technology. It is a
command line program and the interface is not user friendly; for example, there is no
simple option for executing repetitive commands and the user has to type in each
command. SOM_Pak can be downloaded for free; see the Web site at http://www.cis.
hut.fi/research/som_pak/. A screenshot of SOM_Pak is shown in Figure 2.6.
Viscovery SOMine was developed by Eudaptics (www.eudaptics.com). An
important advantage of SOMine is that it allows the user to visualize and analyze a data
25
Figure 2.6 Sammon map of a distance matrix of 20 data items produced by SOM_Pak.
set without any prior statistical knowledge of the data set. The software provides
suggestions as to which data items should be grouped together. The user can use or
modify several parameters to control data processing. A screenshot of a Viscovery
SOMine map is shown in Figure 2.7.
26
Figure 2.7 Screenshot of Viscovery. Three groups (As, Bs, and Cs) are shown in the
map; the dark lines separate groups.
27
2.5 Asymmetric Proximity
2.5.1 Overview
Proximity refers to the similarity (sometimes also refers to dissimilarity or
distance) between two data items. If the proximity between two data items is measured
in Euclidean space (i.e.,
2 2
) ( ) (
j i j i ji ij
y y x x d d ÷ + ÷ = = ), then the proximity is
symmetric. However, the proximity between data items or objects might not be
symmetric. When objects are compared from different perspectives, for example, object
a is said similar to object b in terms of color closeness, while object b is said dissimilar to
object a because of their different shapes, the proximity relationship between these two
objects is asymmetric, i.e.,
ba ab
d d = .
Asymmetric proximity data arises in a number of diverse research areas such as
marketing, psychology, sociology, and information retrieval. Asymmetric proximity data
can usually be found in frequency matrices. For example, brand switching data has been
utilized in marketing to examine the structure of competition within a particular product
class. An example of car-switching data is given in Table 2.3. The rows represent cars
last owned and the columns reveal cars currently owned. Out of all customers, 40 who
last owned a Ford switch to a Honda. Other examples of asymmetric proximity data are
migration rates between countries, frequencies of journal citations, word relationships in
text documents, etc.
In the past, especially from the late 1970s to the early 1990s, analysis of
asymmetry in proximity data had been one of the most provocative research topics in
psychological research areas, in contrast to traditional MDS approaches. Researchers
realized the psychological relevance of asymmetry in proximity data.
28
BMW Ford Honda Toyata GM
BMW 180 40 20 0 10
Ford 20 343 40 30 70
Honda 10 20 120 10 20
Toyata 30 20 30 70 10
GM 10 20 0 20 250
Table 2.3 An example of car switching data.
Not surprisingly, many approaches for asymmetric proximity data were proposed
in the psychological area. Tversky (1977) initially challenged the validity of the
traditional spatial model (i.e., MDS model) regarding the observed violation of
symmetry, minimality, and triangle inequality conditions of the metric axioms in actual
data. Krumhansl (1978) also discussed the problems of traditional spatial model and
proposed her distance-density model as an alternative to Tversky’s (1977) feature-
matching model. Other researchers proposed many different treatments of asymmetric
proximity data. However, there has not been found any model that is not only
mathematically sound but also practically applicable and easily interpretable.
2.5.2 Mathematical modeling for asymmetric proximity data
As many relationships are intrinsically asymmetric (Tversky, 1977) and
increasing attention has been paid to asymmetric proximity data, a number of approaches
have been developed to analyze asymmetry in proximity data. Most of these can be
classified into three categories.
29
Since traditional approaches represent relationships between data items
symmetrically, asymmetry is discarded as noise with respect to the symmetric part of the
proximity data. The symmetric part is extracted and the data are symmetrized. One
common approach to do this is to average corresponding off-diagonal entries (Kruskal,
1964b), i.e., substitute
ij
d by '
ij
d , where
2
'
ji ij
ij
d d
d
+
= , and then apply the symmetric
model, i.e., traditional MDS. For example, Tversky and Hutchinson (1986) analyzed 39
asymmetric proximity data by averaging. Another way of symmetrization is proposed by
Levin and Brown (1979) who derived row multiplicative constants from two least square
procedures to scale rows or columns of the asymmetric matrix to maximize symmetry.
However, symmetrization of asymmetric proximity data may ignore some important
information brought by asymmetry, and the symmetric solution found in the dimensional
spaces does not depict anything about the asymmetry.
Approaches in the second category aim to capture the information of asymmetry
in addition to the symmetric structure of the data. All major models in this category
involve a symmetric component and an asymmetric component. Krumhansl (1978)
specified a distance-density model in which object A and object B are represented in
projected low dimensional space, the similarity between A and B can be interpreted not
only by the interpoint distance but also the density of points in the surrounding
configuration. In other words, asymmetries are accounted for through points around A
and B. Saito (1986) developed an MDS approach in which estimated constants
considered as density constants are added to the symmetric configuration in relation to
the distance-density model. Different from the distance-density model, Constantine and
Gower proposed an approach partitioning an asymmetric matrix into two matrices: S
30
(symmetric) and N (skew-symmetric; i.e., n
ij
= ÷ n
ji
). A singular value decomposition of
N was performed to obtain a least-squares fit to be plotted in low dimensional space (i.e.,
in two dimensions) and an interpretation of asymmetry was provided. Weeks and Bentler
(1982) specified a model in which similarity is represented as distance, and traditional
additive constants are combined to reflect asymmetries. Description of other models for
asymmetric proximity data can be found in the paper of Zielman and Heiser (1996) in
which most of mentioned models for asymmetric proximity can be decomposed into a
symmetric part and a completely asymmetric part (i.e., skew-symmetry). These models
are mathematically elegant and the resulting dimensional configurations are interpretable.
However, these models assume an underlying symmetric component of the data but the
assumption might not fit every case. Besides, software for computing the model
parameters is not available (Zeilman and Heiser, 1996).
Another category of approaches for scaling asymmetric proximities is a graph-
theoretic representation of asymmetric proximity data (Cunningham, 1978; Hutchingson,
1989; Klauer, 1989). In these models, asymmetries are represented as directed distances.
These models do not require an underlying symmetric relationship. The differences
between these graphic models are the representation type, for example, Cunningham
(1978) employed directed trees as representations of proximities, and Hutchingson (1989)
used networks to represent proximities data. One disadvantage of a graph representation
is that the representation is limited to small data sets. Graph representation of large data
sets seems messy because of a large number of arcs.
31
2.5.3 Other approaches of analyzing asymmetric proximity data
In addition to the above three categories of approaches of modeling asymmetric
proximity data, there are other approaches sharing some features of the second and the
third categories (Rodgers and Thompson, 1992; Merino and Munoz, 2001). Rodgers and
Thompson proposed an approach in which asymmetric proximity data have been
preprocessed using the idea of seriation before applying MDS to the data. Seriation is a
procedure that orders data items on a continuum in order to maximize the sum of the
elements above (or below) the main diagonal (Baker and Hubert, 1977). Rodgers and
Thompson used the seriation algorithm and ordered the data items according to number
of dominances over other data items. They defined dominance as: if s
ij
> s
ji
, then i
dominates j. The data items that consistently dominate other data items are placed lower
in the ordering, i.e., the data item that dominates all other data items is placed on the
bottom row in the below diagonal triangular. MDS is fit to the ordered above diagonal or
below diagonal triangular that explicitly exhibits the dominance relationships of data
items and the resulting MDS configuration contains a directed distance.
Merino and Munoz (2001) introduced another approach to scaling asymmetry
proximity data in which asymmetry coefficients derived from the skew-symmetry matrix
are attached to MDS objective functions and MDS scaling was performed on the
symmetrized proximity data. An asymmetry coefficient is defined as a summation of the
standardized similarities of a data item to all other data items. Intuitively, data items that
are similar to more data items will have larger asymmetry coefficients. In some sense,
asymmetry coefficients convey the dominance information of data items, and data items
with large asymmetry coefficients determine the structure of the configuration.
32
Empirically, it shows that the most dominant data items have a tendency to be placed in
the center of a configuration, and this phenomenon may be interpreted as indirect
confirmation of the usefulness of the Euclidean space, which recognized the central role
of the dominant data items (Tversky and Hutchinson, 1986).
The approaches proposed by Rodgers & Thompson (1992), and Merino & Munoz
(2001) are more of an exploratory data analysis and less specific asymmetric models than
the methods described above. The approach by Rodgers & Thompson is flexible and
tractable, i.e., the ordered triangular of data items, and substantive information is more
accessible to data analysts. Merino and Munoz’s approach suggests that it might be
worth incorporating asymmetry coefficients to improve the visual configuration (Merino
and Munoz, 2001). However, both approaches have some disadvantages. Since Rodgers
and Thompson only considered the maximized triangular part (i.e., below the diagonal),
some important information contained in the other triangular part would be lost,
especially if the other triangular part contained many non-zero values. In Merino and
Munoz’s approach, the definition of asymmetry coefficient might not be suitable in every
case, for example, if most of the data items have approximately equal asymmetry
coefficients, i.e., defined as sum of row (or column) similarities, then the defined
coefficients may not reveal dominant information. In this case, it might be better to
define the coefficient as some function of dominance to reveal the centrality status.
Details on centrality are in Tversky and Hutchinson’s paper (1986).
From a visualization perspective, the visual configurations generated from the
upper triangular and lower triangular parts of an asymmetric proximity matrix are
implicitly different. For example, there are two SM maps generated by using the upper
33
triangular part (see Figure 2.8) and the lower triangular part (see Figure 2.9),
respectively, of the asymmetric distance matrix given in Table 2.4. It is not convincing
that maps generated from the upper triangular part are better than maps generated from
the lower part, or vice versa. In addition, if there are no substantive reasons to assume
that the underlying relationships between data items are symmetric, a natural way to
visualize asymmetric proximity data is to simultaneously take into account the
asymmetric parts, i.e., the upper triangular part and the lower triangular part of the
proximity matrix. To assess the quality of maps, we can quantify the rank preservation
by comparing the generated map results with the order relationships of the original
asymmetric dataset. Chapter 3 discusses our proposed modified SM method and
measures for quality assessment.
34
B1 A2 A3 B4 A5 A6 A7 A8 A9 C10
B1 0 9 8 1 5 15 4 3 13 4
A2 20 0 1 20 6 10 4 4 8 3
A3 20 1 0 20 5 9 3 3 7 3
B4 1 9 8 0 5 15 4 3 13 4
A5 20 8 7 20 0 14 3 2 12 1
A6 20 9 8 20 5 0 2 3 1 2
A7 20 7 6 20 3 13 0 1 11 2
A8 20 6 5 20 2 12 1 0 10 2
A9 20 11 10 20 7 2 4 5 0 1
C10 20 20 20 20 20 20 20 20 20 0
Table 2.4 Asymmetric distance matrix taken from the American college selection data
set.
35
Figure 2.8 Sammon map of the upper triangular matrix.
Figure 2.9 Sammon map of the lower triangular matrix.
36
Chapter 3
Constructing Sammon Maps from Asymmetric Data
Sammon maps are one of the most widely used tools in visualization and
clustering. Sammon mapping (SM) projects high-dimensional data onto a 2-dimensional
output map. A Sammon map is usually created for proximity data with a symmetric
distance matrix. However, there are many applications (e.g., American college selection
data) that have asymmetric distance matrices. In this chapter, we discuss the use of SM
to visualize asymmetric proximity data sets. We describe the objective function, the
associated update rule, and our implementation of SM for proximity data with an
asymmetric distance matrix.
37
3.1 Modification of Sammon Mapping Method
SM tries to minimize the following objective function
¿ ¿
¿ ¿
÷
= + =
÷
= + =
÷
=
1
1 1
2 '
1
1 1
) (
1
n
i
n
i j ij
ij ij
n
i
n
i j
ij
d
d d
d
E , (1)
where
ij
d denotes the input distance (usually Euclidean distance) between data items i
and j, i j, i, j = 1, ..., n, in the original space.
'
ij
d denotes the output distance between i
and j in the mapped D-dimensional space and
¿
=
÷ =
D
k
jk ik ij
x x d
1
2 '
] [ , where x
ik
and x
jk
are
decision variables, i = 1,…, n?1, j = i+1,…, n, k = 1,…, D. Note that only half of all
entries in the distance matrix are taken into account because the distance matrix is
assumed to be symmetric.
To minimize the objective function E, Sammon used the steepest gradient descent
procedure to search for a minimum value of E. For convenience, the updating rule for his
procedure is shown again as
2
2
) (
) (
) (
) (
) ( ) 1 (
t x
t E
t x
t E
t x t x
ik
ik
ik ik
c
c
c
c
÷ = + o , (2)
where
ik
x is the k
th
coordinate of the position of i in the mapped space and is the “magic
factor” (Sammon, 1969). The magic factor is a parameter (Apostol and Szpankowski,
1999) that controls the step size for configuration update. Its value is determined
experimentally. It is treated as a constant over all iterations.
The objective function (1) and the associated search procedure work well when
the distance matrix is symmetric. However, problems arise when the distance matrix is
38
asymmetric. Since the objective function E assumes that the input distance matrix is
symmetric, it is inappropriate to use an asymmetric distance matrix as the input. To
overcome this problem, one simple technique is to symmetrize asymmetric distances by
simply averaging, that is, replacing the entry d
ij
with (d
ij
+ d
ji
)/2. Suppose that there are
three data points a, b, c in an asymmetric distance matrix, and their pairwise distances
are: in the upper triangular part, d
a,b
> d
a,c
, and in the lower triangular part, d
b,a
< d
c,a
.
Therefore, it is uncertain that the distance between a and b is greater or less than the
distance between a and c. Using symmetrized distances, the uncertainty of the order
relationships can be resolved. However, Sammon maps generated from symmetrized
distances will lose the asymmetry information (Merino and Munoz, 2001).
To generate maps that better represent and help visualize asymmetric proximity
data sets, it is natural to consider the original asymmetric distance matrix instead of the
symmetrized distance matrix in the objective function. We would like to account for the
upper triangular portion and the lower triangular portion of the asymmetric distance
matrix simultaneously in the optimization process.
The objective function that we propose has two parts denoted by U and L. The
first part (U) takes into account the upper triangular part, while the second part (L) deals
with the lower triangular part of the original asymmetric distance matrix. Our proposed
objective function, denoted by
1
E , is defined by
) (
1
1
L U
C
E + = , (3)
where
¿ ¿
÷
= + =
÷
=
1
1 1
2
) (
n
i
n
i j ij
ij ij
d
d d
U
'
,
39
¿¿
=
÷
=
÷
=
n
i
i
j ij
ij ij
d
d d
L
2
1
1
2 '
) (
,
¿
=
÷ =
D
k
jk ik ij
x x d
1
2 '
] [
and
¿ ¿
= = =
=
n
i
n
i j j
ij
d C
1 1,
,
constrainted by x
ik
, x
jk
= 0, i, j = 1,…,n, k = 1,…,D.
x
ik
, x
jk
are decision variables and represent k
th
coordinates of data items i and j in the
mapped D-dimensional space.
By using U and L, we seek to obtain a configuration of data items such that the
structures in the upper triangular part and the lower triangular part can be considered
separately. We will use the steepest gradient method as the search procedure.
Let ) (
1
t E denote the error value at t
th
iteration and ) (t U and ) (t L denote the error
values of the upper triangular part and the lower triangular part, respectively. Let
'
ij
d (t)
denote the distance between i and j at the t
th
iteration, that is,
¿
=
÷ =
D
k
jk ik ij
t x t x t d
1
2
)] ( ) ( [ ) (
'
and D denotes the dimensionality of the mapped space (usually D is 2). The new q
th
coordinate of data item p at iteration t+1is given by
2
1
2
1
) ( / ) (
) ( / ) (
) ( ) ( ) 1 (
t x t E
t x t E
MF t x t x
pq
pq
pq pq
c c
c c
÷ = + ,
where MF stands for “magic factor”, and the first derivative is
) (
) ( 1
) (
) ( 1
) (
) (
1
t x
t L
C t x
t U
C t x
t E
pq pq pq
c
c
+
c
c
=
c
c
(4)
40
=
'
, 1
'
) ( ) (
2
pj
jq pq
n
p j j pj
pj pj
d
x x
d
d d
C
÷ ÷
÷
¿
= =
+
'
, 1
'
) ( ) (
2
pj
jq pq
n
p j j jp
pj jp
d
x x
d
d d
C
÷ ÷
÷
¿
= =
.
The second derivative is
2
2
2
2
2
1
2
) (
) ( 1
) (
) ( 1
) (
) (
t x
t L
C t x
t U
C t x
t E
pq pq pq
c
c
+
c
c
=
c
c
(5)
= ( )
( )
(
(
¸
(
¸
|
|
.
|
\
| ÷
+
÷
÷ ÷
÷
¿
= =
'
'
'
2
'
, 1
'
1
1 2
pj
pj pj
pj
jq pq
pj pj
n
p j j pj pj
d
d d
d
x x
d d
d d C
+
( )
( )
(
(
¸
(
¸
|
|
.
|
\
| ÷
+
÷
÷ ÷
÷
¿
= =
'
'
'
2
'
, 1
'
1
1 2
pj
pj jp
pj
jq pq
pj jp
n
p j j pj jp
d
d d
d
x x
d d
d d C
.
Note that in the update rule, no two points are allowed to be identical. This prevents the
partial derivatives from “blowing up.”
The minimization problem in (3) is a nonlinear optimization problem and is non-
convex (Klock and Buhmann, 1999). Therefore, we cannot guarantee finding the global
minimum. The best we can do is to obtain a local minimum from each starting solution.
We use the GRG software (2004) to solve the small asymmetric distance matrix given in
Table 3.1 with our proposed objective function (3). Two sets of random starting
configurations are used as initial coordinates for these three data points. Listed in Tables
3.2 and 3.3 are resulting configurations of three data points corresponding to two
different sets of random starting configurations. The resulting objective function values
are given in Tables 3.2 and 3.3. Meanwhile, we use GRG to solve the same asymmetric
problem in the common approach that takes symmetrized distances as inputs. In Tables
3.4 and 3.5, we show the results of configurations and objective function values obtained
41
in the common approach. Since it is an optimization problem seeking a minimum value
for the SM function, it is better to get smaller resulting objective function values. The
shown
Asymmetric A B C
A 0 1 3
B 2 0 2
C 3 4 0
Table 3.1 Asymmetric distance matrix of three data points.
Point x
1
-coordinate x
2
-coordinate
A 0.270706143 0.159258481
B 0.812893328 1.377376338
C 3.26577432 0.331200098
Objective Function Value 0.066666667
Table 3.2 Results in our proposed approach with random starting point 1.
Point x
1
-coordinate x
2
-coordinate
A 3.481685023 4.95005622
B 2.987622292 6.188468288
C 5.448716802 7.215177319
Objective Function Value 0.066666667
Table 3.3 Results in our proposed approach with random starting point 2.
Point x
1
-coordinate x
2
-coordinate
A 0.331987222 0
B 0.289836968 1.499407704
C 3.214502644 0.831327182
Objective Function Value 0.075000006
Table 3.4 Results in the common approach with random starting point 1.
42
Point x
1
-coordinate x
2
-coordinate
A 3.505923937 4.963893963
B 2.008518266 5.052079549
C 2.927993608 7.907699703
Objective Function Value 0.075000012
Table 3.5 Results in the common approach with random starting point 2.
objective function values generated by our approach are smaller than those generated by
the common approach. This confirms our conjecture that our proposed approach
performs better than the common approach in visualizing asymmetric problems, at least
from the perspective of optimization.
3.2 Implementation of the Modified SM Method
In this section, we discuss the implementation procedures and provide small
examples to illustrate them. We discuss problems that we encountered when
implementing the modified Sammon mapping method. As illustrated in the previous
section, GRG can be used to solve asymmetric problems. However, as the size of the
asymmetric problem increases, it becomes burdensome to formulate the proposed
objective function and the computational time increases significantly. A good alternative
to GRG for solving asymmetric problems is C/C++. C/C++ is a commonly used software
for coding. It is machine portable and requires only a small amount of changes to run on
other computers. It is very fast, almost as fast as assembler. It allows structured
43
programming and is very flexible. It is suited to large and complex problems. Therefore,
we implement the proposed modified SM method in C/C++.
We use random values as a starting configuration and employ the steepest
gradient method as an optimization procedure to generate maps for an asymmetric
distance matrix. If there is no improvement in the value of the objective function after a
certain number of iterations, then the algorithm is considered converged and the resulting
configuration is obtained to visualize the asymmetric distance matrix. Since the magic
factor is experimentally determined, multiple experiments are necessary to find an
appropriate value. We determined from the sensitivity analysis that the recommended
magic factor of 0.4 is a good choice in our study for updating the configuration. The
corresponding sensitivity analysis is discussed in the next chapter.
We illustrate our procedure with a small example. The data set denoted by 30A is
an asymmetric distance matrix for 30 schools taken from the American college selection
data set of 100 schools that was constructed using information provided in The Fiske
Guide (Condon et al., 2002). This data set contains pairwise distances between each pair
of 30 American colleges (see Table 3.6). For example, the distance between A5 and A19
is not symmetric, that is, d
A5, A19
= 11 and d
A19, A5
= 6. In this data set, our resulting
Sammon map provides us with a visualization of the asymmetric data set (see Figure 3.1).
In Table 3.7, we give the symmetrized distances of this data set, for example, the entry of
d
A5, A19
equals the entry of d
A19, A5
, which is the average value of these two entries in the
asymmetric distance matrix. Figure 3.2 is a Sammon map generated by the standard
Sammon map with the symmetrized distances. It is obvious that there are some
differences between these two Sammon maps. For example, the distances between pairs
44
Table 3.6 Asymmetric distance matrix of 30 American colleges.
45
Table 3.7 Symmetrized distance matrix of 30 American colleges.
46
Figure 3.1 Sammon map of the asymmetric distance matrix for data set 30A generated
by the modified SM method.
Figure 3.2 Sammon map of the symmetrized distance matrix for data set 30A generated
by the standard SM method.
47
of the triplet A24, A45, and A89 are close to each other in the original asymmetric
distance matrix, where this relationship ideally need be visualized as an equilateral
triangular in the map. This relationship is better represented in the map given by the
modified SM method than in the map given by the standard SM. In Chapter 4, we will
compare Sammon maps generated by different SM methods and discuss the differences
between them and give our tentative recommendation of choosing a better visualization
map.
48
3.3 Performance Measurements of Sammon Maps
In order to assess the quality of maps produced from asymmetric distance data,
we introduce two performance measures. The first is the distance error that measures the
extent to which the pairwise distances projected on the map deviate from the original
pairwise distances. The distance error function, denoted by DE, is defined by
¿
=
÷
=
j i ij
ij ij
d
d d
DE
2 '
) (
,
where
ij
d is the original distance and
'
ij
d is the projected distance. The smaller the value
of DE, the better the map.
The second measure is an order preservation coefficient that indicates how well a
map preserves the distance order relationships of the original asymmetric data. In Table
3.8, we give a small asymmetric distance matrix. The pairwise distance between two
data items may be very different; for example, we see d
ab
= 1 while d
ba
= 2. Since that
d
ab
< d
ac
and d
ba
< d
ca
, the distance between a and b is less than the distance between a
and c. If a Sammon map A preserves more order relationships for an asymmetric distance
matrix than Sammon map B, then A is said to be more accurate than B. We are concerned
about a map’s ability to preserve order relationships. Our proposed order preservation
measure, denoted by OP, is defined by,
data asymmetric the of ips relationsh of number
preserved ips relationsh of number
= OP .
Note that the order relationship can be uncertain. In Table 3.8, we see that d
ab
<
d
cd
and d
ba
> d
dc
. Therefore, it is incomparable if the distance between a and b is greater
than the distance between c and d on the projected map.
49
a b c d
a 0 1 2 3
b 2 0 1 9
c 5 7 0 3
d 6 8 1 0
Table 3.8 Asymmetric distance matrix of four data points.
In Table 3.8, consider the four data items (a, b, c, d) and the twelve non-zero
distance entries (six entries are in the upper triangular part of the matrix and six entries
are in the lower triangular part of the matrix). The total number of relationships is 15,
which is given by 6(6-1)/2. In Table 3.9, we provide all of the order relationships of the
asymmetric distance matrix given in Table 3.8. If six order relationships are preserved in
a Sammon map, the order preservation coefficient equals 40% (that is, 6/15).
If a Sammon map exhibits greater accuracy than another, then it should have a
smaller distance error and a larger order preservation value. We use these two measures
to assess the quality of Sammon maps in our applications.
50
d
ab
< d
ac and d
ba
< d
ca
Distance(a, b) < Distance(a, c)
d
ab
< d
ad and d
ba
< d
da
Distance(a, b) < Distance(a, d)
d
ab
= d
bc but d
ba
< d
cb
Distance(a, b) Distance(b, c)
d
ab
< d
bd and d
ba
< d
db
Distance(a, b) < Distance(b, d)
d
ab
< d
cd but d
ba
> d
dc
Distance(a, b) ? Distance(c, d)
d
ac
< d
ad and d
ca
< d
da
Distance(a, c) < Distance(a, d)
d
ac
> d
bc but d
ca
< d
cb
Distance(a, c) ? Distance(b, c)
d
ac
< d
bd and d
ca
< d
db
Distance(a, c) < Distance(b, d)
d
ac
< d
cd but d
ca
> d
dc
Distance(a, c) ? Distance(c, d)
d
ad
> d
bc but d
da
< d
cb
Distance(a, d) ? Distance(b, c)
d
ad
< d
bd and d
da
< d
db
Distance(a, d) < Distance(b, d)
d
ad
= d
cd but d
da
> d
dc
Distance(a, d) Distance(c, d)
d
bc
< d
bd and d
cb
< d
db
Distance(b, c) < Distance(b, d)
d
bc
< d
cd but d
cb
> d
dc
Distance(b, c) ? Distance(c, d)
d
bd
> d
cd and d
db
> d
dc
Distance(b, d) > Distance(c, d)
Question mark (?) denotes that the relationship between the distances is incomparable.
Table 3.9 Order relationships.
51
Chapter 4
Visualizing American College Selection Data
4.1 Description of the Data Set
The Fiske Guide (Fiske, 1999) is a well-known publication that has been used for
nearly 20 years to help students and parents select the right college. In the 2000 edition
of The Fiske Guide, information on tuition cost, SAT scores, social life, and quality of
life has been provided for over 300 colleges and universities in the United States. The
Fiske Guide also lists a school’s overlaps, that is, the major competitors to which
applicants are also applying in greatest numbers. The overlaps provide students and
parents with possible alternatives when selecting a school. For example, the overlaps of
the University of Pennsylvania are Harvard, Princeton, Yale, Cornell, and Brown.
Students who applied to the University of Pennsylvania also applied to those five schools.
However, the overlaps of Harvard University -- Princeton, Yale, Stanford, MIT, and
Brown -- do not include the University of Pennsylvania. The overlaps of two schools are
not necessarily symmetric.
The American college selection data set is derived from the overlap data of 100
schools in The Fiske Guide. This data set was constructed by Condon et al. (2002).
There were four steps involved in the construction process: building an adjacency matrix,
52
constructing a directed graph, computing distance measures, and modifying the distance
matrix.
In the first step, Condon et al. created a 100 100× 0-1 asymmetric adjacency
matrix S for 100 schools, where entry s
ij
= 1 (row i and column j) indicates that school j is
an overlap of school i. For example, in Table 4.1, we show an 6 6× adjacency matrix for
six universities (Brown, Cornell, Harvard, MIT, Penn, and Stanford). The entry in the
Penn row and the Harvard column is 1, that is, Harvard is an overlap of Penn. In the
second step, Condon et al. converted the adjacency matrix S to a directed graph with 100
nodes (one node for each school) and a directed arc for each non-zero s
ij
entry
,
where i is
the start node, j is the end node, and the directed arc connects i and j. In Figure 4.1, we
converted the 6 6× adjacency matrix, which is shown in Table 4.1, into a directed graph
with six nodes. We see that there is a directed arc starting at the Penn node and ending at
the Harvard node.
In the third step, Condon et al. set the distance of an arc in the directed graph to
one and computed the all-pairs shortest path distance matrix T, where each entry t
ij
is
calculated as the shortest distance from node i to node j. In the final step, the authors
modified the distance matrix for disconnected nodes, that is, they set each entry t
ij
for a
disconnected pair to a value V that is greater than the longest distance in the matrix. It is
necessary to carefully choose the value V -- a large value of V will push the connected
points closer together so that it will be difficult to observe the inner relationships. A
small value of V will result in merging disconnected points.
53
School Brown Cornell Harvard MIT Penn Stanford
Brown 0 1 1 0 0 1
Cornell 1 0 1 0 1 0
Harvard 1 0 0 1 0 1
MIT 0 1 1 0 0 1
Penn 1 1 1 0 0 0
Stanford 1 0 1 1 0 0
Table 4.1 Adjacency matrix for six schools.
Figure 4.1 Directed graph generated from the adjacency matrix.
The American college selection data set generated by Condon et al. contains four
groups of schools denoted by A, B, C, and D (see Table 4.2). There are 74 schools in A,
11 schools in B (these are schools from the southern United States), 8 schools in C (six
schools are from the Ivy League), and 7 schools in D (all from California).
Brown Cornell
MIT
Penn Stanford
Harvard
54
Key School State
A2 Arizona State University AZ
A3 Arizona, University of AZ
A5 Barnard College (Columbia University) NY
A6 Bates College ME
A7 Boston College MA
A8 Boston University MA
A9 Bowdoin College ME
A11 Bryn Mawr College PA
A12 Bucknell University PA
A19 Carleton College MN
A20 Carnegie Mellon University PA
A23 Colby College ME
A24 Colgate University NY
A25 Colorado College CO
A26 Colorado, University of—Boulder CO
A27 Connecticut, University of CT
A29 Delaware, University of DE
A30 Denver, University of CO
A31 Emory University GA
A34 George Mason University VA
A35 Georgetown University DC
A38 Grinnell College IA
A40 Illinois, University of—Urbana-Champaign IL
A41 Indiana University IN
A42 Iowa State University IA
A43 Iowa, University of IA
A44 James Madison University VA
A45 Lafayette College PA
A46 Lehigh University PA
A47 Lewis and Clark College OR
A48 Macalester College MN
A49 Marquette University WI
A50 Mary Washington College VA
A51 Maryland, University of—College Park MD
A53 Massachusetts, University of—Amherst MA
A55 Michigan State University MI
A56 Michigan, University of MI
A57 Middlebury College VT
A58 Minnesota, University of—Twin Cities MN
Table 4.2 One hundred schools selected from The Fiske Guide for analysis.
55
A59 Mount Holyoke College MA
A60 New Hampshire, University of NH
A61 New Jersey, The College of NJ
A62 New York University NY
A63 North Carolina State University NC
A64 North Carolina, University of—Chapel Hill NC
A65 Northeastern University MA
A66 Northwestern University IL
A67 Notre Dame, University of IN
A68 Oberlin College OH
A69 Oregon State University OR
A70 Oregon, University of OR
A71 Pennsylvania State University PA
A73 Pittsburgh, University of PA
A75 Puget Sound, University of WA
A76 Purdue University IN
A77 Reed College OR
A78 Richmond, University of VA
A79 Rutgers University NJ
A80 Smith College MA
A85 Tufts University MA
A86 Vanderbilt University TN
A87 Vassar College NY
A88 Vermont, University of VT
A89 Villanova University PA
A90 Virginia Polytechnic Institute and State University VA
A91 Virginia, University of VA
A92 Wake Forest University NC
A93 Washington University in St. Louis MO
A94 Washington, University of WA
A95 Wellsley College MA
A96 Whitman College WA
A97 Willamette University OR
A98 William and Mary, College of VA
A99 Wisconsin, University of--Madison WI
B1 Alabama, University of --Tuscaloosa AL
B4 Auburn University AL
B21 Charleston, College of SC
B22 Clemson University SC
B32 Florida State University FL
Table 4.2 (Continued).
56
B33 Florida, University of FL
B36 Georgia Institute of Technology GA
B37 Georgia, University of GA
B54 Miami, University of FL
B81 South Carolina, University of SC
B84 Tennessee, University of--Knoxville TN
C10 Brown University RI
C28 Cornell University NY
C39 Harvard University MA
C52 Massachusetts Institute of Technology MA
C72 Pennsylvania, University of PA
C74 Princeton University NJ
C83 Stanford University CA
C100 Yale University CT
D13 California, University of--Berkeley CA
D14 California, University of--Davis CA
D15 California, University of--Irvine CA
D16 California, University of--Los Angeles CA
D17 California, University of--San Diego CA
D18 California, University of--Santa Barbara CA
D82 Southern California, University of CA
Table 4.2 (Continued).
Each of the four groups is a strongly connected component in the directed graph
of 300 schools given in The Fiske Guide. If a group is strongly connected, then there
exists at least one directed path from any school in the group to any of the other schools
in the same group. In other words, any one school is considered to be a competitor to all
of the other schools in the group. The distance between each pair of schools measures
the magnitude of the competitiveness. The shorter the distance, the more competitive the
schools are. For example, if the distance between schools S and T is shorter than the
distance between schools S and R, then T is more likely a competitor of S than R.
57
4.2 Experimental Design
We implement our modified Sammon mapping method in C/C++. Our code reads
in an n × n asymmetric distance matrix and generates random starting coordinates for
each data item i (i = 1,…,n) to be plotted in the mapped D-dimensional (usually, two
dimensional) space. We use the error function E
1
and the associated updating rule given
in Section 3.1 to adjust the coordinates iteratively to minimize the value of the error
function. If no improvement is found after a certain number of iterations, the modified
SM method is considered converged and the obtained configuration is the resulting
Sammon map. Currently, we employ the steepest gradient method as the optimization
procedure when applying the modified SM method to the asymmetric distance matrix.
We realize that the resulting configuration is a local minimum, multiple
experiments with different random starts are necessary to look for an approximate
solution. Meanwhile, we tested several different magic factors ranging from 0.1 to 0.8 in
our experiments. Tables 4.3 and 4.4 provide the sensitivity analysis of the magic factor.
The values listed in the two tables are average values of five experiments with different
random starts on each data set. The minimum average error measure(s) and the
maximum average order preservation coefficient(s) of each data set can be found in
boldface in Tables 4.3 and 4.4 respectively. For example, on the data set of 100 schools,
the minimum error measure is 28829.5600 (see Table 4.3), which is associated with a
magic factor of 0.6. The error measure associated with a magic factor of 0.6 is the
second minimum (28829.5600). Their corresponding average order preservation
coefficients are tied at the value of 0.4720, which is the maximum coefficient obtained
58
Magic
Factor 100Schools 30Schools ASchools BSchools CSchools DSchools
0.1 28830.5200 773.6160 5688.3040 18.3131 6.1831 5.5503
0.2 28835.6200 773.3146 5688.1640 18.3131 6.0645 5.5502
0.3 28846.9600 773.9876 5688.6180 18.3131 6.0677 5.5490
0.4 28830.3200 773.0448 5688.3020 18.3131 6.0645 5.5490
0.5 28831.8400 773.0162 5706.6340 18.3131 6.0677 5.5490
0.6 28829.5600 773.9876 5709.3380 18.4980 6.1799 5.5369
0.7 28831.6200 773.9776 5815.3500 18.4980 6.0677 5.5490
0.8 28835.4400 773.0448 5760.8320 18.3131 6.0645 5.5490
Table 4.3 Average error measures obtained from the modified SM method.
Magic
Factor 100Schools 30Schools Aschools Bschools Cschools Dschools
0.1 0.4720 0.6065 0.5943 0.7562 0.5683 0.5552
0.2 0.4718 0.6064 0.5944 0.7562 0.5677 0.5533
0.3 0.4714 0.6065 0.5943 0.7562 0.5661 0.5524
0.4 0.4720 0.6065 0.5944 0.7562 0.5677 0.5524
0.5 0.4720 0.6065 0.5939 0.7562 0.5661 0.5524
0.6 0.4720 0.6065 0.5940 0.7539 0.5661 0.5524
0.7 0.4719 0.6065 0.5916 0.7539 0.5661 0.5524
0.8 0.4718 0.6065 0.5937 0.7565 0.5677 0.5524
Table 4.4 Average order preservation coefficients obtained from the modified SM
method.
59
for the 100 school data. When considering the overall performance on all six data sets in
terms of both error measures and order preservation coefficients, each tested magic factor
turns to give either the best error measure or the best order preservation coefficient. In
other words, the experiments are not sensitive to the magic factors. Therefore, as
suggested by Sammon (1969), we use 0.4 as the step size to adjust the locations of data
items iteratively in the mapped space.
For each data set examined in the following sections, five experiments with
different random starts are performed. We choose the best maps in terms of error
measures and order preservation measures for comparison. Listed in the tables are
average values of five experiments on each data set.
We point out that the Sammon maps generated with different starting
configurations are usually similar, that is, the relative relationships among schools are
roughly the same in each map. In our experiments, we keep the substitute value of
infinity distance as what is used in Condon’s work, which is slightly larger than the
longest distance in the distance matrix.
We apply our modified Sammon mapping method to six data sets which are:
American college selection data set with 100 schools, each of the four strongly connected
groups (A, B, C, D), and 30 schools selected from A. We experiment with five different
starting configurations for each data set and then compare the average performance of our
modified SM method to that of the standard SM method and that of Merino’s method.
We use the typical resulting Sammon map of each data set to illustrate the similarities
and differences of the resulting maps generated by these three methods.
60
In Figure 4.2, we show the map of all 100 colleges and universities that was
generated by the modified SM method. In Figures 4.3 and 4.4, we show the Sammon
maps of all 100 schools that were generated by the standard SM method and by Merino’s
method respectively with the standard error function (1) given in Section 3.1. In Figures
4.5 to 4.19, we show Sammon maps generated by our modified procedure, the standard
procedure and Merino’s procedure for five data sets (A, 30 schools from A, B, C, and D).
We discuss the maps and results in the next section.
4.3 Discussion of the Results
Shown in Figures 4.2, 4.3, and 4.4 are Sammon maps of all 100 schools generated
by the modified, the standard, and Merino’s methods respectively. In Figure 4.2, the
group of B schools and group of D schools are separated from the group of A schools and
the group of C schools. This is consistent with the existing structure of the data set that
consists of four strongly connected groups of schools. The most interesting phenomenon
is that the group of C schools is located in the center of the map and is surrounded by
some schools of group A, e.g., Tufts University (A85), New York University (A62),
Boston College (A7), Columbia University (A5) and Georgetown University (A35).
From our experimental results, it shows that the more chances a university is considered
as a competitor by other universities, the more likely this university will be placed in the
center or near the center of the map. Boston College (A7) and New York University
(A62) have more chances to be considered competitors by many other universities so that
they, we think they are popular, are placed in the center of the maps (see Figures 4.2, 4.3,
61
Figure 4.2 Map of 100 schools generated by the modified SM method.
Figure 4.3 Map of 100 schools generated by the standard SM method.
62
Figure 4.4 Map of 100 schools generated by Merino’s method.
and 4.4). In Figure 4.2, the location of group C tells us that group C and some popular A
schools have some in common to some extents. For example, group C consists of Ivy
League schools which have high education quality and expensive tuition costs, etc., and
some A schools that surround group C (i.e., New York University) provide qualified
education the same as or no worse than group C schools provide. This relationship
between group C and these schools of group A is reasonable in practice. However, this
relationship cannot be detected from the maps generated by the standard and Merino’s
methods, where there is no clue that schools in group C are competitors of schools in
group A.
63
In addition, in Figure 4.2, Group B is closer to group A than to other groups.
Group D is also closer to group A than to other groups. In the maps generated by the
standard and Merino’s methods (Figures 4.3 and 4.4), we see that the four strongly
connected groups (As, Bs, Cs, Ds) of schools are separated from each other. It is hard to
determine which group(s) is (are) closer to another group; in other words, the
relationships between groups are not as clear as they are in the map generated by the
modified SM method.
Besides, in terms of inner group relationship, schools in each group are pushed
close together in Figures 4.3 and 4.4 so that it becomes difficult to ascertain within-group
relationships from the maps. Most of relationships that can be seen in the maps generated
by the standard and Merino’s methods can also be detected in the map generated by our
modified SM method. For example, in Figures 4.3 and 4.4, the University of Maryland
(A51) is close to schools A73 (University of Pittsburgh), A29 (University of Delaware),
and A34 (George Mason University). These relationships are still preserved in Figure
4.2.
However, the map generated by the modified SM method has its limitations. It
seems that the modified SM method might not represent some local structures as
precisely as other two SM methods. For example, although these four schools that are
considered close competitors by each other can still be thought of forming a cluster (see
Figure 4.2), A6, A23, A9 and A57 are not placed together in Figure 4.2 as closely as they
are, while in maps generated by the other two methods the relationship between these
four schools is represented more clearly.
64
Regarding the differences between the maps generated by the standard and
Merino’s methods, there is no significant evidence that one outperforms the other. This
observation is also confirmed in our performance measurements that are discussed later
in this section. The general structures of the maps generated by these two methods are
similar and it is reasonable to see local differences due to different random starting
configurations and other factors such as stopping criterion etc.
Figures 4.5, 4.6 and 4.7 show the Sammon maps of group A generated by the
modified SM, the standard SM method, and Merino’s method, respectively. In Figures
4.8, 4.9 and 4.10, we show the Sammon maps of the subset of 30 schools from group A
generated by these three methods. As compared to groups B, C, and D, group A and the
30 schools from group A have a relatively large number of schools.
65
Figure 4.5 Map of group A generated by the modified SM method.
Figure 4.6 Map of group A generated by the standard SM method.
66
Figure 4.7 Map of group A generated by the Merino’s method.
The Sammon map of group A generated by the modified SM method has roughly
similar structure as has the Sammon maps generated by the standard SM and Merino’s
methods. For example, the locations of most A schools are roughly same in these three
maps (see Figures 4.5, 4.6 and 4.7). Grinnell College (A38), McAlester College (A48),
Carleton College (A19), and Oberlin College (A68) are close to each other and are
located in right middle/bottom of the maps away from the other A schools, so that they
can be considered a cluster. Wellsley College (A95), Smith College (A80), Mount
Holyoke College (A59), and Bryn Mawr College (A11) can be considered another cluster
for the same reason. Bates College (A6), Colby College (A23), Bowdoin College (A9)
67
and Middlebury College (A57) are a third cluster that is placed in left bottom of the
maps.
The within-group relationships of schools are illustrated similarly in the maps
generated by these three methods. For example, the order relationships between
Middlebury College (A57), Bowdoin College (A9), Colby College (A23) and Bates
College (A6) are obvious, i.e., Middlebury College and Bates College have the longest
pairwise distance of all pairwise distances of these four schools.
There are some local differences between the map generated by the modified
method and the other two methods. For example, the distance between Arizona State
University (A2) and University of Arizona (A3) is smaller than the distance between
Arizona State University and Colorado College (A25) because, in the asymmetric
distance matrix, d
3,2
< d
25,2
and d
2,3
< d
2,25
. In Figure 4.5, it is clear that Arizona State
University is closer to University of Arizona than it is to Colorado College, while in the
map generated by Merino’s method (Figure 4.7), it appears that Arizona State University
is closer to Colorado College.
As another example to show local differences between these three maps. The
distance between Minnesota University (A58) and Marquette University (A49) seems
equal to the distance between Minnesota University and Iowa State University (A42) in
the map generated by the modified method, while in the maps generated by the other two
methods Minnesota University is obviously closer to Iowa State University. However, in
the original asymmetric distance matrix, d
58,49
< d
58,42
, i.e. 1< 4, and d
49,58
> d
42,58
, i.e.
5>1, so that the relationships are not as obvious as they are in maps.
68
For the 30 schools from group A, the Sammon maps generated by the modified,
the standard SM and Merino’s methods are visually different (see Figures 4.8, 4.9 and
4.10). However, we can see that A41, A76, A56, A93, A31, etc. are placed in the same
sequence in these three maps but in different direction, i.e., clockwise in Figures 4.8 and
4.10, and counter-clockwise in Figure 4.9. Therefore, the general structures of the maps
generated by these three SM methods are actually similar. As another example, Reed
College (A77), Carleton College (A19), University of Washington (A94), Lafayette
College (A45), and Carnegie Mellon University (A20), which are located as outliers in
the map generated by the modified method, are still outliers and apart from other schools
of group A in the other two maps.
The maps generated by the modified SM method for groups B, C, and D have few
visual differences from the maps generated by the standard SM and Merino’s methods.
In these three types of maps, schools are scattered about and the relative relationships
among schools are roughly the same. For example, in Figures 4.11, 4.12 and 4.13,
Auburn University (B4) is about the same distance from University of Alabama (B1) and
University of Georgia (B37).
69
Figure 4.8 Map of 30 schools from group A generated by the modified SM method.
Figure 4.9 Map of 30 schools from group A generated by the standard SM method.
70
Figure 4.10 Map of 30 schools from group A generated by Merino’s method.
Figure 4.11 Map of group B generated by the modified SM method.
71
Figure 4.12 Map of group B generated by the standard SM method.
Figure 4.13 Map of group B generated by Merino’s method.
72
There are some differences in the relative relationships among schools in the
maps generated by the modified SM method and the maps obtained by the standard SM
and Merino’s methods. For example, in the asymmetric data set of group C, Harvard
(C39) and Yale (C100) are considered competitors by all other schools so that it is
desired to place these two schools in the center of the maps and let other schools scatter
about. Only in the map (Figure 4.14) generated by the modified SM method Harvard and
Yale are located near the center and surrounded by other schools. UPenn (C72), MIT
(C52), and Cornell (C28) that are less likely considered competitors by other schools are
located in the marginal area of the maps in Figures 4.14, 4.15 and 4.16.
The pairwise distances between Berkeley (D13) and Los Angeles (D16) are the
same with the pairwise distances between Berkeley and San Diego (D17) in the
asymmetric distance matrix and we expect that Berkeley is equally away from school Los
Angeles and San Diego. In Figures 4.17, 4.18 and 4.19, the distances from Berkeley to
Los Angeles and to San Diego are approximately equal. For data sets with small size, the
modified SM method yields results visually similar to those generated by the standard
and Merino’s methods.
73
Figure 4.14 Map of group C generated by the modified SM method.
Figure 4.15 Map of group C generated by the standard SM method.
74
Figure 4.16 Map of group C generated by Merino’s method.
Figure 4.17 Map of group D generated by the modified SM method.
75
Figure 4.18 Map of group D generated by the standard SM method.
Figure 4.19 Map of group D generated by Merino’s method.
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The standard and Merino’s methods take as inputs symmetrized distance matrices,
which are derived from original asymmetric distance matrices by averaging. The
modified SM method takes original asymmetric distance matrices as inputs directly. Two
performance measurements are employed to assess these three methods’ capability of
representing original asymmetric distance data. We use order preservation coefficient to
show how effectively a SM method preserves order relationships of original data. We
use error measurement to indicate how precisely original distances between data are
represented by a SM method. The represented order relationships among schools and the
resulting error measures are expected to be different in the maps generated by the three
SM methods because their input distances and rationales behind the methods are
different.
As shown in Tables 4.5 and 4.6, the order preservation coefficients of maps
generated by the standard and Merino’s SM methods are very close. The error measures
of these two SM methods do not change dramatically (see Tables 4.7 and 4.8). These can
be explained by their taking the same symmetrized distance matrices. Another
explanation may be the asymmetry coefficients introduced in Merino’s method
(described in section 2.5). In these six data sets most of schools have similar chances to
be considered competitors by others and asymmetry coefficients of schools fall in a
narrow range of values. Introducing similar asymmetry coefficients will not make the
objective function better off and therefore asymmetry coefficients in these cases have
little impact on the optimization results.
By incorporating the upper triangular part and the lower triangular part of the
original asymmetric distance matrix into the objective function and the associated
77
Data Set Standard SM Modified SM
% Improvement of Standard over
Modified
100 Schools
0.482 0.472 2.10
30 Schools
0.619 0.607 1.97
A
0.605 0.594 1.71
B 0.759 0.756 0.34
C 0.580 0.568 2.14
D 0.567 0.552 2.59
Table 4.5 Order preservation coefficients for the six data sets (standard vs. modified).
Data Set Merino’s SM Modified SM
% Improvement of Merino’s over
Modified
100 Schools
0.475 0.472 0.70
30 Schools
0.616 0.607 1.51
A
0.603 0.594 1.45
B 0.759 0.756 0.36
C 0.582 0.568 2.52
D 0.567 0.552 2.59
Table 4.6 Order preservation coefficients for the six data sets (Merino’s vs. modified).
updating rules, the search procedure of the modified SM method takes into account the
entire original distance matrix instead of the symmetrized distance matrix and looks for
an optimal configuration for the entire distance matrix. Therefore, the error measures of
maps generated by the modified SM method are significantly smaller than the error
measures of maps generated by the standard and Merino’s SM methods (see Tables 4.7
and 4.8). It seems that the modified SM method reduces the distance error measures
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Data Set Standard SM Modified SM
% Improvement of Modified over
Standard
100 Schools 54906.620 28830.320 47.49
30 Schools 999.067 773.045 22.62
A 6926.616 5688.302 17.88
B 20.192 18.313 9.31
C 6.547 6.065 7.37
D 6.048 5.549 8.25
Table 4.7 Error measures for the six data sets (standard vs. modified).
Data Set Merino’s SM Modified SM
% Improvement of Modified over
Merino’s
100 Schools 59774.640 28830.320 51.77
30 Schools 1024.385 773.045 24.54
A 7096.426 5688.302 19.84
B 19.927 18.313 8.10
C 6.615 6.065 8.33
D 6.073 5.549 8.62
Table 4.8 Error measures for the six data sets (Merino’s vs. modified).
proportionally to the size of data sets. For example, in 100-school data set, the modified
SM method reduced the error measure over the standard SM method by 47.49% (100 ×
(54906.620 – 28830.320)/ 54906.620). In the remaining data sets, the improvements
range from 8.25% to 22.62%. We point out that the values given in the tables of
performance measures are averages from the five experiments on each data set.
The modified SM method produces slightly smaller order preservation
coefficients than the other two SM methods (large coefficients are better). This may be
79
due to asymmetric distances in original data sets. When the whole asymmetric distance
matrix is taken into account, the search procedure goes through the upper triangular part
and the lower triangular part separately to find an improved result. This adds more
constraints to the optimization process and introduces more difficulty to achieve a better
result because some constraints are conflict due to asymmetry.
Although the modified SM method did not do a better job than the standard SM
and Merino’s SM methods in preserving the order relationships in our American college
selection data, the maps generated by the modified SM method are still considered better
than the maps generated by the standard SM method and Merino’s method. The
modified SM method is capable of preserving order relationships with similar accuracy
and reducing the distance errors with significant improvement as compared to the
standard and Merino’s SM method. The maps generated by the modified SM method
show us the intra-group relationships, which were not reflected in the maps generated by
other two methods. For example, it is interesting to see that schools of group C mixed
with several A schools in the maps generated by the modified SM method. In addition,
the generated maps by the modified method are more readable -- schools were not
squeezed as tightly in Figure 4.2 as they were in Figure 4.3.
In our study of analyzing American college selection data using different
Sammon mapping visualization techniques, currently, the modified SM method seems to
be able to generate visualization maps with higher quality as compared to the standard
and Merino’s methods. The modified method yields results that are with significantly
reduced distance errors and reasonably preserved order relationships compared to the
80
results generated by the other two methods. We hope that our modified SM method can
be robust on other applications.
81
Chapter 5
Visualizing Canadian Ranked College Data
The modified SM method was shown to be good at recovering the structure of
American college data at least comparable to the standard and Merino’s methods. In
American college selection data, the overlaps of each school are not ordered so that there
is no indication of ranks among the overlaps. The generated visualization maps are
unable to provide students with further details such as which school among the overlaps
is the closest competitor. If ranking information is incorporated into the data set,
decisions by students can be made more easily and effectively.
Data sets with ranking information can be collected in many fields. In marketing,
ranked data can be gathered from customers who give ranks of different brands of
products of the same category, i.e., car brands. Consider another example in which a
survey is sent to apartment managers in a metropolitan area. The survey asks the
managers to identify top 10 rival apartment buildings. The managers respond with a top
10 listing in which the first building is the most competitive rival, the second building is
next most competitive rival, and so on. For a person seeking an apartment, a visual map
of competitors can help narrow the search effectively and naturally.
82
Our work in this chapter includes building a visual model of asymmetric data sets
that incorporate ranking information. The Canadian ranked college data set is the one
that has ranking information and will be analyzed in the following sections. The modified
SM method will be applied to this data set to see if ranking information is represented
accurately. For comparison, the standard and Merino’s SM methods will also be applied
to the data set.
5.1 Description of Canadian Ranked College Data
The idea of visualizing universities originates with the work of Yin (2002) and
Condon et al. (2002). Yin proposed ViSOM (visualization-induced SOM) to detect
clusters of universities in the United Kingdom. Condon et al. created visual maps of 100
American universities that can be used to view patterns and clusters and gain insights.
We collected data from surveys that were sent to the admission directors of
undergraduate programs in 52 Canadian universities. The directors were asked to list the
five most competitive rivals in terms of overall education quality. We have received
responses from 44 universities and we list these in Table 5.1. The competitors of each
Canadian university are given in Table 5.2.
83
Key School
1 Acadia University
2 Bishop's University
3 Brandon University
4 Brock University
5 Carleton
6 Concordia University
7 Dalhousie University
8 Lakehead University
9 Laval University
10 McGill University
11 McMaster University
12 Memorial University of Newfoundland
13 Mount Saint Vincent University
14 Nipissing University
15 Queens University
16 Simon Fraser University
17 St. Francis Xavier University
18 Univeristy College of Cape Breton
19 Universite de Moncton
20 Universite de Sherbrooke
21 Universite du Quebec a Rimouski
22 Universite du Quebec en Outaouais
23 University of Alberta
24 University of British Columbia
25 University of Calgary
26 University of Guelph
27 University of Lethbridge
28 University of Manitoba
29 University of Montreal
30 University of New Brunswick
31 University of Ontario Institute of Technology
32 University of Ottowa
33 University of Prince Edward Island
34 University of Regina
35 University of Saskatchewan
36 University of Toronto
37 University of Victoria
38 University of Waterloo
Table 5.1 44 Canadian universities collected from surveys.
84
39 University of Western Ontario
40 University of Windsor
41 Wilfrid Laurier
42 Laurentian University
43 University of St. Anne
44 York University
Table 5.1 (continued).
Key School 1
st
2
nd
3
rd
4
th
5
th
1 Acadia University 38 36 10 15 23
2 Bishop's University 1 41 0 0 0
3 Brandon University 28 25 34 35 0
4 Brock University 11 26 41 44 39
5 Carleton 36 32 10 15 38
6 Concordia University 10 29 20 7 0
7 Dalhousie University 10 36 15 24 23
8 Lakehead University 26 4 39 38 0
9 Laval University 29 20 10 6 0
10 McGill University 36 24 15 7 44
11 McMaster University 36 15 38 39 10
12 Memorial University of Newfoundland 7 30 1 17 13
13 Mount Saint Vincent University 1 17 12 0 0
14 Nipissing University 4 41 8 0 0
15 Queens University 36 24 10 26 38
16 Simon Fraser University 24 37 38 26 36
17 St. Francis Xavier University 7 1 0 0 0
18 Univeristy College of Cape Breton 7 17 13 30 0
19 Universite de Moncton 30 9 32 7 43
20 Universite de Sherbrooke 29 9 2 0 0
21 Universite du Quebec a Rimouski 9 20 0 0 0
22 Universite du Quebec en Outaouais 32 20 0 0 0
23 University of Alberta 36 24 10 0 0
24 University of British Columbia 36 10 15 23 39
25 University of Calgary 36 24 15 10 23
Table 5.2 Competitors of 44 Canadian universities.
85
26 University of Guelph 11 38 15 36 39
27 University of Lethbridge 23 25 34 35 37
28 University of Manitoba 23 35 36 15 0
29 University of Montreal 36 10 32 24 23
30 University of New Brunswick 7 12 1 0 0
31 University of Ontario Institute of Technology 39 36 15 44 38
32 University of Ottowa 11 39 15 29 7
33 University of Prince Edward Island 1 17 30 0 0
34 University of Regina 23 24 28 0 0
35 University of Saskatchewan 34 23 25 15 24
36 University of Toronto 39 24 23 15 10
37 University of Victoria 24 16 25 23 0
38 University of Waterloo 36 39 15 41 11
39 University of Western Ontario 36 10 15 38 11
40 University of Windsor 39 11 41 36 15
41 Wilfrid Laurier 39 38 26 11 36
42 Laurentian University 4 40 8 32 5
43 University of St. Anne 7 17 30 0 0
44 York University 36 15 39 41 0
Table 5.2 (continued).
5.2 Modeling Steps
In the Canadian university data set that we have collected, some universities
specified their competitors that were not on the list of the 44 universities. We did not
include universities that did not respond and yet were selected as rivals by other
universities. For example, York University chose University of Toronto as its top
competitor, Ryerson University was the second, and Queens, Western Ontario, and
Wilfrid Laurier were third, fourth, and fifth, respectively. Ryerson University did not
respond to our survey, so it is not on the list. Therefore, in the rival list of York
University, we excluded Ryerson and replaced it with Queens as the second most
86
competitive school, and then moved Western Ontario and Wilfrid Laurier up one rank,
respectively.
In order to create a visual map of the Canadian college data set, we need to
generate a distance matrix from the data set and then input the distance matrix into the
modified SM method. Using procedures that are described in Chapter 4, we start by
creating a 44 44× 0-1 adjacency matrix, where entry s
ij
= 1 (row i and column j) if
school j is a competitor of school i. Next, we adjust the entries with the value of 1 to
reflect the ranking information. In terms of adjacency or distance, the more competitive
school j is to school i, the smaller the value of the s
ij
entry is. The entry of the most
competitive school is 1 by default. We set the gap between entries of two consecutive
competitors in ranking is 0.5, in other words, the entry of the second competitor is 1.5,
and the entry of the third competitor is 2, and so forth.
After adjusting the adjacency matrix, we construct the distance matrix using a
shortest path procedure and replace infinity entries with an appropriate value, i.e., 25%
larger than the longest distance. After the distance matrix is constructed, we apply the
modified SM method that is described in Chapter 3 to Canadian ranked college data
trying to detect some interesting relationships of Canadian schools.
87
5.3 Discussion of the Results
The map shown in Figure 5.1 is generated by the modified SM method. The
distance gap, which is used to separate two consecutive competitors, is set to 0.5 and the
substitute value of infinity distance is 10.5, which is 25% larger than the longest finite
distance i.e. 8.5. It provides us with a general view of the structure of Canadian colleges.
Universities such as Toronto (36), Queens (15), and Western Ontario (39) that are
frequently considered competitors by other universities are located in the center of the
map. Other universities such as Waterloo (38) and British Columbia (24) with high
frequencies are also placed near the center. Therefore, the map tells us that universities
placed in or near the center are those that are considered popular. Besides, it shows in the
map roughly five clusters of universities – one in the center and the other four
surrounding the center.
The map generated by the modified SM method also reveals some ranking
information of competitors of a particular school. For example, the top five competitors
of Queens University (15) are Toronto (36), British Columbia (24), McGill (10), Guelph
(26), and Waterloo (38), where Toronto is the most competitive school to Queens. In
Figure 5.1, among these five competitors, Toronto is the closest to Queens. Although
Waterloo is not the school farthest from Queens, it is the second farthest school among
these five competitors. Guelph is the school farthest from Queens, because the top rival
of Guelph is McMaster and Queens is its third top. The asymmetric characteristic of a
data set makes it very difficult to generate a map that is a completely accurate
representation of the data set. A map that shows roughly similar structures of the data set
can be generated by visualization techniques.
88
Figure 5.1 Sammon map of the Canadian ranked college generated by the modified
method. Gap is set to 0.5.
Figures 5.2 and 5.3 show the maps generated by the standard and Merino
methods. Again the maps generated by these two methods are similar to each other due
to the same reasons that have been discussed in previous chapter. Universities located in
the center of Figure 5.1 are still placed in the center of these two maps. Other
universities scatter about surrounding popular universities.
Differences between the map generated by the modified method and the maps
generated by the other two methods can be summarized as follows. Universities
surrounding the center are separated more evenly in the maps generated by latter two
methods. This makes it harder to detect clusters of universities. Universities in the
center of the maps generated by the latter two methods squeeze more tightly than they are
in the
89
Figure 5.2 Sammon map of the Canadian ranked college generated by the standard
method. Gap is set to 0.5.
Figure 5.3 Sammon map of the Canadian ranked college generated by Merino’s method.
Gap is set to 0.5.
90
map generated by the modified method. Some interesting insights may be hidden in the
maps of Figures 5.2 and 5.3. For example, all five competitors of Dalhousie (7) are
located in the center of maps and therefore it is reasonable to place Dalhousie near the
center. However, most of the universities that consider Dalhousie a competitor are
located in the top middle area of the maps. It is expected that Dalhousie would be near
these universities while keeping near the center universities. Only in the map generated
by the modified SM map the location of Dalhousie reminds us of the relationships while
in other two maps it is not clear that we can detect the relationships between Dalhousie
and these universities that choose Dalhousie as a competitor.
In terms of insights that possibly will be gained from visualization maps, the
modified SM method provides us with a more reasonable map. In terms of performance
measures, the modified SM method most of the time in our experiments does slightly
better than the standard and Merino’s methods in order relationship preservation (see
Tables 5.3 and 5.4) and always outperforms the other two methods in error measurement
(see Tables 5.5 and 5.6).
In order to see if the gap value affects the performance of these three methods, we
experimented with three different gap values: 0.2, 0.5, and 1. 0.5 is the one that is used in
Figures 5.1, 5.2 and 5.3. Given five random starts, we used each gap value on the
Canadian college data. Tables 5.3 and 5.4 provide average order preservation measures
of these three methods, and Tables 5.5 and 5.6 provide average error measures of these
three methods. In Figures 5.4 and 5.5 are given generated maps by the modified method
with gap values of 0.2 and 1.0 respectively. Comparing these two figures to Figure 5.1,
we can see that although there are some local differences between these three maps, the
91
Gap Value Standard SM Modified SM
% Improvement of Modified over
Standard
Gap-0.2
0.555 0.560 0.99
Gap-0.5
0.546 0.548 0.46
Gap-1.0
0.537 0.542 0.89
Table 5.3 Order preservation measures of different gap values of Canadian ranked
college data (standard vs. modified).
Gap Value Merino’s SM Modified SM
% Improvement of Modified over
Merino’s
Gap-0.2
0.559 0.560 0.14
Gap-0.5
0.543 0.548 0.88
Gap-1.0
0.545 0.542 -0.64
Table 5.4 Order preservation measures of different gap values of Canadian ranked
college data (Merino’s vs. modified).
Gap Value Standard SM Modified SM
% Improvement of Modified over
Standard
Gap-0.2
2413.83 2100.84 12.97
Gap-0.5
3921.53 3336.99 14.91
Gap-1.0
7001.58 5414.63 22.67
Table 5.5 Error measures of different gap values of Canadian ranked college data
(standard vs. modified).
92
Gap Value Merino’s SM Modified SM
% Improvement of Modified over
Merino’s
Gap-0.2
2344.01 2100.84 10.37
Gap-0.5
3972.14 3336.99 15.99
Gap-1.0
6675.48 5414.63 18.89
Table 5.6 Error measures of different gap values of Canadian ranked college data
(Merino’s vs. modified).
Figure 5.4 Sammon map of the Canadian ranked college generated by the modified
method. Gap is set to 0.2.
93
Figure 5.5 Sammon map of the Canadian ranked college generated by the modified
method. Gap is set to 1.0.
general structure of the data set is kept similar in each of these maps, i.e., popular
universities in the center and surrounded by other clusters of universities. Different gap
values have effect on constructing maps especially on local details however the general
structure of the data set is not changed dramatically.
To analyze Canadian ranked college data, the modified SM method seems better
than the standard and Merino’s methods. The modified method reduces distance errors
significantly and preserves the order relationships reasonably well compared to other two
methods. The modified method also helps gain some interesting insights that can hardly
be detected in maps generated by other two methods. We have done some sensitivity
analysis of gap values and it seems to us that gap values do not dramatically affect the
general structure of the data set.
94
So far, we have analyzed two visualization applications using the modified
Sammon mapping method. As compared to the modified SM method, the self-organizing
map, which is a neural-network based method, can also be employed to visualize and
analyze data sets. In the following chapters, we will discuss some applications using
SOMs.
95
Chapter 6
Self-Organizing Maps: State Sponsored Murder Data Set
6.1 Introduction
As a neural-network based unsupervised method, self-organizing maps (SOMs)
are mainly used for clustering, which is one of activities of data analysis in data
visualization applications. In our previous applications, we used the Sammon mapping
method to visualize two college data sets and analyze intra-cluster and inter-cluster
relationships among clusters. In this chapter, we will use SOMs to analyze clusters in the
state sponsored murder data set and to analyze the sport records data in the next chapter.
For discovering clustering information hidden in data sets, there are a few
methods that have been proposed in the literature such as hierarchical clustering methods
(single linkage, average linkage, and complete linkage), K-means clustering, and
Kohonen’s self-organizing maps (SOMs) (1995). Among these methods, Kohonen’s
SOM has received increased attention in the literature in recent years. Some recently
proposed clustering methods such as ViSOM (Yin, 2002) are based on Kohonen’s SOM.
Several software packages (i.e., Viscovery SOMine) with SOM-based clustering
procedures have been released. A natural question arises: How well do these software
packages perform? We want to make sure that we choose the right clustering procedure
96
to analyze datasets. Therefore, in the first part of this chapter, we evaluate the
performance of four software implementations of SOM-based clustering methods and
determine the best SOM-based procedure to be used in our clustering analysis. In the
second part of this chapter, we apply the chosen SOM-procedure to the state sponsored
murder data set.
6.2 Evaluating SOM-based Methods
Four clustering implementations are compared based on their performances: Ward
clustering, modified Ward clustering, single linkage clustering, and classic SOM. The
first three clustering methods are implemented in a commercial package, Viscovery
SOMine 4.0 from Eudaptics Software (www.eudaptics.com). These three SOM-based
clustering methods make use of the representation of the data set given by Kohonen’s
SOM scheme. SOM-Ward clustering uses Ward’s classic minimum distance method
(Ward, 1963). In SOM-modified Ward clustering, the classic Ward method is modified
to use a different distance measure (Viscovery, 2002). SOM-single linkage uses an
adaptation of the classic single-linkage clustering algorithm (Viscovery, 2002). Classic
SOM clustering is implemented in a research package, SOM_Pak from the Helsinki
University of Technology (SOM_Pak, 1997).
Mangiameli et al. (1996) conducted a comprehensive evaluation of seven
hierarchical clustering algorithms and the SOM network generated by the commercial
software package NeuralWorks. Our study can be viewed as an extension of their work.
We assess the clustering performance of four procedures in two current SOM software
97
packages. To our knowledge, the performance of these packages has not been reported in
the open literature.
The standard approach to studying the performance of a clustering procedure is to
apply the procedure to a problem for which the clusters are already known. This
approach allows the researcher to measure the method’s success in assigning data points
to their correct clusters. We adopt this approach and evaluate the performance of the
clustering methods on 96 data sets that we construct. The clustering methods that we
evaluate are applied to data sets in which the clusters are well separated. Figure 6.1
presents a two-dimensional plot of a four-cluster data set.
6.2.1 Constructing data sets
Four experimental factors are used to characterize each data set: the number of
clusters (three, four, five, and six), number of dimensions (three and four), number of
data points (50, 100, 150, and 200), and amount of intra-cluster dispersion (low, medium,
and high). Using this design, we construct 4 × 2 × 4 × 3 = 96 data sets, each of which
exhibits both external isolation and internal cohesion (see Cormack (1971), Mangiameli
et al. (1996), and Milligan (1980)). External isolation means that the members of one
cluster are separated from members of another cluster by empty space. Internal cohesion
means that members of the same cluster are similar (close) to each other.
We use a procedure that is similar to the method proposed by Milligan (1980,
1985) to construct data sets. The first step is to determine the cluster lengths and the
cluster boundaries for the first dimension of the variable space. For each cluster, the
98
Figure 6.1 Example of a four-cluster data set.
cluster length is selected randomly from the uniform distribution on the interval (10, 40).
In order to achieve external isolation, the boundaries of adjacent clusters are separated by
an amount selected randomly from the uniform distribution on the interval (0.25, 0.75).
The mean of each cluster is taken to be the midpoint of the cluster’s boundaries, and the
standard deviation of each cluster is set to 0.5.
The second step is to specify the characteristics of the clusters in the remaining
dimensions. We select the cluster lengths randomly from the uniform distribution on the
interval (10, 40) and then select the cluster boundaries randomly. This makes it possible
that cluster boundaries overlap with each other, unlike in the first dimension where
external isolation is guaranteed. The mean of each cluster is taken to be the midpoint of
the cluster’s boundaries. Three levels of intra-cluster dispersion are used here: low,
medium, and high; these are the same levels specified in Mangiameli et al. (1996). At the
Cluster 1
Cluster 2
Cluster 3
Cluster 4
99
low, medium, and high levels of dispersion, the standard deviation of the cluster is set
equal to 1/12, 1/6, and 1/3 times the cluster length, respectively. The level of the intra-
cluster dispersion indicates the density of data points around the cluster boundaries. The
higher the intra-cluster dispersion, the higher is the density of data points near the cluster
boundaries. Thus, internal cohesion decreases as the intra-cluster dispersion increases.
We generate data points in each cluster from a multivariate normal distribution
with mean vector given by the midpoints of the cluster boundaries. The diagonal
elements of the variance-covariance matrix are given by the squares of the standard
deviations, and all of the off-diagonal elements are equal to zero. We discard data points
that fall outside the cluster boundaries.
6.2.2 Measuring performance
In order to evaluate the performance of a clustering procedure, we use two
measures: the cluster recovery rate and the Rand statistic. The cluster recovery rate is
defined to be the proportion of times a clustering procedure correctly recovers the cluster
structure, that is, the percentage of times a procedure correctly determines the cluster
membership of each data point. The Rand statistic (Rand, 1971) is a widely used
performance metric (Milligan, 1981). The definition of the Rand statistic can be
illustrated using the notation given in Table 6.1. Cell A is the number of pairs of points in
the data set that are from the same cluster and are correctly assigned by a clustering
procedure to the same cluster. Cell B is the number of pairs of points that are from
different clusters and are correctly assigned by a clustering procedure to different
100
Table 6.1 Pairwise classification notation.
clusters. In Cells C and D, clustering errors are counted. The Rand statistic provides the
proportion of correct pairwise classifications for the data set and is given by (A + B)/(A +
B + C + D). If the solution generated by a clustering procedure is correct, then the Rand
statistic equals one; if the generated solution is incorrect, then the value of the Rand
statistic will be less than one. Clearly, the larger the value of the Rand statistic, the better
the solution.
6.2.3 Comparison results and conclusions
We applied the three SOM-based clustering procedures in Viscovery (Ward,
modified Ward, and single linkage, all with default settings), the classic SOM clustering
procedure in SOM_Pak (there are no default settings; for each run, we had to specify
values for several parameters found in the package), and the K-means algorithm in
Clementine from SPSS to each of our 96 data sets. We specified the number of clusters in
each data set as input to each procedure.
The cluster recovery rates for the five procedures are given in Table 6.2. We see
Correct Solution
Clustering Procedure
Solution
Pair in Same Cluster Pair Not in Same Cluster
Pair in Same Cluster A C
Pair Not in Same Cluster D B
101
SOM-Ward
SOM-
Modified Ward
SOM-Single
Linkage SOM-Classic K-Means
92.7 91.7 82.3 14.6 80.2
Table 6.2 Cluster recovery rates (in %).
that two of Viscovery’s procedures, SOM-Ward and SOM-modified Ward, recover the
true clusters more than 90% of the time, while Viscovery’s SOM-single linkage and the
K-means algorithm recover the clusters about 80% of the time. SOM-classic performs
poorly, only recovering the clusters about 15% of the time.
In Table 6.3, we show the effect of intra-cluster dispersion on the cluster recovery
rates of the five procedures. As the intra-cluster dispersion increases, thereby reducing
the internal cohesion of the clusters, we see that the cluster recovery rates decrease. At
all three levels of dispersion, SOM-Ward performs the best, closely followed by SOM-
modified Ward.
In addition to the cluster recovery rate, we also examine the performance of the
five procedures by calculating the Rand statistic. The average value of the Rand statistic
is given in Table 6.4. We apply SOM-Ward to the eight data sets with low dispersion
and three clusters, calculate the Rand statistic for each data set, and then average over the
eight data sets (we see that the entry is 1). The average Rand statistic for SOM-Ward for
all 32 data sets with low dispersion is 1 (this is the row average for SOM-Ward in the
first row of Table 6.4).
In Table 6.4, at the low and medium levels of dispersion, we see that all three of
Viscovery’s procedures perform better than SOM-classic and K-means. At the high level
102
Level of Dispersion
Procedure Low Medium High
SOM-Ward 100 94 84
SOM-Modified Ward 100 91 84
SOM-Single Linkage 100 91 56
SOM-Classic 19 16 9
K-Means 88 78 75
Table 6.3 Cluster recovery rates (in %) by level of dispersion.
Number of Clusters Row
Clustering Procedure 3 4 5 6 Average
Low Intra-Cluster Dispersion
SOM-Ward 1.000 1.000 1.000 1.000 1.000
SOM-Modified Ward 1.000 1.000 1.000 1.000 1.000
SOM-Single Linkage 1.000 1.000 1.000 1.000 1.000
SOM-Classic 0.846 0.899 0.911 0.886 0.886
K-Means 1.000 1.000 0.988 0.965 0.988
Medium Intra-Cluster Dispersion
SOM-Ward 0.994 0.989 1.000 1.000 0.996
SOM-Modified Ward 0.995 0.986 1.000 0.997 0.995
SOM-Single linkage 1.000 0.999 1.000 0.999 0.999
SOM-Classic 0.898 0.878 0.893 0.893 0.890
K-Means 0.995 1.000 0.988 0.931 0.979
High Intra-Cluster Dispersion
SOM-Ward 0.914 0.954 0.984 1.000 0.963
SOM-Modified Ward 0.951 0.971 0.996 1.000 0.980
SOM-Single linkage 0.946 0.977 0.992 0.991 0.977
SOM-Classic 0.915 0.931 0.876 0.898 0.905
K-Means 1.000 0.960 0.990 0.950 0.975
Table 6.4 Values of the Rand statistics.
103
of dispersion, the three procedures in Viscovery and K-means perform nearly the same
and SOM-classic is close behind. It appears that as the level of dispersion increases and
when the number of clusters is small (three or four), the performance of each of the
Viscovery procedures deteriorates somewhat (the value of the average Rand statistic
decreases). We note that SOM-classic and K-means seem to be less affected than
Viscovery’s three procedures by increases in intra-cluster dispersion.
Both SOM-classic and K-means require the user to specify the number of clusters
(that is why we input the number of clusters into all five procedures). Viscovery,
however, does not have this requirement. After inputting the data set, Viscovery can
determine the number of clusters and the assignment of points to clusters. This is a
desirable feature of the package since in practice a user usually does not know how many
clusters to specify in advance. We applied Viscovery to each of the 96 data sets and let it
determine the number of clusters. In Table 6.5, we give the cluster recovery rates for the
procedures. For each procedure, the recovery rate drops about 10 percentage points from
the recovery rate generated by Viscovery when cluster size was specified. These results
are still competitive with the recovery rate from K-means (80.2%) when K-means has the
advantage of knowing the true cluster size.
In this study, we assessed the performance of four SOM-based clustering
procedures that are implemented in commercial and research software. The three
procedures in Viscovery SOMine 4.0 performed generally well in clustering. We found
that Viscovery’s procedures performed slightly better than the K-means algorithm and
much better than the procedure in SOM_Pak. In addition, when clusters were well
separated (i.e., exhibited external isolation), the clustering procedures in Viscovery were
104
SOM-Ward SOM-Modified Ward SOM-Single Linkage
83.3 82.3 71.9
Table 6.5 Cluster recovery rates (in %) for Viscovery (number of clusters is not
specified).
fairly effective at determining the appropriate number of clusters in a data set. This
feature may help Viscovery users who are not sure of the number of clusters to
determine.
6.3 Self-Organizing Maps: the State Sponsored Murder Data Set
As we discussed in the previous section, Viscovery has useful features such as
helping determine the number of clusters and recovering cluster structures of data sets.
In this section, we apply Viscovery SOM-Ward procedure to a state sponsored murder
(also called genocide and politicide) data set. Genocides and politicides refer to actions
committed by governing elites or, in the case of civil war, either of the contending
authorities that are intended to destroy a national, ethnical, racial, religious, or political
group (Harff, 2003).
The genocide and politicide data set examined in this section includes 28
historical cases of genocide and politicide that began between 1955 and 2002 in
independent countries with populations greater than 500,000. The genocide and
politicide data were originally studied for identifying independent variables (risk factors
or pre-conditions) to distinguish countries that have genocides and politicides from those
that do not (Harff, 2003). It consists of 25 countries that have or have no prior genocides
105
and politicides and six variables that are identified as risk factors contributing to genocide
and politicide. The genocide and politicide data set under consideration is given in Table
6.6.
Among the six risk factors (variables) listed, upheaval in political context is
defined as an abrupt change in the political community resulted from the formation of a
state or regime through violent actions, defeat in international war, or rewriting of state
boundaries. Minority elite is designed to reveal the information about interethnic
disputes over access to political power. A positive value under the column of ‘Minority
Elite’ (i.e., ‘Yes’) indicates that elite ethnicity is a recurring issue of political conflicts,
which possibly leads to genocide or politicide. Exclusionary ideology refers to a belief
system that establishes some cardinal principle that maintains efforts to restrict,
persecute, or eliminate certain categories of people. Elite with ‘exclusionary ideology’ is
more apt to eliminate groups. The type of regime is another risk factor that has vital
intervening effects to cause genocide and politicide. Elite in autocratical regime is likely
to opt for restricting citizens’ participation, especially political opponents’ participation.
The level of trade openness is also an indicator of genocide and politicide. Historical
records have shown that armed conflicts and adverse regime changes are more likely to
occur in poor countries, especially those countries in Africa and Asia.
Countries in this data set are listed according to their number of positive risk
factors. If a country has many positive risk factors (i.e., six or five risk factors), it will be
ranked high. For example, Iraq is the only country with six positive risk factors, which
indicates that Iraq has the greatest potential to have future genocides and politicides and
therefore Iraq is listed in the first place in the data set. By comparison,
106
Country Name
Prior
Genocides
or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of Regime
Trade
Openness
Iraq Yes High Yes Yes Autocracy Very Low
Afghanistan
(2000) Yes Very High Yes Yes Autocracy Very Low
Afghanistan
(2002) Yes Very high No No No effective regime Very Low
Burma Yes High No Yes Autocracy Very Low
Burundi Yes Very High Yes No Autocracy Low
Rwanda Yes High Yes No Autocracy Low
Congo-Kinshasa Yes Very High Yes No No effective regime Medium
Somalia Yes Very High No No No effective regime Very Low
Sierra No Very High Yes No No effective regime Low
Ethiopia Yes High Yes No Autocracy Medium
Uganda Yes High No No Autocracy Low
Algeria Yes Very High No Yes Autocracy Medium
Liberia No High No No Autocracy Low
Pakistan Yes Medium No No Autocracy Low
China Yes Medium No Yes Autocracy Medium
Sri Lanka Yes High No No Partial democracy High
Philippines Yes Very High No No Democracy High
Colombia No Very High No No Partial democracy Low
Turkey No High No Yes Partial democracy Medium
India No High No No Democracy Low
Israel No Very High No Yes Democracy High
Indonesia Yes Medium No No Partial democracy Medium
Russia Yes Low No No Partial democracy Medium
Nigeria No Low No No Partial democracy High
Nepal No Medium No No Partial democracy Medium
Macedonia No None No No Partial democracy High
Table 6.6 Genocide and politicide data set from Harff (2003).
Macedonia is very unlikely to have future genocides and politicides according to
the fact that Macedonia has no positive risk factors.
We apply Viscovery to the genocide and politicide data set to detect clusters of
countries that have similar pre-conditions that may result in future genocides and
politicides. Viscovery needs numerical values as input, so that the categorical values in
107
the genocide and politicide data set need to be transformed to numerical values. We
transformed the categorical values using a commonly used approach that assigns sorted
numerical values to categorical values based on the description of the categorical values
(Ritter & Kohonen, 1989). The transformed data are given in Table 6.7. The notation for
the transformed genocide and politicide data set is given in Table 6.8. For example, in
the second column (prior genocides and politicides), ‘No’ is assigned by 0 and ‘Yes’ is
assigned by 1.
We input the transformed values into Viscovery and used the software’s default
settings. We obtained the map shown in Figure 6.2. This map has five clusters of
countries. These five clusters are formed based on six variables. For example,
Macedonia, Russia, Nigeria, Nepal, Indonesia, and Sri Lanka are grouped into the same
cluster. These six countries share several common characteristics. For example, they all
are partial democratic countries. Half of them (Macedonia, Nigeria, and Sri Lanka) have
a high level of trade openness. Countries in this cluster have relatively infrequent
political upheavals except Sri Lanka. Half of this group (i.e., Russia, Indonesia and Sri
Lanka) has prior genocides or politicides. Overall, this cluster can be viewed as a group
of countries where genocide and politicide is less likely to take place.
India, Colombia, Philippines, Israel and Turkey are clustered together. Countries
in this cluster almost have no prior genocides or politicides except that the Philippines
has one. Members of this cluster have frequent political upheavals, as their levels of
political upheavals are either ‘High or Very High’.
108
Country Name
Prior Genocides or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of
Regime
Trade
Openness
Iraq 1 3 1 1 1 0
Afghanistan (2000) 1 4 1 1 1 0
Afghanistan (2002) 1 4 0 0 0 0
Burma 1 3 0 1 1 0
Burundi 1 4 1 0 1 1
Rwanda 1 3 1 0 1 1
Congo-Kinshasa 1 4 1 0 0 2
Somalia 1 4 0 0 0 0
Sierra 0 4 1 0 0 1
Ethiopia 1 3 1 0 1 2
Uganda 1 3 0 0 1 1
Algeria 1 4 0 1 1 2
Liberia 0 3 0 0 1 1
Pakistan 1 2 0 0 1 1
China 1 2 0 1 1 2
Sri Lanka 1 3 0 0 2 3
Philippines 1 4 0 0 3 3
Colombia 0 4 0 0 2 1
Turkey 0 3 0 1 2 2
India 0 3 0 0 3 1
Israel 0 4 0 1 3 3
Indonesia 1 2 0 0 2 2
Russia 1 1 0 0 2 2
Nigeria 0 1 0 0 2 3
Nepal 0 2 0 0 2 2
Macedonia 0 0 0 0 2 3
Table 6.7 Transformed genocide and politicide data.
Prior
Genocides
or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of Regime
Trade
Openness
0 = No
1 = Yes
0 = None
1 = Low
2 = Medium
3 = High
4 = Very high
0 = No
1 = Yes
0 = No
1 = Yes
0 = No effective regime
1 = Autocracy
2 = Partial democracy
3 = Democracy
0 = Very low
1 = Low
2 = Medium
3 = High
Table 6.8 Notation for the transformed genocide and politicide data set.
109
Figure 6.2 Resulting SOM map of the genocide and politicide data set.
The level of trade openness of this group of countries is relatively high among the
five clusters: more than half of member countries in this group have active trade
openness. In addition, countries in this cluster are either democratic or partially
democratic. Summaries of the remaining clusters are given in Table 6.9.
The genocide and politicide data set we examine in this section includes the six
variables given in Table 6.6. Some researchers suggest including a country’s per capita
income, which they claim is the best predictor of the ethnic insurgencies and civil wars
and which underlies Harff’s work. We consider gross domestic product (GDP) per capita
and number of prior genocides and politicides. GDP per capita is a purchasing power
parity
110
Country Name
Prior
Genocides or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of
Regime
Trade
Openness
Cluster 1
Macedonia 0 0 0 0 2 3
Nigeria 0 1 0 0 2 3
Russia 1 1 0 0 2 2
Nepal 0 2 0 0 2 2
Indonesia 1 2 0 0 2 2
Sri Lanka 1 3 0 0 2 3
Cluster 2
Pakistan 1 2 0 0 1 1
Uganda 1 3 0 0 1 1
Somalia 1 4 0 0 0 0
Afghanistan
(2002) 1 4 0 0 0 0
Liberia 0 3 0 0 1 1
Cluster 3
Sierra 0 4 1 0 0 1
Congo-Kinshasa 1 4 1 0 0 2
Ethiopia 1 3 1 0 1 2
Burundi 1 4 1 0 1 1
Rwanda 1 3 1 0 1 1
Cluster 4
India 0 3 0 0 3 1
Colombia 0 4 0 0 2 1
Philippines 1 4 0 0 3 3
Israel 0 4 0 1 3 3
Turkey 0 3 0 1 2 2
Cluster 5
China 1 2 0 1 1 2
Algeria 1 4 0 1 1 2
Burma 1 3 0 1 1 0
Iraq 1 3 1 1 1 0
Afghanistan
(2000) 1 4 1 1 1 0
Table 6.9 Cluster profiles of the genocide and politicide data set.
111
basis divided by population. In the modified data set M1 given in Table 6.10, we include
GDP per capita. In Table 6.11, we give modified data set M2 that includes GDP per
capita and number of prior genocides and politicides. The GDP per capita of each
country can be found at www.cia.gov. Of these 25 countries, Israel has the highest GDP
per capita ($19000) while Somalia has the lowest GDP per capita ($550). Due to large
differences in GDP per capita among the countries, it is necessary to scale these values
and place them into several categories. Each GDP value is divided by the maximum
GDP value ($19000) and then classified into one of five categories according to its scaled
GDP value. The categories are given in Table 6.12. For example, the scaled GDP value
of Iraq is 0.1263 (2400/19000), which is given 1 in the transformed modified genocide
and politicide data sets shown in Tables 6.13 and 6.14. The corresponding visual maps
generated by Viscovery are shown in Figures 6.3 and 6.4. The associated significance
values (or cluster indicators) are 62, 38, and 55 for data sets O, M1, and M2 respectively.
We point out that significance values are recommended by Viscovery to help determine
the appropriate number of clusters of a data set. The larger the significance values, the
better the choice of a particular number of clusters. The cluster profiles of the data sets
M1 and M2 are given in Tables 6.15 and 6.16.
112
Country
Name
Prior
Genocides
or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of Regime
Trade
Openness
GDP
per
Capita
(US$)
Iraq Yes High Yes Yes Autocracy Very Low 2400
Afghanistan
(2000) Yes Very High Yes Yes Autocracy Very Low 700
Afghanistan
(2002) Yes Very high No No
No effective
regime Very Low 700
Burma Yes High No Yes Autocracy Very Low 1660
Burundi Yes Very High Yes No Autocracy Low 600
Rwanda Yes High Yes No Autocracy Low 1200
Congo-
Kinshasa Yes Very High Yes No
No effective
regime Medium 610
Somalia Yes Very High No No
No effective
regime Very Low 550
Sierra No Very High Yes No
No effective
regime Low 580
Ethiopia Yes High Yes No Autocracy Medium 750
Uganda Yes High No No Autocracy Low 1260
Algeria Yes Very High No Yes Autocracy Medium 5300
Liberia No High No No Autocracy Low 1100
Pakistan Yes Medium No No Autocracy Low 2100
China Yes Medium No Yes Autocracy Medium 4400
Sri Lanka Yes High No No
Partial
democracy High 3700
Philippines Yes Very High No No Democracy High 4200
Colombia No Very High No No
Partial
democracy Low 6500
Turkey No High No Yes
Partial
democracy Medium 7000
India No High No No Democracy Low 2540
Israel No Very High No Yes Democracy High 19000
Indonesia Yes Medium No No
Partial
democracy Medium 3100
Russia Yes Low No No
Partial
democracy Medium 9300
Nigeria No Low No No
Partial
democracy High 875
Nepal No Medium No No
Partial
democracy Medium 1400
Macedonia No None No No
Partial
democracy High 5000
Table 6.10 Modified genocide and politicide data set 1 (M1).
113
Country Name
Number of
Prior
Genocides
or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of
Regime
Trade
Openness
GDP per
Capita
(US$)
Iraq 2 High Yes Yes Autocracy Very Low 2400
Afghanistan
(2000)
1
Very High Yes Yes Autocracy Very Low 700
Afghanistan
(2002)
1
Very high No No
No effective
regime Very Low 700
Burma 1 High No Yes Autocracy Very Low 1660
Burundi 3 Very High Yes No Autocracy Low 600
Rwanda 2 High Yes No Autocracy Low 1200
Congo-
Kinshasa
2
Very High Yes No
No effective
regime Medium 610
Somalia
1
Very High No No
No effective
regime Very Low 550
Sierra
0
Very High Yes No
No effective
regime Low 580
Ethiopia 1 High Yes No Autocracy Medium 750
Uganda 2 High No No Autocracy Low 1260
Algeria 1 Very High No Yes Autocracy Medium 5300
Liberia 0 High No No Autocracy Low 1100
Pakistan 2 Medium No No Autocracy Low 2100
China 3 Medium No Yes Autocracy Medium 4400
Sri Lanka
1
High No No
Partial
democracy High 3700
Philippines 1 Very High No No Democracy High 4200
Colombia
0
Very High No No
Partial
democracy Low 6500
Turkey
0
High No Yes
Partial
democracy Medium 7000
India 0 High No No Democracy Low 2540
Israel 0 Very High No Yes Democracy High 19000
Indonesia
2
Medium No No
Partial
democracy Medium 3100
Russia
2
Low No No
Partial
democracy Medium 9300
Nigeria
0
Low No No
Partial
democracy High 875
Nepal
0
Medium No No
Partial
democracy Medium 1400
Macedonia
0
None No No
Partial
democracy High 5000
Table 6.11 Modified genocide and politicide data set 2 (M2).
114
Prior Genocides or
Politicides
0 = No; 1 = Yes
Number of Prior
Genocides or Politicides
Actual number of genocides or politicides
Upheaval 0 = None; 1 = Low; 2 = Medium; 3 = High; 4 = Very high
Minority Elite 0 = No; 1 = Yes
Exclusionary Ideology 0 = No; 1 = Yes
Type of Regime
0 = No effective regime; 1 = Autocracy;
2 = Partial democracy; 3 = Democracy
Trade Openness 0 = Very low; 1 = Low; 2 = Medium; 3 = High
GDP per Capita
0 = if the scaled value < 0.1;
1 = if the scaled value >0.1 and < 0.2;
2 = if the scaled value >0.2 and < 0.3;
3 = if the scaled value >0.3 and < 0.4;
4 = if the scaled value >0.4
Table 6.12 Notation of the transformed genocide and politicide data with added
variables.
115
Country Name
Prior
Genocides or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of
Regime
Trade
Openness
GDP per
Capita
(US$)
Iraq 1 3 1 1 1 0 1
Afghanistan
(2000) 1 4 1 1 1 0 0
Afghanistan
(2002) 1 4 0 0 0 0 0
Burma 1 3 0 1 1 0 0
Burundi 1 4 1 0 1 1 0
Rwanda 1 3 1 0 1 1 0
Congo-Kinshasa 1 4 1 0 0 2 0
Somalia 1 4 0 0 0 0 0
Sierra 0 4 1 0 0 1 0
Ethiopia 1 3 1 0 1 2 0
Uganda 1 3 0 0 1 1 0
Algeria 1 4 0 1 1 2 2
Liberia 0 3 0 0 1 1 0
Pakistan 1 2 0 0 1 1 1
China 1 2 0 1 1 2 2
Sri Lanka 1 3 0 0 2 3 1
Philippines 1 4 0 0 3 3 2
Colombia 0 4 0 0 2 1 3
Turkey 0 3 0 1 2 2 3
India 0 3 0 0 3 1 1
Israel 0 4 0 1 3 3 4
Indonesia 1 2 0 0 2 2 1
Russia 1 1 0 0 2 2 4
Nigeria 0 1 0 0 2 3 0
Nepal 0 2 0 0 2 2 0
Macedonia 0 0 0 0 2 3 2
Table 6.13 Transformed modified genocide and politicide data set 1 (M1).
116
Country Name
Number of
Prior
Genocides or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of
Regime
Trade
Openness
GDP per
Capita
(US$)
Iraq 2 3 1 1 1 0 1
Afghanistan
(2000)
1 4 1 1 1 0
0
Afghanistan
(2002)
1 4 0 0 0 0
0
Burma 1 3 0 1 1 0 0
Burundi 3 4 1 0 1 1 0
Rwanda 2 3 1 0 1 1 0
Congo-
Kinshasa 2 4 1 0 0 2 0
Somalia 1 4 0 0 0 0 0
Sierra 0 4 1 0 0 1 0
Ethiopia 1 3 1 0 1 2 0
Uganda 2 3 0 0 1 1 0
Algeria 1 4 0 1 1 2 2
Liberia 0 3 0 0 1 1 0
Pakistan 2 2 0 0 1 1 1
China 3 2 0 1 1 2 2
Sri Lanka 1 3 0 0 2 3 1
Philippines 1 4 0 0 3 3 2
Colombia 0 4 0 0 2 1 3
Turkey 0 3 0 1 2 2 3
India 0 3 0 0 3 1 1
Israel 0 4 0 1 3 3 4
Indonesia 2 2 0 0 2 2 1
Russia 2 1 0 0 2 2 4
Nigeria 0 1 0 0 2 3 0
Nepal 0 2 0 0 2 2 0
Macedonia 0 0 0 0 2 3 2
Table 6.14 Transformed modified genocide and politicide data set 2 (M2).
117
Figure 6.3 Resulting SOM map of the first modified data set (M1).
Figure 6.4 Resulting SOM amp of the second modified data set (M2).
118
Country Name
Prior
Genocides
or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of
Regime
Trade
Openness
GDP per
Capita
US$
Cluster 1
Israel 0 4 0 1 3 3 4
Turkey 0 3 0 1 2 2 3
Colombia 0 4 0 0 2 1 3
Cluster 2
Indonesia 1 2 0 0 2 2 1
Russia 1 1 0 0 2 2 4
Sri Lanka 1 3 0 0 2 3 1
Philippines 1 4 0 0 3 3 2
Cluster 3
Nigeria 0 1 0 0 2 3 0
Nepal 0 2 0 0 2 2 0
Macedonia 0 0 0 0 2 3 2
India 0 3 0 0 3 1 1
Liberia 0 3 0 0 1 1 0
Cluster 4
Pakistan 1 2 0 0 1 1 1
Uganda 1 3 0 0 1 1 0
Afghanistan (2002) 1 4 0 0 0 0 0
Somalia 1 4 0 0 0 0 0
Cluster 5
Algeria 1 4 0 1 1 2 2
China 1 2 0 1 1 2 2
Burma 1 3 0 1 1 0 0
Iraq 1 3 1 1 1 0 1
Afghanistan (2000) 1 4 1 1 1 0 0
Cluster 6
Rwanda 1 3 1 0 1 1 0
Burundi 1 4 1 0 1 1 0
Ethiopia 1 3 1 0 1 2 0
Congo-Kinshasa 1 4 1 0 0 2 0
Sierra 0 4 1 0 0 1 0
Table 6.15 Cluster profiles of the first modified data set (M1).
119
Country Name
Number of Prior
Genocides or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of
Regime
Trade
Openness
GDP per
Capita
US$
Cluster 1
Russia 2 1 0 0 2 2 4
Nigeria 0 1 0 0 2 3 0
Nepal 0 2 0 0 2 2 0
Macedonia 0 0 0 0 2 3 2
Cluster 2
Colombia 0 4 0 0 2 1 3
Israel 0 4 0 1 3 3 4
Turkey 0 3 0 1 2 2 3
India 0 3 0 0 3 1 1
Sri Lanka 1 3 0 0 2 3 1
Philippines 1 4 0 0 3 3 2
Cluster 3
Algeria 1 4 0 1 1 2 2
Iraq 2 3 1 1 1 0 1
Afghanistan
(2000) 1 4 1 1 1 0 0
Burma 1 3 0 1 1 0 0
China 3 2 0 1 1 2 2
Cluster 4
Burundi 3 4 1 0 1 1 0
Rwanda 2 3 1 0 1 1 0
Congo-Kinshasa 2 4 1 0 0 2 0
Sierra 0 4 1 0 0 1 0
Ethiopia 1 3 1 0 1 2 0
Cluster 5
Indonesia 2 2 0 0 2 2 1
Pakistan 2 2 0 0 1 1 1
Uganda 2 3 0 0 1 1 0
Afghanistan
(2002) 1 4 0 0 0 0 0
Liberia 0 3 0 0 1 1 0
Somalia 1 4 0 0 0 0 0
Table 6.16 Cluster profiles of the second modified data set (M2).
120
The number of clusters in each map is either five or six, and the corresponding
significance values are large. This suggests that five or six clusters may be good enough
to represent the actual cluster structure in the data sets. Although visual maps (Figures
6.3 and 6.4) of the two modified data sets (M1, and M2) are different from the map in
Figure 6.2 of the original data set (O), they still have some elements in common.
The cluster made up of China, Algeria, Burma, Iraq, and Afghanistan (2002)
appears in all three maps, as does the cluster of Sierra, Congo, Ethiopia, Burundi, and
Rwanda. Cluster membership of other countries is to some extent similar in these
clustering results. It indicates that the two new variables do not significantly affect the
clustering structure of observed countries. This is possibly due to the small size of the
genocide and politicide data set. The six variables listed in Harff’s work may be enough
to determine the clustering structure of the data set and may also be sufficient to forecast
the future genocides or politicides.
However, in Harff’s listing, Iraq was the country most likely to have future
genocides, or politicides, and followed by Afghanistan, Burma, Burundi, Rwanda, Congo
and Somalia. Algeria and China were not close to Iraq in the list, while in our clustering
results these two countries are always in the same cluster with Iraq and countries such as
Afghanistan and Burma. If from the perspective of clustering, Algeria and China would
be ranked close to Iraq, Burma, and Afghanistan, rather than several ranks down from
them in the list.
There might be several factors causing differences between our clustering results
and Harff’s results. One could be the weight assigned to each variable during
121
preprocessing, where all variables are equally weighted. Another reason might be lack of
further domain knowledge.
Comparing our clustering results with Harff’s work, we conclude that countries
in the same cluster are equally likely to have future genocides or politicides. This is
different from what we see in Tables 6.6 and 6.7, which has the forecasted ordering of
countries for possible future genocides or politicides (Harff, 2003).
Our current result might be helpful to future genocide and politicide research.
The variables ‘GDP per capital’ and ‘number of prior genocides and politicides’ have
been shown to be of little influence in determining the clustering structures of the data
sets, though possibly due to the small size of the data sets. This is not in favor of other
scholars’ suggestion that per capita income is the best indicator of genocides and
politicides, and therefore somehow reinforces Harff’s conclusion regarding identification
of risks factors. Moreover, the listing differences may raise research questions such as if
clustering offers a reasonable alternative view of the genocide and politicide data set, and
to what extent will clustering contribute to forecasting countries’ genocides and
politicides in the coming years.
122
Chapter 7
Self-Organizing Maps: Best Values in Colleges
The prospect of graduating with a lot of debt may be one of the many worries that
concern new college students. It becomes necessary for those college applicants and their
parents to critically screen colleges to see which colleges give students the best value for
their money. Kiplinger’s Personal Finance gave a best value list of 100 public colleges
(Kiplinger, 2003) and a best value list of 100 private colleges (Kiplinger, 2004) that
combine great academics and reasonable costs. In each list, the colleges judged to be the
best value is ranked number one with lower rank values preferred.
As we discussed in previous chapters, SOMs are capable of discovering the
hidden structures to help a decision maker understand a data set. SOMs have been
widely used to cluster data and gain insight into data sets using visual maps. In this
chapter we will apply Viscovery SOMine (2002) to analyze public and private colleges to
determine which colleges are the best values and compare our rankings to those given by
Kiplinger.
123
7.1 Data Description and Preprocessing
We used two data sets from Kiplinger’s Personal Finance (2003 and 2004): a
public college data set and a private college data set. The public college data set contains
the top 100 public colleges considered to be the best values. Similarly, the private
college data set contains the top 100 private colleges considered to be the best values. A
total of 11 variables are included in each data set and seven of them are common to both
data sets. The seven common variables are listed in Table 7.1. The four additional
variables are listed in Table 7.2.
Except for the variable Enrollment, other variables in the two data sets indicate
either the academic quality or the financial cost of a school. Typically, college applicants
look for suitable colleges in terms of both academic quality and financial cost. The
variable Enrollment tells us the number of students who were enrolled in the current
academic year. College applicants can get a rough idea of how many students may be
admitted by each college. However, the variable Enrollment does not provide us the
percentage of applicants who may be admitted – which is known as Admission Rate. The
variable Admission Rate reflects a school’s academic quality and it affects applicants’
decisions. Typically, a low admission rate implies a high academic quality. Compared to
the variable Enrollment, the variable Admission Rate speaks about schools’ academic
quality more directly. Therefore, we included the variable Admission Rate in our
datasets and didn’t include the variable Enrollment. In addition, we did not include the
variable Aid from Grants and Cost after Non-need-based Aid in the private college data.
Values in the Aid from Grants variable tell us that at least 70% students in 83 out of 100
private colleges were awarded aid from grants, and at least 50% students were awarded in
124
Variable Description
Enrollment Number of full-time and part-time undergraduates enrolled at the
college during 2002-2003 academic year
Admission Rate Percentage of applicants who were offered admission
SAT or ACT
Percentage of the 2002-03 freshman class that scored above 600
on the verbal and math parts, separated by slash, or the
percentage that scored above 24 on the ACT
Student/Faculty
Ratio
Average number of students for each faculty member
4-yr. Graduate Rate
Percentage of 1996-97 freshmen who earned a bachelor’s degree
in four years or fewer
6-yr. Graduate Rate
Percentage of 1996-97 freshmen who earned a bachelor’s degree
within six years
Average Debt at
Graduation
Average debt a student accumulates before graduation
Table 7.1 Seven common variables in Kiplinger’s public and private data sets.
Data Set Variable Description
In-State Total
Costs
Overall cost for residents
In-State Costs
After Aid
Overall cost for residents after subtracting
average need-based award
Total Out-of-State
Costs
Overall cost for out-of-state students
Public
College
Out-of-State Costs
after Aid
Overall cost for out-of-state students after
subtracting average need-based award
Total Cost Overall cost for college students
Cost After Need-
based Aid
Total cost in 2003-04 academic year after
subtracting the average need-based award
Aid from Grants
Percentage of the average aid package that came
from grants or scholarships
Private
College
Cost after Non-
need-based Aid
2003-04 cost for a student after subtracting non-
need-based award
Table 7.2 Four additional variables in Kiplinger’s public and private data sets.
125
98% private colleges. These statistics indicate that the majority of private schools are
able to support many of their students. Therefore, the variable Aid from Grants may not
help discover private colleges that are the best values. We did not include the variable
Cost after Non-need-based Aid in the private college dataset because the Cost after Non-
need-based Aid is not available in more than 25% of 100 schools in the data set. Other
variables such as Total Cost and Cost after Need-based Aid provide applicants financial
cost information.
After selecting appropriate variables for each data set, we show the public and
private college data in Tables 7.3 and 7.4, respectively.
126
Rank School Name
SAT or
ACT
(%)
Admis.
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-
State
Total
Costs
($)
In-State
Costs
After
Aid ($)
Total
Out-of-
State
Costs ($)
Out-of-
State
Costs
After
Aid ($)
Avg.
Debt at
Grad.
($)
1
University of North
Carolina at Chapel Hill 65/76 35 14 65 79 11,290 6,673 23,138 16,465 11,156
2 University of Virginia 77/84 39 16 81 91 12,640 8,737 28,610 19,873 13,536
3
College of William
and Mary 83/85 35 12 80 89 13,024 6,668 27,724 21,056 19,762
4 University of Georgia 54/59 65 13 46 66 10,534 5,245 21,310 16,065 12,906
5 University of Florida 58/67 58 21 49 77 10,611 6,125 21,639 15,514 14,449
6
New College of
Florida 93/74 65 11 47 72 10,947 5,613 24,185 18,572 16,645
7
Georgia Institute of
Technology 75/95 59 14 18 69 11,340 6,022 23,266 17,244 17,221
8
University of Illinois
at Urbana, Champaign 79 60 13 52 76 14,410 8,045 25,446 17,401 14,791
9
Truman State
University 82 79 15 39 62 10,609 7,604 14,409 6,805 14,382
10
Virginia Polytechnic
Institute and State
University 39/52 65 15 36 72 10,122 5,613 20,006 14,393 16,229
11
North Carolina State
University 39/60 59 15 25 60 10,688 5,508 22,536 17,028 15,476
12
University of
Delaware 41/53 48 12 54 72 13,416 7,666 22,946 15,280 13,610
13
University of
Wisconsin, Madison 86 71 14 41 77 13,391 7,842 27,401 19,559 15,904
14
University of
Michigan, Ann Arbor 83 49 15 61 82 16,671 8,661 33,473 25,463 16,825
15
University of
California, San Diego 47/73 41 19 43 78 16,000 9,043 24,105 17,148 13,275
16
University of
California, Berkeley 65/78 25 17 48 83 17,265 8,982 25,584 17,301 14,990
17
University of
Washington 42/54 68 11 40 70 13,835 6,926 24,991 18,082 14,500
18
New Mexico Institute
of Mining and
Technology 75 63 13 12 40 9,714 2,708 16,151 9,145 9,500
19
University of
Wisconsin, La Crosse 60 65 21 23 58 10,425 7,327 20,102 12,775 14,306
20
University of Texas at
Austin 51/65 61 19 39 71 14,391 9,111 20,999 11,888 16,400
21
University of
Oklahoma 69 89 21 19 51 10,139 6,618 16,652 10,034 16,886
22 University of Kansas 55 67 19 26 55 9,673 6,266 17,149 10,883 17,347
23
University of North
Carolina at Asheville 48/40 67 14 31 48 8,929 6,308 17,754 11,446 14,547
24
State University of
New York at
Binghamton 51/73 42 19 69 80 13,587 9,214 19,537 10,323 13,915
25
Colorado School of
Mines 89 67 12 31 61 13,780 8,532 26,970 18,438 17,500
26 Auburn University 51 83 16 40 68 10,276 7,308 18,736 11,428 18,585
Table 7.3 Kiplinger’s (2003) public college data.
127
27
Colorado State
University 56 77 17 29 62 10,689 6,700 21,161 14,461 16,042
28 College of New Jersey 61/71 48 12 59 80 16,686 13,915 21,261 7,346 5,490
29
Michigan State
University 57 67 18 31 66 12,743 8,670 22,703 14,033 18,663
30
Appalachian State
University 26/30 64 19 31 60 7,913 4,785 16,834 12,049 13,000
31
Iowa State University
of Science and
Technology 58 89 16 24 62 11,588 8,736 20,930 12,194 17,119
32
State University of
New York College at
Geneseo 65/73 49 19 67 79 12,840 10,840 18,790 7,950 15,000
33
Texas A&M
University 36/49 68 21 27 69 11,899 7,057 18,979 11,922 15,670
34
University of Texas at
Dallas 45/59 53 20 30 53 12,075 7,787 19,155 11,368 NA
35
University of North
Carolina at
Wilmington 18/26 55 16 34 60 9,715 6,375 19,290 12,915 13,583
36
University of
Maryland, College
Park 64/77 43 13 33 63 16,304 12,223 22,881 10,658 15,566
37
University of
California, Los
Angeles 62/75 24 17 40 81 17,616 9,975 26,006 16,031 12,775
38
Louisiana State
University and
Agricultural and
Mechanical College 53 77 21 23 58 10,126 7,406 15,426 8,020 17,569
39
University of
Tennessee 49 58 18 24 56 11,681 6,706 20,763 14,057 21,689
40 University of Iowa 59 84 15 34 64 11,763 9,460 22,055 12,595 15,335
41
Rutgers, The State
University of New
Jersey, New
Brunswick 39/55 55 14 44 72 16,519 10,044 23,033 12,989 15,270
42 Clemson University 44/61 52 16 35 69 14,618 11,121 22,216 11,095 14,347
43
University of Colorado
at Boulder 65 80 16 38 64 11,937 7,877 28,253 20,376 16,737
44
University of
Kentucky 54 82 17 27 58 9,432 5,594 16,112 10,518 NA
45 University of Arkansas 59 86 17 20 45 10,706 7,144 17,456 13,894 14,029
46
Mary Washington
College 62/49 60 17 65 75 12,287 9,033 20,035 11,002 13,100
47
Oklahoma State
University 49 92 19 22 56 10,677 7,478 16,623 9,145 15,580
48
Kansas State
University 49 58 20 18 45 9,626 7,104 16,990 9,886 17,000
49
University of Northern
Iowa 39 80 16 30 64 10,634 7,632 17,592 9,960 15,786
50
University of
Mississippi 46 80 19 29 48 11,257 6,607 16,167 9,560 14,459
51
James Madison
University 33/40 58 17 59 78 13,145 9,320 21,367 12,047 11,786
52
University of
California, Davis 42/62 63 19 28 75 16,521 10,334 23,814 13,480 13,507
Table 7.3 (Continued).
128
53 Miami University 84 77 17 61 80 15,833 12,467 25,603 13,136 17,579
54 Purdue University 29/46 76 16 28 64 14,691 8,723 26,311 17,588 15,677
55
Mississippi State
University 50 74 16 19 48 11,130 8,174 16,036 7,862 15,081
56
University of
Nebraska, Lincoln 55 90 19 15 51 10,687 7,277 18,269 10,992 15,682
57
Florida State
University 34/39 70 22 38 64 10,994 7,035 22,022 14,987 16,372
58
University of
California, Irvine 24/55 57 18 34 72 15,635 8,507 22,450 13,944 12,513
59
University of
Missouri, Columbia 68 88 18 32 65 13,208 8,138 22,655 14,517 17,137
60
University of
Minnesota, Morris 61 82 14 50 76 13,477 8,427 13,477 5,050 9,208
61 University of Alabama 46 85 18 31 61 11,197 8,136 18,357 10,221 18,978
62
University of South
Carolina 29/36 70 17 31 58 11,795 8,794 21,133 12,339 15,260
63
St. Mary's College of
Maryland 67/60 59 12 58 67 16,908 12,908 23,228 10,320 17,125
64
Michigan
Technological
University 68 92 11 22 63 14,135 9,339 25,025 15,686 15,711
65
University of
California, Santa
Barbara 38/55 51 19 44 73 16,154 10,231 24,246 14,015 NA
66
University of
Minnesota, Twin
Cities Campus 64 74 15 17 53 13,910 7,902 25,540 17,638 NA
67
Mississippi University
for Women 56 65 13 21 43 8,649 8,649 13,316 4,667 13,500
68
California Polytechnic
State University, San
Luis Obispo 39/64 39 19 17 66 11,781 10,429 15,268 4,839 12,842
69
University of
Wyoming 42 95 15 22 54 10,863 6,833 16,713 9,880 18,311
70
George Mason
University 26/29 66 16 25 48 11,602 7,597 21,442 13,845 14,143
71
University of Central
Florida 30/36 62 24 25 49 10,839 8,353 21,867 13,514 14,927
72 Ohio State University 69 74 14 25 59 15,249 11,315 25,113 13,798 15,011
73
Illinois State
University 47 81 19 28 55 12,971 7,199 17,441 10,242 13,921
74
University at Buffalo,
The State University
of New York 28/41 61 14 32 56 13,422 9,956 19,372 9,416 16,255
75 Salisbury University 26/35 50 17 50 68 12,895 9,506 19,783 10,277 14,773
76
University of
Massachusetts
Amherst 32/38 58 19 41 61 14,480 9,714 23,333 13,619 15,321
77 University of Vermont 35/38 71 14 48 67 17,116 8,258 30,168 21,910 22,425
78 College of Charleston 47/47 60 14 32 52 14,008 11,214 21,270 10,056 15,135
79
Indiana University
Bloomington 27/33 81 20 40 65 13,129 8,704 24,164 15,460 16,930
80
Pennsylvania State
University University
Park Campus 40/60 57 17 43 80 17,017 12,872 26,639 13,767 17,900
Table 7.3 (Continued).
129
81
State University of
New York at Albany 26/35 56 21 52 66 13,574 9,599 19,524 9,925 15,108
82 University of Arizona 27/33 86 19 29 55 11,163 8,506 19,933 11,427 17,340
83 Towson University 18/26 58 19 30 56 12,776 8,651 20,422 11,771 15,530
84
Rutgers, The State
University of New
Jersey, Camden 19/24 54 11 21 60 16,189 10,111 22,703 12,592 15,223
85
University of
Maryland, Baltimore
County 46/63 63 17 28 53 16,516 12,946 23,368 10,422 14,500
86
University of
Connecticut 33/42 62 17 23 70 14,413 9,141 25,197 16,056 16,093
87
University of
Pittsburgh 48/56 55 17 35 60 17,025 12,723 26,337 13,614 20,154
88
State University of
New York College at
Fredonia 22/25 53 18 47 66 11,782 8,961 17,732 8,771 12,430
89
Stony Brook
University, State
University of New
York 25/50 54 18 30 51 13,645 9,704 19,595 9,891 15,747
90
State University of
New York at New
Paltz 30/29 40 17 21 52 11,276 8,276 16,176 7,900 15,000
91 University of Maine 22/29 79 15 29 56 12,780 8,162 21,480 13,318 17,917
92
University of New
Hampshire 25/32 77 14 48 71 15,779 13,693 26,139 12,446 20,700
93
University of
Missouri, Rolla 83 92 14 10 52 14,326 9,416 23,144 13,728 17,991
94
University of
California, Santa Cruz 37/39 80 19 40 64 16,877 9,309 24,250 14,941 13,282
95 Rowan University 22/33 44 14 37 63 15,416 10,632 20,812 10,180 NA
96
University of Illinois
at Chicago 43 63 15 9 37 14,299 6,399 23,995 17,596 17,000
97 University of Oregon 30/31 86 18 36 59 12,795 9,586 24,231 14,645 22,783
98 Texas Tech University 23/29 69 20 22 51 12,968 9,878 20,048 10,170 13,805
99 Ohio University 49 75 20 43 70 16,514 13,102 24,737 11,635 15,285
100 UC Riverside 16/34 86 19 39 64 16,751 9,932 24,530 17,711 13,226
Table 7.3 (Continued).
130
Rank School Name
Admission
Rate (%)
SAT or
ACT
(%)
Student/
faculty
Ratio
4-Year
Grad.
Rate (%)
6-Year
Grad.
Rate (%)
Total
Costs ($)
Cost After
Need-
Based Aid
($)
Average
Debt at
Graduation
($)
1
California Institute of
Technology 21 99/100 3 71 85 32,682 10,981 10,244
2 Rice University 24 89/92 5 68 89 28,350 14,779 12,705
3 Williams College 23 93/93 8 89 94 36,550 14,737 12,316
4 Swarthmore College 24 94/98 8 86 92 38,676 17,386 12,759
5 Amherst College 18 94/92 9 84 94 38,492 14,453 11,544
6 Webb Institute 42 100/100 7 79 83 8,079 5,579 5,700
7 Yale University 8 96/97 7 88 95 38,432 15,729 19,228
8
Washington and Lee
University 31 89/89 11 86 89 30,225 15,452 15,634
9 Harvard University 11 90*/90* 8 86 97 38,831 17,456 10,465
10 Stanford University 13 93/95 7 77 93 38,875 17,746 15,782
11 Princeton University 11 95/97 5 91 97 40,169 18,325 12,000
12
Massachusetts
Institute of
Technology 16 95/100 6 82 91 39,213 19,609 22,855
13 Pomona College 23 98/97 9 83 88 38,130 17,411 15,600
14 Emory University 42 89/94 7 82 87 37,272 19,657 17,675
15 Columbia University 12 91/93 7 83 93 39,493 17,778 15,331
16 Duke University 25 91/94 11 88 93 40,080 19,996 20,025
17 Davidson College 34 86/89 10 89 91 34,706 21,455 13,697
18 Wellesley College 47 88/89 9 84 88 37,419 17,526 15,697
19 Vassar College 31 93/89 9 81 87 37,870 19,404 17,170
20 Haverford College 32 89/90 8 89 92 38,928 17,826 15,253
21
Northwestern
University 33 88/92 7 83 92 38,817 20,376 14,551
22 Bowdoin College 25 87/92 10 83 90 38,663 17,773 15,307
23
University of
Pennsylvania 21 91/96 6 83 91 39,040 20,596 20,247
24
Johns Hopkins
University 35 85/93 8 81 88 39,188 19,142 13,600
25 Cooper Union 14 81/83 7 57 78 14,652 11,167 9,250
26
Washington
University 24 93/98 7 75 86 39,253 20,700 NA
27 Dartmouth College 23 92/96 9 87 95 38,898 19,546 NA
28
Claremont McKenna
College 28 89/95 7 82 86 37,730 17,988 16,914
29
University of Notre
Dame 34 83/91 12 88 95 35,392 18,011 25,595
30 Colgate University 34 80/86 10 85 89 38,820 18,856 12,984
31 The Colorado College 53 66/70 9 72 79 35,275 16,516 13,500
32
University of
Richmond 41 74/83 10 79 84 31,679 17,588 16,115
33
Georgetown
University 21 87/89 11 86 91 39,182 24,382 20,000
34 Brown University 17 86/90 8 79 94 40,248 20,838 21,700
35 Carleton College 35 88/89 9 82 86 35,288 21,677 14,543
36 Lafayette College 36 63/78 11 79 84 35,713 15,147 17,380
Table 7.4 Kiplingers’ (2004) private college data.
131
37 Middlebury College 27 93/96 11 81 87 39,532 18,288 21,751
38 Grinnell College 65 87/85 10 78 84 31,460 16,585 13,854
39
Illinois Wesleyan
University 48 96 12 76 81 30,780 18,858 17,722
40 Bates College 28 90/91 10 82 87 38,932 18,258 17,045
41 Cornell University 29 85/92 9 82 90 38,974 23,122 15,587
42 Wesleyan University 28 89/92 9 76 81 39,127 21,401 23,753
43 Colby College 33 84/89 11 85 88 38,699 18,168 17,270
44 Bucknell University 39 72/84 12 83 87 36,165 19,165 16,000
45 Kenyon College 52 87/81 9 80 84 36,273 17,905 20,850
46 Centre College 78 89 11 71 73 28,529 15,842 14,300
47 Rhodes College 70 95 11 71 73 30,080 18,899 15,100
48 Macalester College 44 87/88 10 71 77 32,847 16,394 NA
49 Barnard College 34 88/88 10 72 84 37,940 17,826 14,030
50 Brandeis University 42 88/88 8 79 85 39,101 22,257 NA
51
College of the Holy
Cross 43 71/76 11 88 90 36,851 23,846 16,063
52 Harvey Mudd College 37 97/100 9 75 83 38,880 22,041 20,219
53
Wake Forest
University 41 79/86 10 77 87 36,079 21,196 24,769
54 Bryn Mawr College 50 86/75 9 76 80 37,890 18,609 NA
54 Wheaton College 54 84/83 11 70 84 27,076 17,341 15,864
55 Tufts University 27 81/90 9 81 88 39,173 20,115 15,499
56 Oberlin College 33 87/80 10 63 76 37,688 21,081 13,926
57
Mount Holyoke
College 52 80/70 10 75 79 38,668 19,268 14,200
58 Furman University 58 65/70 11 74 81 29,430 16,296 17,741
59 St. Olaf College 73 84 13 71 75 29,879 17,458 18,806
60
Brigham Young
University 73 86 18 31 73 9,663 7,621 11,000
61 Lehigh University 44 59/85 11 70 84 35,670 19,123 16,972
62 Smith College 53 72/66 9 76 80 37,937 18,466 19,911
63 Beloit College 70 82 11 60 72 30,264 17,452 14,942
64 Taylor University 78 74 15 71 75 24,723 15,678 15,117
65 Union College 45 56/71 11 75 80 36,455 18,431 15,725
66 Hamilton College 35 77/82 10 79 84 38,463 19,474 16,856
67 DePauw University 61 55/60 11 75 79 32,150 15,531 14,481
68 Hillsdale College 82 74 11 53 71 23,353 13,853 14,500
69 Knox College 72 72 12 67 74 30,894 15,494 16,920
70
University of Southern
California 30 79/91 10 51 73 37,968 21,606 20,619
71 Trinity College 36 73/79 9 77 83 38,890 19,667 17,000
72 Trinity University 69 71/80 11 65 75 27,086 16,706 NA
73
Gustavus Adolphus
College 77 70 13 72 75 27,820 17,609 17,400
74 Vanderbilt University 46 83/90 9 78 84 38,847 20,971 24,023
75 Whitman College 50 80/81 10 60 71 33,776 21,176 15,000
76 Scripps College 58 85/78 12 63 68 36,500 17,984 12,941
77
Franklin and Marshall
College 62 62/71 11 78 83 36,580 20,925 19,656
Table 7.4 (Continued).
132
78 Saint Louis University 72 72 12 52 67 29,983 16,902 14,989
79
Carnegie Mellon
University 38 75/95 11 61 77 38,460 24,689 19,195
80 Lawrence University 68 83 11 58 68 32,875 17,882 18,311
81 Connecticut College 35 83/84 11 75 81 37,057 16,930 17,250
82
Case Western Reserve
University 78 74/85 8 49 75 32,802 18,323 21,830
84 Dickinson College 51 65/64 13 74 78 36,600 19,753 17,586
85 Kalamazoo College 73 91 12 60 69 30,917 17,947 20,000
86
Saint John's
University 87 66 13 67 74 27,272 19,544 20,680
87 Boston College 34 78/85 13 0 86 37,745 24,470 16,732
88 Reed College 55 95/86 10 45 67 37,900 18,804 16,758
89 Bard College 36 85/67 9 59 71 38,282 20,558 15,400
90
University of
Rochester 50 79/87 12 65 76 37,246 20,297 NA
91 New York University 28 87/86 11 65 74 40,105 28,282 21,495
92 Villanova University 47 57/74 13 79 84 36,560 26,463 28,217
93 Skidmore College 46 70/69 11 71 75 38,838 21,023 15,560
94
Rose-Hulman Institute
of Technology 65 62/91 13 58 71 32,625 28,677 27,000
95 St. John's College 71 95/75 8 63 71 36,635 21,940 20,753
96 Babson College 48 50/77 13 77 81 38,443 21,316 NA
97
Rhode Island School
of Design 32 49/55 11 0 87 34,472 26,447 21,125
98
Rensselaer
Polytechnic Institute 70 68/92 17 48 75 39,200 22,360 24,590
99
Sarah Lawrence
College 40 79/44 6 51 66 42,121 22,847 14,864
100
The George
Washington
University 40 68/72 14 62 73 40,240 25,866 NA
Table 7.4 (Continued).
As shown in Tables 7.3 and 7.4, there are some missing values, denoted by NA or
“0”, in the Average Debt columns. For example, there are five NAs in the Average Debt
column in the public college data and nine “0”s in the same column in the private college
data. Values in the SAT or ACT fields are inconsistent with values in other fields
because of their value format. For example, the SAT or ACT value for George
Washington University in the private college data (see Table 7.4) is 68/72, which is not a
single value needed for a SOM method.
133
To meet the numerical input requirements of an SOM method, the data sets have
to be preprocessed. In the Average Debt column, we used the average value of the
column to replace the missing values. For example, we replaced the missing values with
the average value of 15480 in the public college data. The average value of 16957 was
used to replace the missing values in the private college data. In the SAT or ACT
column, many entries that are not in a single value format (i.e., percentages above 600 on
the verbal and math parts of SAT are separated by a slash). We used the average
percentage instead of the two individual percentages as input. For example, for George
Washington University, the average percentage value of 70 for the two individual
percentages, that is, 68 and 72, in the SAT or ACT column was used as input. After
preprocessing data in the two data sets, we applied Viscovery on the public and private
college data sets to see which groups of colleges are real bargains.
7.2 Discussion of Results
The Viscovery SOM map of the public college data is given in Figure 7.1. There
are three clusters separated by solid lines. The summary of the clusters is given in Table
7.5. Cluster membership of each public college is given in Table 7.6.
Cluster A has 18 schools. The average SAT or ACT percentage of schools in
cluster A is 70%, which is the highest among three clusters, indicating that these schools
have a good academic atmosphere. Other academic quality variables provide the same
insight. For example, schools in cluster A have the lowest average admission rate (53%),
the highest average student/faculty ratio (15), the highest average 4-year and 6-year
134
Figure 7.1 Map for the public college data set.
Cluster
SAT
or
ACT
(%)
Admis.
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-State
Total
Costs ($)
In-State
Costs
After Aid
($)
Total Out-
of-State
Costs ($)
Out-of-State
Costs After
Aid ($)
Avg. Debt
at Grad.
($)
A 70 53 15 49 75 13,657 7,605 25,845 18,432 15,500
B 48 62 16 43 69 15,238 10,682 22,737 12,158 14,915
C 47 72 17 27 57 11,469 7,737 19,384 11,634 15,798
Table 7.5 Summary of clusters of the public colleges.
135
Rank
School
Name
SAT
or
ACT
(%)
Admis..
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-
State
Total
Costs
($)
In-State
Costs
After Aid
($)
Total
Out-of-
State
Costs ($)
Out-of-
State
Costs
After
Aid ($)
Avg.
Debt at
Grad.
($)
Cluster A
1
University of
North Carolina
at Chapel Hill 71 35 14 65 79 11,290 6,673 23,138 16,465 11,156
2
University of
Virginia 81 39 16 81 91 12,640 8,737 28,610 19,873 13,536
3
College of
William and
Mary 84 35 12 80 89 13,024 6,668 27,724 21,056 19,762
4
University of
Georgia 57 65 13 46 66 10,534 5,245 21,310 16,065 12,906
5
University of
Florida 63 58 21 49 77 10,611 6,125 21,639 15,514 14,449
6
New College
of Florida 84 65 11 47 72 10,947 5,613 24,185 18,572 16,645
7
Georgia
Institute of
Technology 85 59 14 18 69 11,340 6,022 23,266 17,244 17,221
8
University of
Illinois at
Urbana,
Champaign 79 60 13 52 76 14,410 8,045 25,446 17,401 14,791
12
University of
Delaware 47 48 12 54 72 13,416 7,666 22,946 15,280 13,610
13
University of
Wisconsin,
Madison 86 71 14 41 77 13,391 7,842 27,401 19,559 15,904
14
University of
Michigan, Ann
Arbor 83 49 15 61 82 16,671 8,661 33,473 25,463 16,825
15
University of
California, San
Diego 60 41 19 43 78 16,000 9,043 24,105 17,148 13,275
16
University of
California,
Berkeley 72 25 17 48 83 17,265 8,982 25,584 17,301 14,990
17
University of
Washington 48 68 11 40 70 13,835 6,926 24,991 18,082 14,500
25
Colorado
School of
Mines 89 67 12 31 61 13,780 8,532 26,970 18,438 17,500
37
University of
California, Los
Angeles 69 24 17 40 81 17,616 9,975 26,006 16,031 12,775
43
University of
Colorado at
Boulder 65 80 16 38 64 11,937 7,877 28,253 20,376 16,737
77
University of
Vermont 37 71 14 48 67 17,116 8,258 30,168 21,910 22,425
Average 70 53 15 49 75 13,657 7,605 25,845 18,432 15,500
Table 7.6 Cluster profiles of the public colleges.
136
Rank
School
Name
SAT or
ACT
(%)
Admis.
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-
State
Total
Costs
($)
In-State
Costs
After Aid
($)
Total
Out-of-
State
Costs ($)
Out-of-
State
Costs
After
Aid ($)
Avg.
Debt at
Grad.
($)
Cluster B
20
University of
Texas at
Austin 58 61 19 39 71 14,391 9,111 20,999 11,888 16,400
24
State
University of
New York at
Binghamton 62 42 19 69 80 13,587 9,214 19,537 10,323 13,915
28
College of
New Jersey 66 48 12 59 80 16,686 13,915 21,261 7,346 5,490
32
State
University of
New York
College at
Geneseo 69 49 19 67 79 12,840 10,840 18,790 7,950 15,000
36
University of
Maryland,
College Park 71 43 13 33 63 16,304 12,223 22,881 10,658 15,566
41
Rutgers, The
State
University of
New Jersey,
New
Brunswick 47 55 14 44 72 16,519 10,044 23,033 12,989 15,270
42
Clemson
University 53 52 16 35 69 14,618 11,121 22,216 11,095 14,347
46
Mary
Washington
College 56 60 17 65 75 12,287 9,033 20,035 11,002 13,100
51
James
Madison
University 37 58 17 59 78 13,145 9,320 21,367 12,047 11,786
52
University of
California,
Davis 52 63 19 28 75 16,521 10,334 23,814 13,480 13,507
53
Miami
University 84 77 17 61 80 15,833 12,467 25,603 13,136 17,579
54
Purdue
University 38 76 16 28 64 14,691 8,723 26,311 17,588 15,677
58
University of
California,
Irvine 40 57 18 34 72 15,635 8,507 22,450 13,944 12,513
60
University of
Minnesota,
Morris 61 82 14 50 76 13,477 8,427 13,477 5,050 9,208
63
St. Mary's
College of
Maryland 64 59 12 58 67 16,908 12,908 23,228 10,320 17,125
65
University of
California,
Santa Barbara 47 51 19 44 73 16,154 10,231 24,246 14,015 15,480
74
University at
Buffalo, The
State
University of
New York 35 61 14 32 56 13,422 9,956 19,372 9,416 16,255
Table 7.6 (Continued).
137
Rank
School
Name
SAT or
ACT
(%)
Admis.
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-
State
Total
Costs
($)
In-State
Costs
After Aid
($)
Total
Out-of-
State
Costs ($)
Out-of-
State
Costs
After
Aid ($)
Avg.
Debt at
Grad.
($)
Cluster B (Continued)
76
University of
Massachusetts
Amherst 35 58 19 41 61 14,480 9,714 23,333 13,619 15,321
78
College of
Charleston 47 60 14 32 52 14,008 11,214 21,270 10,056 15,135
79
Indiana
University
Bloomington 30 81 20 40 65 13,129 8,704 24,164 15,460 16,930
80
Pennsylvania
State
University
University
Park Campus 50 57 17 43 80 17,017 12,872 26,639 13,767 17,900
84
Rutgers, The
State
University of
New Jersey,
Camden 22 54 11 21 60 16,189 10,111 22,703 12,592 15,223
85
University of
Maryland,
Baltimore
County 55 63 17 28 53 16,516 12,946 23,368 10,422 14,500
86
University of
Connecticut 38 62 17 23 70 14,413 9,141 25,197 16,056 16,093
87
University of
Pittsburgh 52 55 17 35 60 17,025 12,723 26,337 13,614 20,154
92
University of
New
Hampshire 29 77 14 48 71 15,779 13,693 26,139 12,446 20,700
94
University of
California,
Santa Cruz 38 80 19 40 64 16,877 9,309 24,250 14,941 13,282
95
Rowan
University 28 44 14 37 63 15,416 10,632 20,812 10,180 15,480
99
Ohio
University 49 75 20 43 70 16,514 13,102 24,737 11,635 15,285
100 UC Riverside 25 86 19 39 64 16,751 9,932 24,530 17,711 13,226
Average 48 62 16 43 69 15,238 10,682 22,737 12,158 14,915
Table 7.6 (Continued).
138
Rank
School
Name
SAT or
ACT
(%)
Admis.
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-
State
Total
Costs
($)
In-State
Costs
After Aid
($)
Total
Out-of-
State
Costs ($)
Out-of-
State
Costs
After
Aid ($)
Avg.
Debt at
Grad.
($)
Cluster C
9
Truman State
University 82 79 15 39 62 10,609 7,604 14,409 6,805 14,382
10
Virginia
Polytechnic
Institute and
State
University 46 65 15 36 72 10,122 5,613 20,006 14,393 16,229
11
North
Carolina State
University 50 59 15 25 60 10,688 5,508 22,536 17,028 15,476
18
New Mexico
Institute of
Mining and
Technology 75 63 13 12 40 9,714 2,708 16,151 9,145 9,500
19
University of
Wisconsin, La
Crosse 60 65 21 23 58 10,425 7,327 20,102 12,775 14,306
21
University of
Oklahoma 69 89 21 19 51 10,139 6,618 16,652 10,034 16,886
22
University of
Kansas 55 67 19 26 55 9,673 6,266 17,149 10,883 17,347
23
University of
North
Carolina at
Asheville 44 67 14 31 48 8,929 6,308 17,754 11,446 14,547
26
Auburn
University 51 83 16 40 68 10,276 7,308 18,736 11,428 18,585
27
Colorado
State
University 56 77 17 29 62 10,689 6,700 21,161 14,461 16,042
29
Michigan
State
University 57 67 18 31 66 12,743 8,670 22,703 14,033 18,663
30
Appalachian
State
University 28 64 19 31 60 7,913 4,785 16,834 12,049 13,000
31
Iowa State
University of
Science and
Technology 58 89 16 24 62 11,588 8,736 20,930 12,194 17,119
33
Texas A&M
University 43 68 21 27 69 11,899 7,057 18,979 11,922 15,670
34
University of
Texas at
Dallas 52 53 20 30 53 12,075 7,787 19,155 11,368 15,480
35
University of
North
Carolina at
Wilmington 22 55 16 34 60 9,715 6,375 19,290 12,915 13,583
Table 7.6 (Continued).
139
Rank
School
Name
SAT or
ACT
(%)
Admis.
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-
State
Total
Costs
($)
In-State
Costs
After Aid
($)
Total
Out-of-
State
Costs ($)
Out-of-
State
Costs
After
Aid ($)
Avg.
Debt at
Grad.
($)
Cluster C (Continued)
38
Louisiana
State
University and
Agricultural
and
Mechanical
College 53 77 21 23 58 10,126 7,406 15,426 8,020 17,569
39
University of
Tennessee 49 58 18 24 56 11,681 6,706 20,763 14,057 21,689
40
University of
Iowa 59 84 15 34 64 11,763 9,460 22,055 12,595 15,335
44
University of
Kentucky 54 82 17 27 58 9,432 5,594 16,112 10,518 15,480
45
University of
Arkansas 59 86 17 20 45 10,706 7,144 17,456 13,894 14,029
47
Oklahoma
State
University 49 92 19 22 56 10,677 7,478 16,623 9,145 15,580
48
Kansas State
University 49 58 20 18 45 9,626 7,104 16,990 9,886 17,000
49
University of
Northern Iowa 39 80 16 30 64 10,634 7,632 17,592 9,960 15,786
50
University of
Mississippi 46 80 19 29 48 11,257 6,607 16,167 9,560 14,459
55
Mississippi
State
University 50 74 16 19 48 11,130 8,174 16,036 7,862 15,081
56
University of
Nebraska,
Lincoln 55 90 19 15 51 10,687 7,277 18,269 10,992 15,682
57
Florida State
University 37 70 22 38 64 10,994 7,035 22,022 14,987 16,372
59
University of
Missouri,
Columbia 68 88 18 32 65 13,208 8,138 22,655 14,517 17,137
61
University of
Alabama 46 85 18 31 61 11,197 8,136 18,357 10,221 18,978
62
University of
South
Carolina 33 70 17 31 58 11,795 8,794 21,133 12,339 15,260
64
Michigan
Technological
University 68 92 11 22 63 14,135 9,339 25,025 15,686 15,711
66
University of
Minnesota,
Twin Cities
Campus 64 74 15 17 53 13,910 7,902 25,540 17,638 15,480
67
Mississippi
University for
Women 56 65 13 21 43 8,649 8,649 13,316 4,667 13,500
68
California
Polytechnic
State
University,
San Luis
Obispo 52 39 19 17 66 11,781 10,429 15,268 4,839 12,842
Table 7.6 (Continued).
140
Rank
School
Name
SAT or
ACT
(%)
Admis.
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-
State
Total
Costs
($)
In-State
Costs
After Aid
($)
Total
Out-of-
State
Costs ($)
Out-of-
State
Costs
After
Aid ($)
Avg.
Debt at
Grad.
($)
Cluster C (Continued)
69
University of
Wyoming 42 95 15 22 54 10,863 6,833 16,713 9,880 18,311
70
George
Mason
University 28 66 16 25 48 11,602 7,597 21,442 13,845 14,143
71
University of
Central
Florida 33 62 24 25 49 10,839 8,353 21,867 13,514 14,927
72
Ohio State
University 69 74 14 25 59 15,249 11,315 25,113 13,798 15,011
73
Illinois State
University 47 81 19 28 55 12,971 7,199 17,441 10,242 13,921
75
Salisbury
University 31 50 17 50 68 12,895 9,506 19,783 10,277 14,773
81
State
University of
New York at
Albany 31 56 21 52 66 13,574 9,599 19,524 9,925 15,108
82
University of
Arizona 30 86 19 29 55 11,163 8,506 19,933 11,427 17,340
83
Towson
University 22 58 19 30 56 12,776 8,651 20,422 11,771 15,530
88
State
University of
New York
College at
Fredonia 24 53 18 47 66 11,782 8,961 17,732 8,771 12,430
89
Stony Brook
University,
State
University of
New York 38 54 18 30 51 13,645 9,704 19,595 9,891 15,747
90
State
University of
New York at
New Paltz 30 40 17 21 52 11,276 8,276 16,176 7,900 15,000
91
University of
Maine 26 79 15 29 56 12,780 8,162 21,480 13,318 17,917
93
University of
Missouri,
Rolla 83 92 14 10 52 14,326 9,416 23,144 13,728 17,991
96
University of
Illinois at
Chicago 43 63 15 9 37 14,299 6,399 23,995 17,596 17,000
97
University of
Oregon 31 86 18 36 59 12,795 9,586 24,231 14,645 22,783
98
Texas Tech
University 26 69 20 22 51 12,968 9,878 20,048 10,170 13,805
Average 47 72 17 27 57 11,469 7,737 19,384 11,634 15,798
Table 7.6 (Continued).
141
graduation rates (49% and 75%) among three clusters. The financial cost is another
important concern for students as to choosing schools of the best values. Although the
average total costs for out-of-state students of cluster A schools (i.e., Total Out-of-State
Cost and Out-of-State Costs after Aid) are the highest, when cluster A schools are
compared to cluster B schools and cluster C schools, the average In-State Total Costs
($13657) is the second highest and the average In-State Costs after Aid ($7605) is the
lowest. For cluster A, average of the variable Average Debt at Graduation ($15500) is
the second highest and it is only about $600 greater than the lowest average debt
($14915) of cluster B schools. Therefore, cluster A schools are the group of schools that
not only have excellent education quality but also are financially comparable for students.
Perhaps those in-state students who have excellent academic performance are more
willing to consider cluster A schools because of the lowest average In-State Costs after
Aid.
Cluster C has 52 schools. The overall academic quality of schools in this cluster
is worse than schools in the other two clusters. For example, the average SAT or ACT
percentage of 47% is the lowest, the 4-year and 6-year graduation rates are the lowest,
i.e., 27% and 57% respectively, and the Student/faculty Ratio of 17 is the highest.
However, the average total costs for both residents and non-residents are the lowest
among the three groups. The average In-State Total Costs is $11469, which is about
$2200 less than cluster A schools.
The average Out-of-State Total Costs for cluster B schools is $19384, which is
about $6500 less than that of schools in cluster A. The average total costs after aid for in-
142
state and out-of-state students are also low. It tells that cluster C schools may be good
choices for those students who care more about financial costs.
Cluster B has 30 colleges that have good academic quality, i.e., the second highest
average values in SAT/ACT percentage, Admission Rate, Student/faculty Ratio, 4-year
Graduation Rate, and 6-year Graduation Rate. This cluster of schools has the highest
average costs ($15238) for in-state students, the second highest average costs ($22737)
for non-resident students, and the lowest Average Debt at Graduation ($14915). Schools
in the cluster B may be considered as alternatives to schools in cluster A and schools in
cluster C because they are comparable to cluster A schools in terms of good education
quality and similar to cluster C schools in terms of less expensive financial costs.
We did not include the Kiplinger’s rank variable to generate our SOM maps. The
Rank variable is used as school labels in the maps. Fifteen schools in cluster A are from
the top 25 public schools in Kiplinger’s list, such as University of North Carolina at
Chapel Hill, University of Virginia, College of William & Mary, University of Georgia
(Table 7.6). Two (i.e., UCLA and University of Colorado-Boulder) of the rest schools
come from the middle range, which is between No.26 and No.75, and the last one
(University of Vermont) is ranked No.77. Cluster A schools have the highest education
quality and require reasonable financial expenditures. Cluster B schools have better
academic quality and higher financial expenditures for students than cluster C schools.
When looking at individual schools, we found some schools belonging to the
same cluster have different ranks in Kiplinger’s list. For example, University of
Colorado at Boulder and Colorado School of Mines are in the same cluster A. They have
comparable education quality and the financial costs in these two schools are close. In
143
Kiplinger’s ranking, University of Colorado ranks No.43, which is 18 places lower than
Colorado school of Mines whose rank is No.25. Additional examples can be found in
cluster B schools and cluster C schools. For example, University of California at
Riverside and University of Clemson are members of cluster B. However, UC Riverside
is ranked No.100 and Clemson is ranked No.42. Although there are some differences
between them, most of their educational and financial measures are close. For example,
their 4-year and 6-year graduation rates and Average Debt at Graduation are close.
Therefore University of Clemson and UC Riverside should have close rankings. In
cluster C, University of Maine and Iowa State University have the same situation. Maine
is ranked No.31 while Iowa State is ranked No.91 despite their close educational and
financial qualities as shown in Table 7.6.
The Viscovery SOM map for the private colleges is given in Figure 7.2. There
are five clusters and the summary of the clusters is given in Table 7.7. Cluster
memberships of colleges are provided in Table 7.8.
Cluster A has two member colleges: Webb Institute (6) and Copper Union (25).
Both colleges have excellent educational qualities: the lowest average Student/faculty
Ratio (7), the highest average SAT or ACT percentage (91%) among the five clusters,
and the highest average 4-year and 6-year graduation rates (68% and 81% respectively)
as well. In addition, the financial costs in both colleges are very low. The average total
cost of both schools is $11366, less than half of the second lowest average total cost
($31335) of cluster E schools. The average total cost after need-based aid is the lowest
($8373) among five clusters. The average debt at graduation is the lowest ($7475), which
144
Figure 7.2 Map for the private college data set.
Cluster
Admission
Rate (%)
SAT or
ACT
(%)
Student/
faculty
Ratio
4-Year
Grad. Rate
(%)
6-Year
Grad. Rate
(%)
Total Costs
($)
Total Costs after
Need-based Aid
($)
Avg. Debt at
Grad. ($)
A 28 91 7 68 81 11,366 8,373 7,475
B 27 91 8 82 90 38,051 18,949 16,914
C 46 77 11 77 82 35,574 18,567 16,608
D 43 76 13 47 78 37,486 25,429 21,770
E 65 78 11 61 73 31,115 17,749 16,387
Table 7.7 Summary of clusters of the public colleges.
145
Rank School Name
Admiss.
Rate (%)
SAT or
ACT %
Student/
faculty
Ratio
4-Year
Grad.
Rate %
6-Year
Grad.
Rate %
Total
Costs
Cost After
Need- Based
Aid
Average
Debt
Cluster A
6 Webb Institute 42 100 7 79 83 8,079 5,579 5,700
25 Cooper Union 14 82 7 57 78 14,652 11,167 9,250
Average 28 91 7 68 81 11366 8373 7475
Cluster B
1
California Institute of
Technology 21 100 3 71 85 32,682 10,981 10,244
2 Rice University 24 91 5 68 89 28,350 14,779 12,705
3 Williams College 23 93 8 89 94 36,550 14,737 12,316
4 Swarthmore College 24 96 8 86 92 38,676 17,386 12,759
5 Amherst College 18 93 9 84 94 38,492 14,453 11,544
7 Yale University 8 97 7 88 95 38,432 15,729 19,228
9 Harvard University 11 90 8 86 97 38,831 17,456 10,465
10 Stanford University 13 94 7 77 93 38,875 17,746 15,782
11 Princeton University 11 96 5 91 97 40,169 18,325 12,000
12
Massachusetts Institute
of Technology 16 98 6 82 91 39,213 19,609 22,855
13 Pomona College 23 98 9 83 88 38,130 17,411 15,600
14 Emory University 42 92 7 82 87 37,272 19,657 17,675
15 Columbia University 12 92 7 83 93 39,493 17,778 15,331
16 Duke University 25 93 11 88 93 40,080 19,996 20,025
17 Davidson College 34 88 10 89 91 34,706 21,455 13,697
18 Wellesley College 47 89 9 84 88 37,419 17,526 15,697
19 Vassar College 31 91 9 81 87 37,870 19,404 17,170
20 Haverford College 32 90 8 89 92 38,928 17,826 15,253
21 Northwestern University 33 90 7 83 92 38,817 20,376 14,551
22 Bowdoin College 25 90 10 83 90 38,663 17,773 15,307
23
University of
Pennsylvania 21 94 6 83 91 39,040 20,596 20,247
24
Johns Hopkins
University 35 89 8 81 88 39,188 19,142 13,600
26 Washington University 24 96 7 75 86 39,253 20,700 16,957
27 Dartmouth College 23 94 9 87 95 38,898 19,546 16,957
28
Claremont McKenna
College 28 92 7 82 86 37,730 17,988 16,914
29
University of Notre
Dame 34 87 12 88 95 35,392 18,011 25,595
30 Colgate University 34 83 10 85 89 38,820 18,856 12,984
33 Georgetown University 21 88 11 86 91 39,182 24,382 20,000
34 Brown University 17 88 8 79 94 40,248 20,838 21,700
35 Carleton College 35 89 9 82 86 35,288 21,677 14,543
Table 7.8 Cluster profiles of the private college data.
146
Rank School Name
Admiss.
Rate (%)
SAT or
ACT %
Student/
faculty
Ratio
4-Year
Grad.
Rate %
6-Year
Grad.
Rate %
Total
Costs
Cost After
Need- Based
Aid
Average
Debt
Cluster B (Continued)
37 Middlebury College 27 95 11 81 87 39,532 18,288 21,751
40 Bates College 28 91 10 82 87 38,932 18,258 17,045
41 Cornell University 29 89 9 82 90 38,974 23,122 15,587
42 Wesleyan University 28 91 9 76 81 39,127 21,401 23,753
43 Colby College 33 87 11 85 88 38,699 18,168 17,270
50 Brandeis University 42 88 8 79 85 39,101 22,257 16,957
52 Harvey Mudd College 37 99 9 75 83 38,880 22,041 20,219
53 Wake Forest University 41 83 10 77 87 36,079 21,196 24,769
56 Tufts University 27 86 9 81 88 39,173 20,115 15,499
75 Vanderbilt University 46 87 9 78 84 38,847 20,971 24,023
Average 27 91 8 82 90 38,051 18,949 16,914
Cluster C
8
Washington and Lee
University 31 89 11 86 89 30,225 15,452 15,634
31 The Colorado College 53 68 9 72 79 35,275 16,516 13,500
32 University of Richmond 41 79 10 79 84 31,679 17,588 16,115
36 Lafayette College 36 71 11 79 84 35,713 15,147 17,380
38 Grinnell College 65 86 10 78 84 31,460 16,585 13,854
39
Illinois Wesleyan
University 48 96 12 76 81 30,780 18,858 17,722
44 Bucknell University 39 78 12 83 87 36,165 19,165 16,000
45 Kenyon College 52 84 9 80 84 36,273 17,905 20,850
48 Macalester College 44 88 10 71 77 32,847 16,394 16,957
49 Barnard College 34 88 10 72 84 37,940 17,826 14,030
51
College of the Holy
Cross 43 74 11 88 90 36,851 23,846 16,063
54 Bryn Mawr College 50 81 9 76 80 37,890 18,609 16,957
55 Wheaton College 54 84 11 70 84 27,076 17,341 15,864
58 Mount Holyoke College 52 75 10 75 79 38,668 19,268 14,200
62 Lehigh University 44 72 11 70 84 35,670 19,123 16,972
63 Smith College 53 69 9 76 80 37,937 18,466 19,911
66 Union College 45 64 11 75 80 36,455 18,431 15,725
67 Hamilton College 35 80 10 79 84 38,463 19,474 16,856
72 Trinity College 36 76 9 77 83 38,890 19,667 17,000
78
Franklin and Marshall
College 62 67 11 78 83 36,580 20,925 19,656
82 Connecticut College 35 84 11 75 81 37,057 16,930 17,250
84 Dickinson College 51 65 13 74 78 36,600 19,753 17,586
93 Skidmore College 46 70 11 71 75 38,838 21,023 15,560
96 Babson College 48 64 13 77 81 38,443 21,316 16,957
Average 46 77 11 77 82 35,574 18,567 16,608
Table 7.8 (Continued).
147
Rank School Name
Admiss.
Rate (%)
SAT or
ACT %
Student/
faculty
Ratio
4-Year
Grad.
Rate %
6-Year
Grad.
Rate %
Total
Costs
Cost After
Need- Based
Aid
Average
Debt
Cluster D
71
University of Southern
California 30 85 10 51 73 37,968 21,606 20,619
80
Carnegie Mellon
University 38 85 11 61 77 38,460 24,689 19,195
87 Boston College 34 82 13 0 86 37,745 24,470 16,732
91 New York University 28 87 11 65 74 40,105 28,282 21,495
92 Villanova University 47 66 13 79 84 36,560 26,463 28,217
94
Rose-Hulman Institute of
Technology 65 77 13 58 71 32,625 28,677 27,000
97
Rhode Island School of
Design 32 52 11 0 87 34,472 26,447 21,125
98
Rensselaer Polytechnic
Institute 70 80 17 48 75 39,200 22,360 24,590
100
The George Washington
University 40 70 14 62 73 40,240 25,866 16,957
Average 43 76 13 47 78 37,486 25,429 21,770
Table 7.8 (Continued).
148
Rank School Name
Admiss.
Rate (%)
SAT or
ACT %
Student/
faculty
Ratio
4-Year
Grad.
Rate %
6-Year
Grad.
Rate %
Total
Costs
Cost After
Need- Based
Aid
Average
Debt
Cluster E
46 Centre College 78 89 11 71 73 28,529 15,842 14,300
47 Rhodes College 70 95 11 71 73 30,080 18,899 15,100
57 Oberlin College 33 84 10 63 76 37,688 21,081 13,926
59 Furman University 58 68 11 74 81 29,430 16,296 17,741
60 St. Olaf College 73 84 13 71 75 29,879 17,458 18,806
61
Brigham Young
University 73 86 18 31 73 9,663 7,621 11,000
64 Beloit College 70 82 11 60 72 30,264 17,452 14,942
65 Taylor University 78 74 15 71 75 24,723 15,678 15,117
68 DePauw University 61 58 11 75 79 32,150 15,531 14,481
69 Hillsdale College 82 74 11 53 71 23,353 13,853 14,500
70 Knox College 72 72 12 67 74 30,894 15,494 16,920
73 Trinity University 69 76 11 65 75 27,086 16,706 16,957
74
Gustavus Adolphus
College 77 70 13 72 75 27,820 17,609 17,400
76 Whitman College 50 81 10 60 71 33,776 21,176 15,000
77 Scripps College 58 82 12 63 68 36,500 17,984 12,941
79 Saint Louis University 72 72 12 52 67 29,983 16,902 14,989
81 Lawrence University 68 83 11 58 68 32,875 17,882 18,311
83
Case Western Reserve
University 78 80 8 49 75 32,802 18,323 21,830
85 Kalamazoo College 73 91 12 60 69 30,917 17,947 20,000
86 Saint John's University 87 66 13 67 74 27,272 19,544 20,680
88 Reed College 55 91 10 45 67 37,900 18,804 16,758
89 Bard College 36 76 9 59 71 38,282 20,558 15,400
90 University of Rochester 50 83 12 65 76 37,246 20,297 16,957
95 St. John's College 71 85 8 63 71 36,635 21,940 20,753
99 Sarah Lawrence College 40 62 6 51 66 42,121 22,847 14,864
Average 65 78 11 61 73 31,115 17,749 16,387
Table 7.8 (Continued).
149
is nearly $9000 less than the second lowest average debt at graduation ($16387) of the
cluster E. Therefore, cluster A schools can be considered as schools of great value.
Cluster B has 40 schools. From the average Admission Rate, the average SAT or
ACT, and other academic-related variables (i.e., SAT/ACT, Student/Faculty Ratio, 4-year
and 6-year Graduation Rates), we see that most of schools in cluster B have excellent
education quality. All of Ivy League schools are included. Most of them are financially
expensive, which is shown by the highest average Total Costs ($38051) (see Table 7.7)
among the five clusters. However, if the need-based aid is taken into account, the
average total cost is reduced to $18949 and the average Average Debt at Graduation is
$16914, which makes cluster B schools comparable to schools in clusters C, D, and E. If
students care more about educational quality, then schools in cluster B are worthwhile.
There are 24 colleges in cluster C. Compared to cluster B schools, schools in
cluster C are financially less expensive. For example, the average Total Cost ($35574) is
about $2400 less than that of cluster B schools. However, the education quality of
schools in this cluster is not as exceptional as schools in cluster B. For example, in
cluster C, the average SAT or ACT percentage is less than the average SAT or ACT
percentage of cluster B by 14 points. In addition, the Student/faculty Ratio and, the 4-
year and 6-year graduation rates, are all lower than those of cluster B. Therefore, cluster
C schools provide relatively good academic quality and ask for reasonable financial
sacrifice.
There are nine schools in cluster D. The overall academic quality of these schools
is not as good as the overall academic quality of schools in Cluster C, while the financial
costs of cluster D schools are much higher than the financial costs of cluster C. As
150
shown in Table 7.7, the average academic-related measures are slightly lower than those
of cluster C schools. For example, the average 6-year graduation rate of cluster D (78%)
is a little worse than the average 6-year graduation of cluster C (82%). The average
financial costs in cluster D are the highest among the five clusters. For example, the
average Cost after Need-based Aid of this cluster is $25429 which is about $6500 higher
than the second highest of cluster B ($18949). In terms of educational quality and
financial consideration, schools in cluster D might be considered less competitive than
schools in clusters A, B, and C.
Cluster E has 25 colleges. Compared to the other four clusters, the academic
quality of schools in this cluster is not as good as schools in other four clusters. The
average 4-year and 6-year graduation rates are the lowest among five clusters. However,
the average financial cost of schools in cluster E is the second lowest. Since the
academic quality of cluster E is close to or slightly worse than that of cluster D and the
financial cost of cluster E is much lower than that of cluster D, schools in cluster E might
be considered alternatives of schools in cluster D for students who have financial
concerns.
Although our analysis of cluster structures agrees with Kiplinger’s list in most
cases, there are some discrepancies, especially when examining individual schools within
each cluster. For example, in cluster B, Vanderbilt University has similar academic and
financial measures as Wake Forest (53) and Wesleyan (42). Vanderbilt is ranked No. 75
in Kiplinger’s list, where Vanderbilt’s Kiplinger rank is 23 places and 32 places lower
than Wake Forest and Wesleyan, respectively. Barnard College (49) and Connecticut
College (82) in cluster C is another example. These two schools have comparable
151
academic measures and close financial measures (see Table 7.8). However, the
difference between their Kiplinger’s ranks is 47. This kind of information can not be
discovered on Kiplinger’s list. With the help of the SOM visual maps, alternatives can
easily be found by examining a school’s neighbors.
Our results of the public and private college data sets have shown that
Viscovery’s SOM map helps identify alternatives of a particular school, which may not
be detected from Kiplingers’ rankings. For example, Figure 7.1 gives us a visual map of
the hidden cluster structure of the public college data. On this map, three clusters are
clearly visible, where alternatives of a school can be easily recognized. Colorado State
(27) has six neighbors: Virginia Polytechnic Institute and State University (10), North
Carolina State University (11), Auburn University (26), Michigan State University (29),
Iowa State University (31), and University of Missouri at Columbia (59). If we look for
Colorado State’s alternatives solely on the Kiplinger’s list, we are very likely to think
about schools with close ranks to Colorado State as its alternatives. However, not every
alternative has a close rank to Colorado State, such as North Caroline State. Therefore,
Viscovery can help college students identify alternatives, gain more insights from the
data, and facilitate them to make a better decision.
The differences between our results and Kiplinger’s list do not mean that the
Kiplinger’s list is of no value, although our analysis shows that the Kiplinger’s list can be
misleading. If we could combine the results obtained from our maps and Kiplinger’s
rankings, we might have better way to analyze and explain the data set to help students.
152
Chapter 8
Summary and Future Work
In order to visualize a data set with an asymmetric distance matrix, the standard
SM method takes as inputs the symmetrized distance matrix by averaging entries in the
asymmetric distance matrix. This approach is the simplest way to represent asymmetric
data onto a 2-dimentional map. However, some interesting information hidden in
asymmetric data may be ignored due to averaging. Merino et al.’s method introduces
into the standard SM method an asymmetry coefficient that is expected to reflect
asymmetric information. However, when the data set under consideration is not very
asymmetric, the asymmetry coefficient defined by Merino et al. has little influence on the
resulting map. The map obtained from Merino’s method is very similar to the one
obtained from the standard SM method. Our modified SM method takes into account the
upper triangular part and the lower triangular part of an asymmetric distance matrix
simultaneously. It is reasonable to expect that the modified SM method may outperform
the standard and Merino’s method to some extents.
We applied the modified method to two asymmetric data sets: American college
selection data and Canadian ranked college data. From the results obtained on these two
data sets, we found that the modified SM method always did a fairly good job at reducing
153
distance errors and performed reasonably well at preserving order relationships at least
comparable to the standard and Merino’s methods. Since asymmetric proximity data
arise in other research areas such as marketing, psychology, sociology, etc, one research
problem could be how to use our modified SM method to visualize these data sets. If
such maps could be generated with reasonable interpretability, they might be used to
discover relationships between data items that may hardly be detected by other methods,
and therefore assist the analysis of the asymmetric data sets in different research and
business disciplines.
In terms of helping detect hidden structures, clustering has been widely used in
the applications of data visualization. We have found that the clustering procedures in
Viscovery outperformed the K-means clustering method and the classic SOM method.
We have applied Viscovery to the state-sponsored murder data set and got some
interesting results. Meanwhile, through analyzing 200 public and private colleges with
Viscovery, we have generated several clusters for the public college data set and the
private college data set, respectively. These college clustering results are not quite
similar to Kiplinger’s results. In practice, there are lots of ranking lists trying to give
readers the idea of which is the best/worst or which is most likely to happen. However,
the ranking lists such as Kiplinger’s list do not include alternative information that
readers want to look for. Maybe in addition to their original ranking lists, SOM maps
showing the clustering information should be included as well to better deliver useful
information to readers to help make their decisions.
154
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Self-organization is a process where some form of global order or coordination arises out of the local interactions between the components of an initially disordered system.
ABSTRACT
Title of dissertation: DATA VISUALIZATION OF ASYMMETRIC DATA
USING SAMMON MAPPING AND APPLICATIONS OF
SELF-ORGANIZING MAPS
Haiyan Li, Doctor of Philosophy, 2005
Dissertation directed by: Professor Bruce L. Golden
Department of Decision and Information Technologies
Data visualization can be used to detect hidden structures and patterns in data sets
that are found in data mining applications. However, although efficient data visualization
algorithms to handle data sets with asymmetric proximities have been proposed, we
develop an improved algorithm in this dissertation.
In the first part of the proposal, we develop a modified Sammon mapping
approach that uses the upper triangular part and the lower triangular part of an
asymmetric distance matrix simultaneously. Our proposed approach is applied to two
asymmetric data sets: an American college selection data set, and a Canadian college
selection data set which contains rank information. When compared to other approaches
that are used in practice, our modified approach generates visual maps that have smaller
distance errors and provide more reasonable representations of the data sets.
In data visualization, self-organizing maps (SOM) have been used to cluster
points. In the second part of the proposal, we assess the performance of several software
implementations of SOM-based methods. Viscovery SOMine is found to be helpful in
determining the number of clusters and recovering the cluster structure of data sets. A
genocide and politicide data set is analyzed using Viscovery SOMine, followed by
another analysis on the public and private college data sets with the goal to find out
schools with best values.
DATA VISUALIZATION OF ASYMMETRIC DATA USING SAMMON MAPPING
AND APPLICATIONS OF SELF-ORGANIZING MAPS
by
Haiyan Li
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2005
Advisory Committee:
Professor Bruce L. Golden, Chair
Professor Edward A. Wasil
Assistant Professor Paul F. Zantek
Assistant Professor Steven Gabriel
Professor Ali Haghani
©Copyright by
Haiyan Li
2005
ii
ACKNOWLEDGEMENTS
The work with this dissertation has been extensive and trying, but in the first
place exciting, instructive, and fun. Without help, support, and encouragement from
several persons, I would never have been able to finish this work.
First of all, I would like to thank my advisor, Dr. Bruce Golden, for his generous
time and commitment. It is not often that one has an advisor that always contributes the
time for listening to the little problems and roadblocks that unavoidably crop up in the
course of performing research. Throughout my doctoral work he encouraged me to
develop independent thinking and research skills. He continually stimulated my
analytical thinking. His technical and editorial advices were essential to the completion
of this dissertation proposal and have taught me innumerable lessons and insights on the
workings of academic research in general.
I am also appreciative and thankful for the support and patience of Dr. Edward
Wasil, who has read through my draft copies, listened to my complaining and encouraged
me every step of the way. His insightful comments were crucial for editing the drafts
into the dissertation proposal. His timeliness in returning the drafts did not go unnoticed.
I am indebted to Dr. Paul Zantek who has been a source of support throughout my
graduate studies. His data mining class has been instrumental in shaping my
understanding of clustering and data visualization. I am very grateful to Dr. Steven
iii
Gabriel for the many valuable comments that helped shape this work. I would like to
express my appreciation to Dr. Ali Haghani for gladly serving on my dissertation
committee.
Thanks also to all my colleagues at the Department of Decision and Information
Technologies for providing a good working atmosphere, especially Stacy Calo for
assisting me with collecting survey and data entry during my late months of pregnancy.
At last, to my family thanks for supporting me with your love and understanding.
iv
TABLE OF CONTENTS
LIST OF TABLES………………………………………………………………………..vi
LIST OF FIGURES ......................................................................................................... viii
Chapter 1 Introduction................................................................................................. 1
1.1 Motivation and the Problem of Interest ...................................................... 1
1.2 Summary of Objectives............................................................................... 6
1.3 Dissertation Organization ........................................................................... 7
Chapter 2 Literature Review........................................................................................ 9
2.1 Data Visualization..................................................................................... 10
2.2 Multidimensional Scaling......................................................................... 13
2.2.1 Overview................................................................................................... 13
2.2.2 Problem definition and stress function ..................................................... 15
2.3 Sammon Mapping..................................................................................... 17
2.3.1 Overview................................................................................................... 17
2.3.2 SM algorithm............................................................................................ 18
2.3.3 Implementation ......................................................................................... 19
2.4 Self-organizing Maps................................................................................ 22
2.4.1 Overview................................................................................................... 22
2.4.2 SOM algorithm......................................................................................... 23
2.4.3 SOM software packages ........................................................................... 24
2.5 Asymmetric Proximity.............................................................................. 27
2.5.1 Overview................................................................................................... 27
2.5.2 Mathematical modeling for asymmetric proximity data........................... 28
2.5.3 Other approaches of analyzing asymmetric proximity data ..................... 31
Chapter 3 Constructing Sammon Maps from Asymmetric Data............................... 36
3.1 Modification of Sammon Mapping Method ............................................. 37
v
3.2 Implementation of the Modified SM Method........................................... 42
3.3 Performance Measurements of Sammon Maps ........................................ 48
Chapter 4 Visualizing American College Selection Data ......................................... 51
4.1 Description of the Data Set ....................................................................... 51
4.2 Experimental Design................................................................................. 57
4.3 Discussion of the Results.......................................................................... 60
Chapter 5 Visualizing Canadian Ranked College Data............................................. 81
5.1 Description of Canadian Ranked College Data ........................................ 82
5.2 Modeling Steps ......................................................................................... 85
5.3 Discussion of the Results.......................................................................... 87
Chapter 6 Self-Organizing Maps: State Sponsored Murder Data Set ....................... 95
6.1 Introduction............................................................................................... 95
6.2 Evaluating SOM-based Methods .............................................................. 96
6.2.1 Constructing data sets ............................................................................... 97
6.2.2 Measuring performance ............................................................................ 99
6.2.3 Comparison results and conclusions....................................................... 100
6.3 Self-Organizing Maps: the State Sponsored Murder Data Set ............... 104
Chapter 7 Self-Organizing Maps: Best Values in Colleges .................................... 122
7.1 Data Description and Preprocessing....................................................... 123
7.2 Discussion of Results.............................................................................. 133
Chapter 8 Future Work............................................................................................ 152
References …………………………………………………………………………..154
vi
LIST OF TABLES
Table 1.1 Asymmetric proximity matrix of schools A, B, and C. ....................... 4
Table 1.2 Example of a 3 × 3 asymmetric distance matrix.................................. 6
Table 2.1 Matrix of distances among 10 buildings............................................ 14
Table 2.2 Stress guidelines suggested by Kruskal (1964a). .............................. 17
Table 2.3 An example of car switching data. .................................................... 28
Table 2.4 Asymmetric distance matrix taken from the American college
selection data ..................................................................................... 34
Table 3.1 Asymmetric distance matrix of three data points. ............................. 41
Table 3.2 Results in our proposed approach with random starts 1. ................... 41
Table 3.3 Results in our proposed approach with random starts 2. ................... 41
Table 3.4 Results in the common approach with random starts 1. .................... 41
Table 3.5 Results in the common approach with random starts 2. .................... 42
Table 3.6 Asymmetric distance matrix of 30 American colleges. ..................... 44
Table 3.7 Symmetrized distance matrix of 30 American colleges. ................... 45
Table 3.8 Asymmetric distance matrix of four data points................................ 49
Table 3.9 Order relationships............................................................................. 50
Table 4.1 Adjacency matrix for six schools. ..................................................... 53
Table 4.2 One hundred schools selected from The Fiske Guide for analysis. ... 54
Table 4.3 Average error measures obtained from the modified SM method. ... 58
Table 4.4 Average order preservation coefficients obtained from the modified
SM method......................................................................................... 58
Table 4.5 Order preservation coefficients for the six data sets (standard vs.
modified)............................................................................................ 77
Table 4.6 Order preservation coefficients for the six data sets (Merino’s vs.
modified)............................................................................................ 77
Table 4.7 Error measures for the six data sets (standard vs. modified). ............ 78
Table 4.8 Error measures for the six data sets (Merino’s vs. modified)............ 78
Table 5.1 44 Canadian universities collected from surveys. ............................. 83
Table 5.2 Competitors of 44 Canadian universities........................................... 84
Table 5.3 Order preservation measures of different gap values of Canadian
ranked................................................................................................ 91
Table 5.4 Order preservation measures of different gap values of Canadian
ranked................................................................................................ 91
Table 5.5 Error measures of different gap values of Canadian ranked college
data.................................................................................................... 91
Table 5.6 Error measures of different gap values of Canadian ranked college
data (Merino’s vs. modified)............................................................. 92
vii
Table 6.1 Pairwise classification notation. ...................................................... 100
Table 6.2 Cluster recovery rates (in %). .......................................................... 101
Table 6.3 Cluster recovery rates (in %) by level of dispersion........................ 102
Table 6.4 Values of the Rand statistics............................................................ 102
Table 6.5 Cluster recovery rates (in %) for Viscovery. ……………………..104
Table 6.6 Genocide and politicide data set from Harff (2003). ....................... 106
Table 6.7 Transformed genocide and politicide data....................................... 108
Table 6.8 Notation for the transformed genocide and politicide data set. ....... 108
Table 6.9 Cluster profiles of the genocide and politicide data set. .................. 110
Table 6.10 Modified genocide and politicide data set 1 (M1)........................... 112
Table 6.11 Modified genocide and politicide data set 2 (M2)........................... 113
Table 6.12 Notation of the transformed genocide and politicide data with
added variables……………………………………………………..114
Table 6.13 Transformed modified genocide and politicide data set 1 (M1)...... 115
Table 6.14 Transformed modified genocide and politicide data set 2 (M2)...... 116
Table 6.15 Cluster profiles of the first modified data set (M1). ........................ 118
Table 6.16 Cluster profiles of the second modified data set (M2). ................... 119
Table 7.1 Seven common variables in Kiplinger’s public and private data
sets .………………………………………………………………..124
Table 7.2 Four additional variables in Kiplinger’s public and private data
sets. .................................................................................................. 124
Table 7.3 Kiplinger’s (2003) public college data. ........................................... 126
Table 7.4 Kiplingers’ (2004) private college data. .......................................... 130
Table 7.5 Summary of clusters of the public colleges. .................................... 134
Table 7.6 Cluster profiles of the public colleges. ............................................ 135
Table 7.7 Summary of clusters of the public colleges. .................................... 144
Table 7.8 Cluster profiles of the private college data. ..................................... 145
viii
LIST OF FIGURES
Figure 2.1 Example using principle components analysis. ................................. 11
Figure 2.2 Illustration of a principal curve.......................................................... 13
Figure 2.3 Example of multidimensional scaling. .............................................. 14
Figure 2.4 Sammon map of the 10 building........................................................ 18
Figure 2.5 Illustration of a self-organizing map.................................................. 23
Figure 2.6 Sammon map produced by SOM_Pak............................................... 25
Figure 2.7 Screenshot of Viscovery. Three groups (As, Bs, and Cs) are
shown in the....................................................................................... 26
Figure 2.8 Sammon map of the upper triangular matrix. .................................... 35
Figure 2.9 Sammon map of the lower triangular matrix. .................................... 35
Figure 3.1 Sammon map of the asymmetric distance matrix for data set 30A
generated by the modified SM method. ............................................ 46
Figure 3.2 Sammon map of the symmetrized distance matrix for data set
30A generated by the standard SM method. ..................................... 46
Figure 4.1 Directed graph generated from the adjacency matrix........................ 53
Figure 4.2 Map of 100 schools generated by the modified SM method. ............ 61
Figure 4.3 Map of 100 schools generated by the standard SM method. ............. 61
Figure 4.4 Map of 100 schools generated by Merino’s method. ........................ 62
Figure 4.5 Map of group A generated by the modified SM method................... 65
Figure 4.6 Map of group A generated by the standard SM method.................... 65
Figure 4.7 Map of group A generated by the Merino’s method. ........................ 66
Figure 4.8 Map of 30 schools from group A generated by the modified SM
method. .............................................................................................. 69
Figure 4.9 Map of 30 schools from group A generated by the standard SM
method. .............................................................................................. 69
Figure 4.10 Map of 30 schools from group A generated by Merino’s method. ... 70
Figure 4.11 Map of group B generated by the modified SM method. .................. 70
Figure 4.12 Map of group B generated by the standard SM method. ................... 71
Figure 4.13 Map of group B generated by Merino’s method. .............................. 71
Figure 4.14 Map of group C generated by the modified SM method. .................. 73
Figure 4.15 Map of group C generated by the standard SM method. ................... 73
Figure 4.16 Map of group C generated by Merino’s method. .............................. 74
Figure 4.17 Map of group D generated by the modified SM method................... 74
Figure 4.18 Map of group D generated by the standard SM method.................... 75
Figure 4.19 Map of group D generated by Merino’s method. .............................. 75
Figure 5.1 Sammon map of the Canadian ranked college generated by the
modified method................................................................................ 88
Figure 5.2 Sammon map of the Canadian ranked college generated by the
standard method................................................................................. 89
ix
Figure 5.3 Sammon map of the Canadian ranked college generated by Merino’s
method. .............................................................................................. 89
Figure 5.4 Sammon map of the Canadian ranked college generated by the
modified method. ............................................................................... 92
Figure 5.5 Sammon map of the Canadian ranked college generated by the
modified method. ............................................................................... 93
Figure 6.1 Example of a four-cluster data set. .................................................... 98
Figure 6.2 Resulting SOM map of the genocide and politicide data set. .......... 109
Figure 6.3 Resulting SOM map of the first modified data set (M1). ................ 117
Figure 6.4 Resulting SOM amp of the second modified data set (M2). ........... 117
Figure 7.1 Map for the public college data set. ................................................. 134
Figure 7.2 Map for the private college data set................................................. 144
1
Chapter 1
Introduction
In this chapter, we provide background information on data visualization and
describe the motivation and objectives of our research.
1.1 Motivation and the Problem of Interest
Data visualization is the process of “representing data as a visual image” (Latham,
1995) in which an image is created using a combination of points, lines, coordinate
systems, numbers, symbols, words, shadings, and colors to represent different measured
quantities (Tufte, 1983).
Data visualization is often used to make apparent any pattern in a data set that is
large in size or dimensionality. For example, analyzing increasing amounts of data, such
as customer data, to discover hidden patterns is a major problem facing businesses and
organizations today. Visualization, together with other data mining techniques such as
clustering and classification, can be employed to generate a data map that serves as a
guide and provides the user with insights, i.e., detecting customer purchase patterns. The
ability to show patterns attracts decision makers to use data visualization as a tool to get a
2
better understanding of the data set and then make better decisions. For example,
consider the problem facing each high-school senior: selecting an undergraduate
American university or college to pursue a bachelor’s degree. Students can consult
rankings in popular publications and reference books such as The Fiske Guide, which
provides information on 300 universities and colleges. The information contained in
these publications and books is not easy to assimilate. Condon et al. (2002) built a model
of American universities that enables a student to visualize the data. Information that
cannot be revealed in lists and tables, such as similarities between the universities, can be
directly quantified according to the distances between universities on visual maps. These
maps can assist students in identifying similar universities to consider.
As data visualization receives more and more attention, a variety of methods for
visualization have been proposed including self-organizing maps (SOM), multi-
dimensional scaling (MDS), and Sammon mapping (SM). These methods have been
widely used in data visualization, as they are easy to implement and have modest
computational requirements. Our proposed work is closely related to the Sammon
mapping method, which we will describe in later chapters. SOM, MDS, and SM usually
deal with the problems that have symmetric structures, for example, symmetric distance
matrices. These methods take the table format of data sets as input, where rows are
observations and columns are attributes. MDS and SM also accept a distance matrix as
input but they require a symmetric distance matrix.
3
A proximity matrix or a similarity matrix contains the pairwise distances or
similarities between all pairs of data items in a data set. In a proximity matrix, the
distances are usually assumed to be symmetric. However, in practice, there are many
interesting problems in which asymmetric proximities arise, especially in marketing or
human behavior surveys. Asymmetric proximity is one type of proximity in which the
pairwise distances are not symmetric. For example, in terms of teaching quality, the
president of university A thinks that university B is the most competitive rival, while the
president of university B thinks that the closest competitor is university C and not
university A, and the president of university C thinks that university A is the closest
competitor. The corresponding proximity matrix is shown in Table 1.1. If school j is a
competitor of school i, then d
ij
= 1, otherwise, d
ij
= 0. Clearly, the matrix is asymmetric,
i.e. entry (i, j) = entry (j, i) for some i and j, i = j.
Since visualization methods normally work on symmetric cases, a question that
needs to be answered is how to visualize asymmetric cases such as the matrix in Table
1.1. A visualization method that can handle asymmetry may need to be developed, or
some modification may need to be made to an existing visualization method. A simple
modification could be made by averaging off-diagonal entries to create a symmetric
matrix. However, replacing asymmetric distances with an averaged distance alters the
structure of the data set in a way that may result in a less accurate representation.
In order to maintain the asymmetric information of the data set, Merino and
Munoz (2001) developed asymmetry coefficients. Mathematically, they defined an
asymmetry coefficient of a data observation as a summation of the standardized
similarities of the data observation to all of the other data observations (details such as
4
Competitor
School
A B C
A -- 1 0
B 0 -- 1
C 1 0 --
Table 1.1 Asymmetric proximity matrix of schools A, B, and C.
transforming distances to similarities are described in the next chapter). Merino and
Munoz incorporated asymmetry coefficients into the MDS and SM objective functions
and corresponding search procedures. Their approach dealt with the symmetrized
distance matrix of data observations. Based on their computational results, Merino and
Munoz observed that data observations with large asymmetry coefficients were more
influential in determining the structure of a map. However, data observations that are
similar to many other data observations are usually dominant in forming the structure of a
visual map, regardless of whether asymmetry coefficients are introduced into the MDS or
SM methods. When most of the data observations have similar asymmetry coefficients,
there is little or no impact the coefficients have on influencing the structure of a map.
In order to visualize asymmetric cases, we can examine the upper triangular part
and the lower triangular part of the matrix simultaneously rather than study the
symmetrized distance matrix derived by averaging corresponding upper and lower entries
in the asymmetric distance matrix, or arithmetically calculate asymmetric coefficients.
We expect that the visual maps generated by taking into account the upper triangular part
and the lower triangular part of the matrix simultaneously are a better representation of
the asymmetric data, and hopefully produce more insight into the data sets. For example,
5
we apply our approach to a small 3 × 3 asymmetric distance matrix given in Table 1.2.
We use the GRG Solver, an optimization software developed by Frontline Systems
(www.solver. com), to solve the small problem in our proposed approach and in one of
the traditional approaches (i.e., the standard Sammon mapping approach which takes the
average values of asymmetric parts as input) for comparison. Our approach seeks to
optimize a Sammon mapping objective function directly on the whole asymmetric
distance matrix. The common approach aims to solve the same optimization problem on
the symmetrized distance matrix of the asymmetric data. The results show that our
proposed approach is better than the one in the common approach (detailed discussions
are given in Chapter 3) even in this tiny data set with only three data points. In addition,
an asymmetric case with ranking information will be analyzed to extend our research on
visualization of asymmetric problems.
Once an algorithm is implemented, it is necessary to check that if the algorithm
works effectively as compared to other algorithms. Meanwhile, it is helpful to see if the
algorithm is reliable on different data sets. We will apply our algorithm to twodata sets:
American college selection data and Canadian college data. The proximity matrix of
each data set is asymmetric. The American college selection data are gathered from the
Fiske Guide (1999). Canadian college data are collected from a survey that we
conducted. Detailed information about each data set will be given in later chapters.
Another part of our study focuses on applications of data visualization.
Clustering is such an application of data visualization that is widely used to detect hidden
patterns
6
Asymm A B C
A 0 1 3
B 2 0 2
C 3 4 0
Table 1.2 Example of a 3 × 3 asymmetric distance matrix.
that cannot be observed directly from enormous amounts of data. Several SOM-based
software implementations for clustering are available either commercially or free of
charge, such as Som_Pak (1997) and Viscovery (2002). However, there is little
information indicating which implementation works better in practice. In our work, we
will evaluate the performance of four software implementations of SOM-based clustering
methods. Based on our evaluation, we will analyze several applications, such as
clustering a state murder data set (Harff, 2003) and finding out colleges with best values
(2003, 2004).
1.2 Summary of Objectives
A summary of the research work that has been done is as follows.
Firstly, we developed a new visualization method that uses data with asymmetric
proximities.
Second, we implemented our visualization method using a gradient descent
algorithm.
7
Third, we applied our method to twodata sets: American college selection and
Canadian college selection and compared our results with the results generated by the
standard method and the Merino’s method.
Fourthly, we assessed four SOM-based clustering methods and analyzed
clustering applications: state-sponsored data and college data with the best values.
1.3 Dissertation Organization
In Chapter 2, an overview of the literature on data visualization, including
multidimensional scaling, Sammon mapping, and self-organizing maps, is provided. We
define asymmetric distances and discuss several techniques that are used to handle an
asymmetric distance matrix for visualization.
In Chapter 3, our modified Sammon mapping algorithm and its implementation
are described and evaluated.
In Chapter 4, our modified Sammon mapping method is applied to American
college selection data. We give the construction procedures of the data set. We discuss
the results and compare our results to results generated by the commonly used standard
SM method and Merino’s method.
In Chapter 5, problems with ranking information are introduced. An example is
given to show the process of constructing a distance matrix that incorporates ranking
information. Canadian college selection data is analyzed by using our modified SM
method. We discuss our preliminary results and provide insights gained from our work.
8
In Chapter 6, the performances of four SOM-based clustering software
implementations are evaluated. We analyze an application (i.e., state-sponsored data)
using Viscovery.
In Chapter 7, we apply Viscovery to public and private college data to find out
colleges with best values. The visual maps of the public college data and the private
college data are given and the results are discussed.
In Chapter 8, we summarize our research and point out our future work.
9
Chapter 2
Literature Review
Data visualization is one technique that can help researchers and business
decision makers discover patterns in a data set. Data visualization has received lots of
attention in the literature. Many methods (e.g., multidimensional scaling and Sammon
mapping) and systems have been proposed and implemented in research and business
areas like biomedical science, marketing, and financial services.
In this chapter, we provide an overview of the relevant literature in data
visualization. First, we survey the papers that pertain to data visualization. Second, we
examine papers on multidimensional scaling, Sammon mapping, and self-organizing
maps. Third, we discuss several approaches dealing with asymmetric proximity data.
10
2.1 Data Visualization
There are many visualization methods that have been proposed for illustrating
structures and multivariate relationships among data items. These methods can be
classified into two categories: linear visualization methods and nonlinear visualization
methods. In this section, we will discuss some of these methods.
Principle component analysis (PCA) (Hotelling, 1933) is a standard method in
data analysis. Principle components are a set of variables that define a projection that
encapsulates the maximum amount of variation in a dataset and is orthogonal (and
therefore uncorrelated) to the previous principle component of the same (see Figure 2.1).
Projection pursuit (Friedman, 1987) tries to show the best visual representation that
reveals as much of the non-normally distributed structure of the data set as possible. A
neural network implementation of this method is given by Fyfe and Baddeley (1995).
PCA cannot take into account nonlinear structures and projection pursuit cannot
project the nonlinear structures onto a low-dimensional display if the data set has many
dimensions and is highly nonlinear. Several approaches have been proposed to project
nonlinear, high-dimensional structures onto a low-dimensional space. The most common
methods allocate each individual data point onto a lower dimensional display and then
optimize the display so that the distances between the points are as close as possible to
the original distances. These methods differ in the selection of the objective function and
the optimization approach.
Multidimensional scaling (MDS) refers to a group of methods that use proximities
among data points to produce a representation of the data set (Kruskal, 1964a; Shepard,
1962). The representation consists of a geometric configuration of the points on a map
11
Figure 2.1 Example using principle components analysis.
where each point corresponds to one of the data items. MDS is widely used in
behavioral, economic, and social sciences to analyze the pairwise proximities of the data
points (e.g., similarity of brands in a market survey). MDS is discussed in greater detail
in a later section.
Another nonlinear visualization method is Sammon mapping (Sammon, 1969).
Sammon mapping (SM) is closely related to MDS. It tries to optimize an objective
function in order to preserve the relative pairwise distance between data points. Details
on MDS and Sammon mapping are provided in Sections 2.2 and 2.3.
Principal curves are a nonlinear generalization of PCA that projects a data set
onto a nonlinear manifold after a linear manifold of the data set has been generated using
PCA. Here, a manifold (or surface) refers to a topological space on which every point
12
has a neighborhood sharing some essential features of the data set (i.e., a sphere is a
manifold). Principal curves were first defined as self-consistent smooth curves (Hastie
and Stuetzle, 1989) that pass through the middle of a d-dimensional probability
distribution or data cloud (see Figure 2.2). Extensions of principal curves use
multidimensional base functions to construct a nonlinear manifold, such as adaptive
principal surfaces (LeBlanc and Tibshirani, 1994). Another popular approach is to use
variants of the self-organizing map (SOM) algorithm (e.g., apply self-organizing maps to
extract principal curves and extended principal curves from data (Der et al., 1998)). The
self-organizing map is an efficient tool for the visualization of high-dimensional data sets
(Vesanto, 1999).
Nonlinear visualization methods are computationally very intensive for large data
sets. The triangulation method (Lee et al., 1977) can be used to reduce the computational
complexity. An important property of the triangulation method is that the distances to its
nearest two neighbors can be preserved exactly when inserting a new data item. Using
the triangulation method, data items will be projected onto a map one by one and the
nearest neighbor distances can always be preserved, that is, the generated map is based on
a subset of distances in the original space. The projection process is, thus, substantially
faster than nonlinear visualization methods. However, the generated map from the
triangulation method may not be as accurate as the map from nonlinear visualization
methods since the projection preserves only a part of distances.
13
Figure 2.2 Illustration of a principal curve.
2.2 Multidimensional Scaling
2.2.1 Overview
As we mentioned in Section 2.1, multidimensional scaling is a collection of
visualization methods that project proximity data onto lower dimensional space. In
general, the goal of multidimensional scaling is to provide a visual representation of
proximities in a set of investigated objects. Proximity data are always represented as
distances. For example, in Table 2.1, each entry represents the pairwise distance between
two buildings.
To better illustrate the idea of MDS, consider an example that visualizes the
locations of 10 buildings. Given the symmetric matrix of distances among 10 buildings
(see Table 2.1), MDS produces the map given in Figure 2.3.
14
A B C D E F G H I J
A 0 14.5 14 1 12.5 17.5 12 11.5 16.5 12
B 14.5 0 1 14.5 7 9.5 5.5 5 9.5 11.5
C 14 1 0 14 6 8.5 4.5 4 8.5 11.5
D 1 14.5 14 0 12.5 17.5 12 11.5 16.5 12
E 12.5 7 6 12.5 0 9.5 3 2 9.5 10.5
F 17.5 9.5 8.5 17.5 9.5 0 7.5 7.5 1.5 11
G 12 5.5 4.5 12 3 7.5 0 1 7.5 11
H 11.5 5 4 11.5 2 7.5 1 0 7.5 11
I 16.5 9.5 8.5 16.5 9.5 1.5 7.5 7.5 0 10.5
J 12 11.5 11.5 12 10.5 11 11 11 10.5 0
Table 2.1 Matrix of distances among 10 buildings.
A
B C
D
E
F
G
H
I
J
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
-8 -6 -4 -2 0 2 4 6 8 10
Figure 2.3 Example of multidimensional scaling.
15
In this example, the relationship between the original distances among data points
and resulting distances shown on the map is positive, that is, the smaller the original
distance, the closer the resulting distance between points, and vice versa. If the original
proximity data had been represented as similarities, the relationship would have been
negative which means the smaller the similarity between two data items, the farther apart
in the map they would be. In our study, proximity data are represented as distances
among data items.
2.2.2 Problem definition and stress function
From a slightly more technical point of view, for a set of observed distances
between every pair of N items, multidimensional scaling methods aim to find a visual
representation of the items in lower dimensional space such that the resulting distances
among items match the original distances as closely as possible. The metric version of
MDS aims to find configurations for data items where the resulting distances are as close
as possible to the original distances of data items. Nonmetric MDS tries to keep the rank
orders of the distances among data items as close as possible to the original rank orders.
We consider only nonmetric MDS in this dissertation.
The mathematical definition of MDS now follows. Given N items and a
corresponding distance matrix where entry d
ij
is the pairwise distance between data items
i and j, MDS seeks to find vector configurations ] ,..., 1 : [
*
p k x x
ik i
= =
-
and
] ,..., 1 : [
*
p k x x
jk j
= =
-
for data items i and j (i, j = 1,…,N, i j) in a p-dimensional space
(p N – 1), such that the Euclidian distance between
-
i
x and
-
j
x
16
¿
=
÷ = ÷ =
p
k
jk ik j i ij
x x x x d
1
2 * * * * *
) ( ,
ij ij j i
d d ~ ¬
<
*
.
approximates the corresponding distance
ij
d , for all pairs of data items i and j.
Since the proximity matrix is assumed to be symmetric, it is sufficient to take into
account each pair of data items i and j just once. However, it may not be possible to
perfectly represent the original distances (Johnson and Wichern, 1998) in a given lower
dimensional space. Therefore, a numerical measure is needed to indicate the closeness
and the measure is called a stress function.
Kruskal’s stress (Kruskal, 1964a), known as Stress formula 1 or Stress 1 for short,
measures the extent to which a representation deviates from a perfect match and is
defined as
¿
¿
<
<
÷
j i
ij
j i
ij ij
d
d d
2
2 *
) (
.
If the stress is zero, then the resulting pairwise distances are exactly the same as
the pairwise distances in the original distance matrix. However, in order to be useful, it
is not necessary to require a zero stress value as long as a certain amount of distortion is
tolerable. Kruskal (1964a) provides guidelines for the amount of stress to tolerate (see
Table 2.2).
Multidimensional scaling has been applied in many areas--the literature is vast
and dispersed over many periodicals and books. We will not attempt to give an overview
of the developments to date, and refer the reader elsewhere for details (Borg and
Groenen, 1997; Cox and Cox, 1994).
17
Stress Goodness of fit
0.2 Poor
0.1 Fair
0.05 Good
0.025 Excellent
0 Perfect
Table 2.2 Stress guidelines suggested by Kruskal (1964a).
2.3 Sammon Mapping
2.3.1 Overview
Frequently, a Sammon map is used for data exploration. A practical area of
Sammon mapping is in the visualization of protein structures based on measures of
similarity between molecules. For example, Sammon maps have been used to analyze
protein sequence relationships (Agrafiotis, 1997).
Like other MDS visualization methods, SM deals with proximity data. We are
given distances between every pair of data items (we possibly have no direct access to
any high-dimensional data but we do have access to a measure of distances between
every two data items). SM tries to reconstruct the original data based solely on the given
distance matrix. For example, given the distances between two buildings, SM can be
used to construct a map for the coordinates of the building themselves. A demonstration
of SM is presented in Figure 2.4 using the distance matrix shown in Table 2.1.
18
Figure 2.4 Sammon map of the 10 building.
2.3.2 SM algorithm
SM is an unsupervised nonlinear method that tries to preserve relative distances
(Lerner et al., 1998). Here “unsupervised” means no targeted information or outcome to
predict. To preserve the inherent structure, the algorithm that generates a Sammon map
employs a nonlinear transformation of the observed distances among data items when
mapping data items from a high-dimensional space onto a low-dimensional space.
If we denote the distance (usually Euclidean distance) between two data items i
and j, i j, in the original dimensional space by
ij
d and the distance in the required
projected space by
'
ij
d , the error function of SM is given by
19
¿ ¿
¿ ¿
= + =
= + =
÷
=
n
i
n
i j ij
ij ij
n
i
n
i j
ij
d
d d
d
E
1 1
2 '
1 1
) (
1
.
In the error function E, the smaller the error value, the better the map we get.
However, in practice, we are often unlikely to obtain perfect maps especially when the
data set is large and in high-dimensional space. Therefore, approximate preservation is
the likely result when we project high-dimensional data onto a low-dimensional plane.
The error function of SM is similar to Kruskal’s S stress function (see Section
2.2.2) except that each squared difference of distances is divided by the corresponding
input distance rather than the squared input distance. In other words, the only difference
between SM and MDS is that the errors in distance preservation are normalized with the
distance in the original space. Because of the normalization, SM places greater emphasis
on smaller distances rather than on larger distances; this is different from Kruskal’s MDS
which treats small and large distances roughly the same.
2.3.3 Implementation
SM can also be viewed as an optimization problem and its error function can be
minimized using several available techniques. Sammon solved the minimization problem
using steepest gradient descent that is also referred to as pseudo-Newton minimization
(Becker and Le Cun, 1989). This optimization procedure can be achieved by iteratively
using the following rule
2 '
2
'
' '
) (
) (
) (
) (
) ( ) 1 (
t x
t E
t x
t E
t x t x
ik
ik
ik ik
c
c
c
c
÷ = + o .
20
Note that
'
ik
x is the k
th
coordinate of the position of data item i in the required
projected space, and is called “magic factor” which controls the step size of updating
coordinates.
We point out that Sammon (1969) suggested a value of 0.3 to 0.4 for . However,
since is experimentally determined, the suggested value may not be optimal for all
problems. Therefore, multiple experiments are necessary in order to find an appropriate
value of to minimize E.
Because a second derivative is used in the denominator, the update rule can be
unstable at some points where the second derivative is very small. To avoid the
instability, an alternative minimization technique, called normal gradient descent, has
been used, where
) (
) (
) ( ) 1 (
'
' '
t x
t E
t x t x
ik
ik ik
c
c
÷ = + o .
When employing the gradient descent procedure to search for a minimum error
value, a local minimum in the error surface could be obtained. Therefore, several
experiments with different random initializations may be necessary. Another problem is
the computational requirement of SM is O(n
2
). The pairwise distances and the
derivatives have to be computed each iteration. Therefore, as the number of data items
increases, the computational time increases dramatically. To lower the computational
overhead, the Hamming metric has been used as a distance measure instead of the
Euclidean metric (White, 1972). This showed some improvement in the computational
efficiency, but the resulting maps could be distorted when the input space is the
Euclidean space (Chien, 1978) and the interpretation becomes more complex.
21
So far, all of the problems that we have discussed are assumed to have symmetric
proximity data, that is, the distance between data items i and j denoted by d
ij
is exactly
same as the distance between data items j and i denoted by d
ji
. However, in practice,
there are lots of interesting problems that have asymmetric proximity data, which means
that the distance d
ij
may be different from the distance d
ji
.
To our knowledge, there are few papers that discuss the visualization of
asymmetric proximity data sets. One of our major research goals is to develop a method
that visualizes asymmetric proximity data sets. Our objective function and search
procedure for implementing SM with asymmetric data are discussed in Chapter 3.
22
2.4 Self-organizing Maps
2.4.1 Overview
The self-organizing map (SOM) invented by Kohonen in the early 1980s is a type
of neural network based on the idea of self-organized or unsupervised learning (Kohonen,
1995). This means that the algorithm has no targeted information or outcome to predict.
Consequently, SOMs are ideal for clustering where no requirement of output fields is
defined. However, SOM can also be employed to visualize high-dimensional data items
(Flexer, 1999).
Being a stable and flexible method for clustering and visualization, SOM has been
used for a wide range of purposes, ranging from controlling industry processes to
analyzing gene data (Kaski et al., 2001). Many applications of SOM are given in the
survey by Oja et al. (2003).
An illustration of the mechanism of SOM is shown in Figure 2.5. The 3 3× map
consists of two layers: the input layer, and the output layer, which is often referred to as
the output map. All the input neurons are connected to all the output neurons, and
weights are assigned to each connection (not shown).
23
Figure 2.5 Illustration of a self-organizing map.
2.4.2 SOM algorithm
The self-organizing map trains by locating data items, one by one, to the output
map. When an input data item is presented to an output neuron, its characteristics are
compared with those of all output neurons, which are given initial weights. The neuron
with the characteristics that are most similar to that of the input data item is chosen to
represent the input data item; at the same time, the surrounding neurons of this chosen
neuron are adjusted to be more similar to the chosen neuron in order to attract input data
items similar to the mapped data item. This chosen neuron has a better chance, as
compared to other neurons, of representing input data items that have similar patterns,
and its neighbors are gradually able to represent similar input data items.
Each input data item is attracted to one and only one neuron, while each neuron
can attract one or more data items. Each neuron i has a reference vector m
i
= [m
i1
, …,
m
in
], which is used to represent an input data item, where n is the dimensionality of the
Input layer
Output layer
24
input space. When locating an input data item x on the output map, the neuron w is the
winner selected based on the minimum Euclidean distance, that is,
{ }
i
i
m x w ÷ = min , where .. is Euclidean distance.
During the training process, the winning neuron, also called the best matching
unit, and its neighbors are allowed to modify their reference vectors as close to the
current input data item as possible. The general form of the modification is given by
)] ( ) ( )[ ( ) ( ) ( ) 1 ( t m t x t h t t m t m
i wi i i
÷ + = + o ,
where ) (t o is the learning rate that controls the training speed and ) (t h
wi
is the
neighborhood function centered on the winning neuron w (this function indicates the
radius of neighborhood set) and ) (t x is the input at time t. Initially, the neighborhood
function is set to a large value; this value decreases monotonically with time, as does the
learning rate.
2.4.3 SOM software packages
In this section, we focus on two SOM software packages that are publicly
available: SOM_Pak and Viscovery SOMine.
SOM_Pak was developed at the Helsinki University of Technology. It is a
command line program and the interface is not user friendly; for example, there is no
simple option for executing repetitive commands and the user has to type in each
command. SOM_Pak can be downloaded for free; see the Web site at http://www.cis.
hut.fi/research/som_pak/. A screenshot of SOM_Pak is shown in Figure 2.6.
Viscovery SOMine was developed by Eudaptics (www.eudaptics.com). An
important advantage of SOMine is that it allows the user to visualize and analyze a data
25
Figure 2.6 Sammon map of a distance matrix of 20 data items produced by SOM_Pak.
set without any prior statistical knowledge of the data set. The software provides
suggestions as to which data items should be grouped together. The user can use or
modify several parameters to control data processing. A screenshot of a Viscovery
SOMine map is shown in Figure 2.7.
26
Figure 2.7 Screenshot of Viscovery. Three groups (As, Bs, and Cs) are shown in the
map; the dark lines separate groups.
27
2.5 Asymmetric Proximity
2.5.1 Overview
Proximity refers to the similarity (sometimes also refers to dissimilarity or
distance) between two data items. If the proximity between two data items is measured
in Euclidean space (i.e.,
2 2
) ( ) (
j i j i ji ij
y y x x d d ÷ + ÷ = = ), then the proximity is
symmetric. However, the proximity between data items or objects might not be
symmetric. When objects are compared from different perspectives, for example, object
a is said similar to object b in terms of color closeness, while object b is said dissimilar to
object a because of their different shapes, the proximity relationship between these two
objects is asymmetric, i.e.,
ba ab
d d = .
Asymmetric proximity data arises in a number of diverse research areas such as
marketing, psychology, sociology, and information retrieval. Asymmetric proximity data
can usually be found in frequency matrices. For example, brand switching data has been
utilized in marketing to examine the structure of competition within a particular product
class. An example of car-switching data is given in Table 2.3. The rows represent cars
last owned and the columns reveal cars currently owned. Out of all customers, 40 who
last owned a Ford switch to a Honda. Other examples of asymmetric proximity data are
migration rates between countries, frequencies of journal citations, word relationships in
text documents, etc.
In the past, especially from the late 1970s to the early 1990s, analysis of
asymmetry in proximity data had been one of the most provocative research topics in
psychological research areas, in contrast to traditional MDS approaches. Researchers
realized the psychological relevance of asymmetry in proximity data.
28
BMW Ford Honda Toyata GM
BMW 180 40 20 0 10
Ford 20 343 40 30 70
Honda 10 20 120 10 20
Toyata 30 20 30 70 10
GM 10 20 0 20 250
Table 2.3 An example of car switching data.
Not surprisingly, many approaches for asymmetric proximity data were proposed
in the psychological area. Tversky (1977) initially challenged the validity of the
traditional spatial model (i.e., MDS model) regarding the observed violation of
symmetry, minimality, and triangle inequality conditions of the metric axioms in actual
data. Krumhansl (1978) also discussed the problems of traditional spatial model and
proposed her distance-density model as an alternative to Tversky’s (1977) feature-
matching model. Other researchers proposed many different treatments of asymmetric
proximity data. However, there has not been found any model that is not only
mathematically sound but also practically applicable and easily interpretable.
2.5.2 Mathematical modeling for asymmetric proximity data
As many relationships are intrinsically asymmetric (Tversky, 1977) and
increasing attention has been paid to asymmetric proximity data, a number of approaches
have been developed to analyze asymmetry in proximity data. Most of these can be
classified into three categories.
29
Since traditional approaches represent relationships between data items
symmetrically, asymmetry is discarded as noise with respect to the symmetric part of the
proximity data. The symmetric part is extracted and the data are symmetrized. One
common approach to do this is to average corresponding off-diagonal entries (Kruskal,
1964b), i.e., substitute
ij
d by '
ij
d , where
2
'
ji ij
ij
d d
d
+
= , and then apply the symmetric
model, i.e., traditional MDS. For example, Tversky and Hutchinson (1986) analyzed 39
asymmetric proximity data by averaging. Another way of symmetrization is proposed by
Levin and Brown (1979) who derived row multiplicative constants from two least square
procedures to scale rows or columns of the asymmetric matrix to maximize symmetry.
However, symmetrization of asymmetric proximity data may ignore some important
information brought by asymmetry, and the symmetric solution found in the dimensional
spaces does not depict anything about the asymmetry.
Approaches in the second category aim to capture the information of asymmetry
in addition to the symmetric structure of the data. All major models in this category
involve a symmetric component and an asymmetric component. Krumhansl (1978)
specified a distance-density model in which object A and object B are represented in
projected low dimensional space, the similarity between A and B can be interpreted not
only by the interpoint distance but also the density of points in the surrounding
configuration. In other words, asymmetries are accounted for through points around A
and B. Saito (1986) developed an MDS approach in which estimated constants
considered as density constants are added to the symmetric configuration in relation to
the distance-density model. Different from the distance-density model, Constantine and
Gower proposed an approach partitioning an asymmetric matrix into two matrices: S
30
(symmetric) and N (skew-symmetric; i.e., n
ij
= ÷ n
ji
). A singular value decomposition of
N was performed to obtain a least-squares fit to be plotted in low dimensional space (i.e.,
in two dimensions) and an interpretation of asymmetry was provided. Weeks and Bentler
(1982) specified a model in which similarity is represented as distance, and traditional
additive constants are combined to reflect asymmetries. Description of other models for
asymmetric proximity data can be found in the paper of Zielman and Heiser (1996) in
which most of mentioned models for asymmetric proximity can be decomposed into a
symmetric part and a completely asymmetric part (i.e., skew-symmetry). These models
are mathematically elegant and the resulting dimensional configurations are interpretable.
However, these models assume an underlying symmetric component of the data but the
assumption might not fit every case. Besides, software for computing the model
parameters is not available (Zeilman and Heiser, 1996).
Another category of approaches for scaling asymmetric proximities is a graph-
theoretic representation of asymmetric proximity data (Cunningham, 1978; Hutchingson,
1989; Klauer, 1989). In these models, asymmetries are represented as directed distances.
These models do not require an underlying symmetric relationship. The differences
between these graphic models are the representation type, for example, Cunningham
(1978) employed directed trees as representations of proximities, and Hutchingson (1989)
used networks to represent proximities data. One disadvantage of a graph representation
is that the representation is limited to small data sets. Graph representation of large data
sets seems messy because of a large number of arcs.
31
2.5.3 Other approaches of analyzing asymmetric proximity data
In addition to the above three categories of approaches of modeling asymmetric
proximity data, there are other approaches sharing some features of the second and the
third categories (Rodgers and Thompson, 1992; Merino and Munoz, 2001). Rodgers and
Thompson proposed an approach in which asymmetric proximity data have been
preprocessed using the idea of seriation before applying MDS to the data. Seriation is a
procedure that orders data items on a continuum in order to maximize the sum of the
elements above (or below) the main diagonal (Baker and Hubert, 1977). Rodgers and
Thompson used the seriation algorithm and ordered the data items according to number
of dominances over other data items. They defined dominance as: if s
ij
> s
ji
, then i
dominates j. The data items that consistently dominate other data items are placed lower
in the ordering, i.e., the data item that dominates all other data items is placed on the
bottom row in the below diagonal triangular. MDS is fit to the ordered above diagonal or
below diagonal triangular that explicitly exhibits the dominance relationships of data
items and the resulting MDS configuration contains a directed distance.
Merino and Munoz (2001) introduced another approach to scaling asymmetry
proximity data in which asymmetry coefficients derived from the skew-symmetry matrix
are attached to MDS objective functions and MDS scaling was performed on the
symmetrized proximity data. An asymmetry coefficient is defined as a summation of the
standardized similarities of a data item to all other data items. Intuitively, data items that
are similar to more data items will have larger asymmetry coefficients. In some sense,
asymmetry coefficients convey the dominance information of data items, and data items
with large asymmetry coefficients determine the structure of the configuration.
32
Empirically, it shows that the most dominant data items have a tendency to be placed in
the center of a configuration, and this phenomenon may be interpreted as indirect
confirmation of the usefulness of the Euclidean space, which recognized the central role
of the dominant data items (Tversky and Hutchinson, 1986).
The approaches proposed by Rodgers & Thompson (1992), and Merino & Munoz
(2001) are more of an exploratory data analysis and less specific asymmetric models than
the methods described above. The approach by Rodgers & Thompson is flexible and
tractable, i.e., the ordered triangular of data items, and substantive information is more
accessible to data analysts. Merino and Munoz’s approach suggests that it might be
worth incorporating asymmetry coefficients to improve the visual configuration (Merino
and Munoz, 2001). However, both approaches have some disadvantages. Since Rodgers
and Thompson only considered the maximized triangular part (i.e., below the diagonal),
some important information contained in the other triangular part would be lost,
especially if the other triangular part contained many non-zero values. In Merino and
Munoz’s approach, the definition of asymmetry coefficient might not be suitable in every
case, for example, if most of the data items have approximately equal asymmetry
coefficients, i.e., defined as sum of row (or column) similarities, then the defined
coefficients may not reveal dominant information. In this case, it might be better to
define the coefficient as some function of dominance to reveal the centrality status.
Details on centrality are in Tversky and Hutchinson’s paper (1986).
From a visualization perspective, the visual configurations generated from the
upper triangular and lower triangular parts of an asymmetric proximity matrix are
implicitly different. For example, there are two SM maps generated by using the upper
33
triangular part (see Figure 2.8) and the lower triangular part (see Figure 2.9),
respectively, of the asymmetric distance matrix given in Table 2.4. It is not convincing
that maps generated from the upper triangular part are better than maps generated from
the lower part, or vice versa. In addition, if there are no substantive reasons to assume
that the underlying relationships between data items are symmetric, a natural way to
visualize asymmetric proximity data is to simultaneously take into account the
asymmetric parts, i.e., the upper triangular part and the lower triangular part of the
proximity matrix. To assess the quality of maps, we can quantify the rank preservation
by comparing the generated map results with the order relationships of the original
asymmetric dataset. Chapter 3 discusses our proposed modified SM method and
measures for quality assessment.
34
B1 A2 A3 B4 A5 A6 A7 A8 A9 C10
B1 0 9 8 1 5 15 4 3 13 4
A2 20 0 1 20 6 10 4 4 8 3
A3 20 1 0 20 5 9 3 3 7 3
B4 1 9 8 0 5 15 4 3 13 4
A5 20 8 7 20 0 14 3 2 12 1
A6 20 9 8 20 5 0 2 3 1 2
A7 20 7 6 20 3 13 0 1 11 2
A8 20 6 5 20 2 12 1 0 10 2
A9 20 11 10 20 7 2 4 5 0 1
C10 20 20 20 20 20 20 20 20 20 0
Table 2.4 Asymmetric distance matrix taken from the American college selection data
set.
35
Figure 2.8 Sammon map of the upper triangular matrix.
Figure 2.9 Sammon map of the lower triangular matrix.
36
Chapter 3
Constructing Sammon Maps from Asymmetric Data
Sammon maps are one of the most widely used tools in visualization and
clustering. Sammon mapping (SM) projects high-dimensional data onto a 2-dimensional
output map. A Sammon map is usually created for proximity data with a symmetric
distance matrix. However, there are many applications (e.g., American college selection
data) that have asymmetric distance matrices. In this chapter, we discuss the use of SM
to visualize asymmetric proximity data sets. We describe the objective function, the
associated update rule, and our implementation of SM for proximity data with an
asymmetric distance matrix.
37
3.1 Modification of Sammon Mapping Method
SM tries to minimize the following objective function
¿ ¿
¿ ¿
÷
= + =
÷
= + =
÷
=
1
1 1
2 '
1
1 1
) (
1
n
i
n
i j ij
ij ij
n
i
n
i j
ij
d
d d
d
E , (1)
where
ij
d denotes the input distance (usually Euclidean distance) between data items i
and j, i j, i, j = 1, ..., n, in the original space.
'
ij
d denotes the output distance between i
and j in the mapped D-dimensional space and
¿
=
÷ =
D
k
jk ik ij
x x d
1
2 '
] [ , where x
ik
and x
jk
are
decision variables, i = 1,…, n?1, j = i+1,…, n, k = 1,…, D. Note that only half of all
entries in the distance matrix are taken into account because the distance matrix is
assumed to be symmetric.
To minimize the objective function E, Sammon used the steepest gradient descent
procedure to search for a minimum value of E. For convenience, the updating rule for his
procedure is shown again as
2
2
) (
) (
) (
) (
) ( ) 1 (
t x
t E
t x
t E
t x t x
ik
ik
ik ik
c
c
c
c
÷ = + o , (2)
where
ik
x is the k
th
coordinate of the position of i in the mapped space and is the “magic
factor” (Sammon, 1969). The magic factor is a parameter (Apostol and Szpankowski,
1999) that controls the step size for configuration update. Its value is determined
experimentally. It is treated as a constant over all iterations.
The objective function (1) and the associated search procedure work well when
the distance matrix is symmetric. However, problems arise when the distance matrix is
38
asymmetric. Since the objective function E assumes that the input distance matrix is
symmetric, it is inappropriate to use an asymmetric distance matrix as the input. To
overcome this problem, one simple technique is to symmetrize asymmetric distances by
simply averaging, that is, replacing the entry d
ij
with (d
ij
+ d
ji
)/2. Suppose that there are
three data points a, b, c in an asymmetric distance matrix, and their pairwise distances
are: in the upper triangular part, d
a,b
> d
a,c
, and in the lower triangular part, d
b,a
< d
c,a
.
Therefore, it is uncertain that the distance between a and b is greater or less than the
distance between a and c. Using symmetrized distances, the uncertainty of the order
relationships can be resolved. However, Sammon maps generated from symmetrized
distances will lose the asymmetry information (Merino and Munoz, 2001).
To generate maps that better represent and help visualize asymmetric proximity
data sets, it is natural to consider the original asymmetric distance matrix instead of the
symmetrized distance matrix in the objective function. We would like to account for the
upper triangular portion and the lower triangular portion of the asymmetric distance
matrix simultaneously in the optimization process.
The objective function that we propose has two parts denoted by U and L. The
first part (U) takes into account the upper triangular part, while the second part (L) deals
with the lower triangular part of the original asymmetric distance matrix. Our proposed
objective function, denoted by
1
E , is defined by
) (
1
1
L U
C
E + = , (3)
where
¿ ¿
÷
= + =
÷
=
1
1 1
2
) (
n
i
n
i j ij
ij ij
d
d d
U
'
,
39
¿¿
=
÷
=
÷
=
n
i
i
j ij
ij ij
d
d d
L
2
1
1
2 '
) (
,
¿
=
÷ =
D
k
jk ik ij
x x d
1
2 '
] [
and
¿ ¿
= = =
=
n
i
n
i j j
ij
d C
1 1,
,
constrainted by x
ik
, x
jk
= 0, i, j = 1,…,n, k = 1,…,D.
x
ik
, x
jk
are decision variables and represent k
th
coordinates of data items i and j in the
mapped D-dimensional space.
By using U and L, we seek to obtain a configuration of data items such that the
structures in the upper triangular part and the lower triangular part can be considered
separately. We will use the steepest gradient method as the search procedure.
Let ) (
1
t E denote the error value at t
th
iteration and ) (t U and ) (t L denote the error
values of the upper triangular part and the lower triangular part, respectively. Let
'
ij
d (t)
denote the distance between i and j at the t
th
iteration, that is,
¿
=
÷ =
D
k
jk ik ij
t x t x t d
1
2
)] ( ) ( [ ) (
'
and D denotes the dimensionality of the mapped space (usually D is 2). The new q
th
coordinate of data item p at iteration t+1is given by
2
1
2
1
) ( / ) (
) ( / ) (
) ( ) ( ) 1 (
t x t E
t x t E
MF t x t x
pq
pq
pq pq
c c
c c
÷ = + ,
where MF stands for “magic factor”, and the first derivative is
) (
) ( 1
) (
) ( 1
) (
) (
1
t x
t L
C t x
t U
C t x
t E
pq pq pq
c
c
+
c
c
=
c
c
(4)
40
=
'
, 1
'
) ( ) (
2
pj
jq pq
n
p j j pj
pj pj
d
x x
d
d d
C
÷ ÷
÷
¿
= =
+
'
, 1
'
) ( ) (
2
pj
jq pq
n
p j j jp
pj jp
d
x x
d
d d
C
÷ ÷
÷
¿
= =
.
The second derivative is
2
2
2
2
2
1
2
) (
) ( 1
) (
) ( 1
) (
) (
t x
t L
C t x
t U
C t x
t E
pq pq pq
c
c
+
c
c
=
c
c
(5)
= ( )
( )
(
(
¸
(
¸
|
|
.
|
\
| ÷
+
÷
÷ ÷
÷
¿
= =
'
'
'
2
'
, 1
'
1
1 2
pj
pj pj
pj
jq pq
pj pj
n
p j j pj pj
d
d d
d
x x
d d
d d C
+
( )
( )
(
(
¸
(
¸
|
|
.
|
\
| ÷
+
÷
÷ ÷
÷
¿
= =
'
'
'
2
'
, 1
'
1
1 2
pj
pj jp
pj
jq pq
pj jp
n
p j j pj jp
d
d d
d
x x
d d
d d C
.
Note that in the update rule, no two points are allowed to be identical. This prevents the
partial derivatives from “blowing up.”
The minimization problem in (3) is a nonlinear optimization problem and is non-
convex (Klock and Buhmann, 1999). Therefore, we cannot guarantee finding the global
minimum. The best we can do is to obtain a local minimum from each starting solution.
We use the GRG software (2004) to solve the small asymmetric distance matrix given in
Table 3.1 with our proposed objective function (3). Two sets of random starting
configurations are used as initial coordinates for these three data points. Listed in Tables
3.2 and 3.3 are resulting configurations of three data points corresponding to two
different sets of random starting configurations. The resulting objective function values
are given in Tables 3.2 and 3.3. Meanwhile, we use GRG to solve the same asymmetric
problem in the common approach that takes symmetrized distances as inputs. In Tables
3.4 and 3.5, we show the results of configurations and objective function values obtained
41
in the common approach. Since it is an optimization problem seeking a minimum value
for the SM function, it is better to get smaller resulting objective function values. The
shown
Asymmetric A B C
A 0 1 3
B 2 0 2
C 3 4 0
Table 3.1 Asymmetric distance matrix of three data points.
Point x
1
-coordinate x
2
-coordinate
A 0.270706143 0.159258481
B 0.812893328 1.377376338
C 3.26577432 0.331200098
Objective Function Value 0.066666667
Table 3.2 Results in our proposed approach with random starting point 1.
Point x
1
-coordinate x
2
-coordinate
A 3.481685023 4.95005622
B 2.987622292 6.188468288
C 5.448716802 7.215177319
Objective Function Value 0.066666667
Table 3.3 Results in our proposed approach with random starting point 2.
Point x
1
-coordinate x
2
-coordinate
A 0.331987222 0
B 0.289836968 1.499407704
C 3.214502644 0.831327182
Objective Function Value 0.075000006
Table 3.4 Results in the common approach with random starting point 1.
42
Point x
1
-coordinate x
2
-coordinate
A 3.505923937 4.963893963
B 2.008518266 5.052079549
C 2.927993608 7.907699703
Objective Function Value 0.075000012
Table 3.5 Results in the common approach with random starting point 2.
objective function values generated by our approach are smaller than those generated by
the common approach. This confirms our conjecture that our proposed approach
performs better than the common approach in visualizing asymmetric problems, at least
from the perspective of optimization.
3.2 Implementation of the Modified SM Method
In this section, we discuss the implementation procedures and provide small
examples to illustrate them. We discuss problems that we encountered when
implementing the modified Sammon mapping method. As illustrated in the previous
section, GRG can be used to solve asymmetric problems. However, as the size of the
asymmetric problem increases, it becomes burdensome to formulate the proposed
objective function and the computational time increases significantly. A good alternative
to GRG for solving asymmetric problems is C/C++. C/C++ is a commonly used software
for coding. It is machine portable and requires only a small amount of changes to run on
other computers. It is very fast, almost as fast as assembler. It allows structured
43
programming and is very flexible. It is suited to large and complex problems. Therefore,
we implement the proposed modified SM method in C/C++.
We use random values as a starting configuration and employ the steepest
gradient method as an optimization procedure to generate maps for an asymmetric
distance matrix. If there is no improvement in the value of the objective function after a
certain number of iterations, then the algorithm is considered converged and the resulting
configuration is obtained to visualize the asymmetric distance matrix. Since the magic
factor is experimentally determined, multiple experiments are necessary to find an
appropriate value. We determined from the sensitivity analysis that the recommended
magic factor of 0.4 is a good choice in our study for updating the configuration. The
corresponding sensitivity analysis is discussed in the next chapter.
We illustrate our procedure with a small example. The data set denoted by 30A is
an asymmetric distance matrix for 30 schools taken from the American college selection
data set of 100 schools that was constructed using information provided in The Fiske
Guide (Condon et al., 2002). This data set contains pairwise distances between each pair
of 30 American colleges (see Table 3.6). For example, the distance between A5 and A19
is not symmetric, that is, d
A5, A19
= 11 and d
A19, A5
= 6. In this data set, our resulting
Sammon map provides us with a visualization of the asymmetric data set (see Figure 3.1).
In Table 3.7, we give the symmetrized distances of this data set, for example, the entry of
d
A5, A19
equals the entry of d
A19, A5
, which is the average value of these two entries in the
asymmetric distance matrix. Figure 3.2 is a Sammon map generated by the standard
Sammon map with the symmetrized distances. It is obvious that there are some
differences between these two Sammon maps. For example, the distances between pairs
44
Table 3.6 Asymmetric distance matrix of 30 American colleges.
45
Table 3.7 Symmetrized distance matrix of 30 American colleges.
46
Figure 3.1 Sammon map of the asymmetric distance matrix for data set 30A generated
by the modified SM method.
Figure 3.2 Sammon map of the symmetrized distance matrix for data set 30A generated
by the standard SM method.
47
of the triplet A24, A45, and A89 are close to each other in the original asymmetric
distance matrix, where this relationship ideally need be visualized as an equilateral
triangular in the map. This relationship is better represented in the map given by the
modified SM method than in the map given by the standard SM. In Chapter 4, we will
compare Sammon maps generated by different SM methods and discuss the differences
between them and give our tentative recommendation of choosing a better visualization
map.
48
3.3 Performance Measurements of Sammon Maps
In order to assess the quality of maps produced from asymmetric distance data,
we introduce two performance measures. The first is the distance error that measures the
extent to which the pairwise distances projected on the map deviate from the original
pairwise distances. The distance error function, denoted by DE, is defined by
¿
=
÷
=
j i ij
ij ij
d
d d
DE
2 '
) (
,
where
ij
d is the original distance and
'
ij
d is the projected distance. The smaller the value
of DE, the better the map.
The second measure is an order preservation coefficient that indicates how well a
map preserves the distance order relationships of the original asymmetric data. In Table
3.8, we give a small asymmetric distance matrix. The pairwise distance between two
data items may be very different; for example, we see d
ab
= 1 while d
ba
= 2. Since that
d
ab
< d
ac
and d
ba
< d
ca
, the distance between a and b is less than the distance between a
and c. If a Sammon map A preserves more order relationships for an asymmetric distance
matrix than Sammon map B, then A is said to be more accurate than B. We are concerned
about a map’s ability to preserve order relationships. Our proposed order preservation
measure, denoted by OP, is defined by,
data asymmetric the of ips relationsh of number
preserved ips relationsh of number
= OP .
Note that the order relationship can be uncertain. In Table 3.8, we see that d
ab
<
d
cd
and d
ba
> d
dc
. Therefore, it is incomparable if the distance between a and b is greater
than the distance between c and d on the projected map.
49
a b c d
a 0 1 2 3
b 2 0 1 9
c 5 7 0 3
d 6 8 1 0
Table 3.8 Asymmetric distance matrix of four data points.
In Table 3.8, consider the four data items (a, b, c, d) and the twelve non-zero
distance entries (six entries are in the upper triangular part of the matrix and six entries
are in the lower triangular part of the matrix). The total number of relationships is 15,
which is given by 6(6-1)/2. In Table 3.9, we provide all of the order relationships of the
asymmetric distance matrix given in Table 3.8. If six order relationships are preserved in
a Sammon map, the order preservation coefficient equals 40% (that is, 6/15).
If a Sammon map exhibits greater accuracy than another, then it should have a
smaller distance error and a larger order preservation value. We use these two measures
to assess the quality of Sammon maps in our applications.
50
d
ab
< d
ac and d
ba
< d
ca
Distance(a, b) < Distance(a, c)
d
ab
< d
ad and d
ba
< d
da
Distance(a, b) < Distance(a, d)
d
ab
= d
bc but d
ba
< d
cb
Distance(a, b) Distance(b, c)
d
ab
< d
bd and d
ba
< d
db
Distance(a, b) < Distance(b, d)
d
ab
< d
cd but d
ba
> d
dc
Distance(a, b) ? Distance(c, d)
d
ac
< d
ad and d
ca
< d
da
Distance(a, c) < Distance(a, d)
d
ac
> d
bc but d
ca
< d
cb
Distance(a, c) ? Distance(b, c)
d
ac
< d
bd and d
ca
< d
db
Distance(a, c) < Distance(b, d)
d
ac
< d
cd but d
ca
> d
dc
Distance(a, c) ? Distance(c, d)
d
ad
> d
bc but d
da
< d
cb
Distance(a, d) ? Distance(b, c)
d
ad
< d
bd and d
da
< d
db
Distance(a, d) < Distance(b, d)
d
ad
= d
cd but d
da
> d
dc
Distance(a, d) Distance(c, d)
d
bc
< d
bd and d
cb
< d
db
Distance(b, c) < Distance(b, d)
d
bc
< d
cd but d
cb
> d
dc
Distance(b, c) ? Distance(c, d)
d
bd
> d
cd and d
db
> d
dc
Distance(b, d) > Distance(c, d)
Question mark (?) denotes that the relationship between the distances is incomparable.
Table 3.9 Order relationships.
51
Chapter 4
Visualizing American College Selection Data
4.1 Description of the Data Set
The Fiske Guide (Fiske, 1999) is a well-known publication that has been used for
nearly 20 years to help students and parents select the right college. In the 2000 edition
of The Fiske Guide, information on tuition cost, SAT scores, social life, and quality of
life has been provided for over 300 colleges and universities in the United States. The
Fiske Guide also lists a school’s overlaps, that is, the major competitors to which
applicants are also applying in greatest numbers. The overlaps provide students and
parents with possible alternatives when selecting a school. For example, the overlaps of
the University of Pennsylvania are Harvard, Princeton, Yale, Cornell, and Brown.
Students who applied to the University of Pennsylvania also applied to those five schools.
However, the overlaps of Harvard University -- Princeton, Yale, Stanford, MIT, and
Brown -- do not include the University of Pennsylvania. The overlaps of two schools are
not necessarily symmetric.
The American college selection data set is derived from the overlap data of 100
schools in The Fiske Guide. This data set was constructed by Condon et al. (2002).
There were four steps involved in the construction process: building an adjacency matrix,
52
constructing a directed graph, computing distance measures, and modifying the distance
matrix.
In the first step, Condon et al. created a 100 100× 0-1 asymmetric adjacency
matrix S for 100 schools, where entry s
ij
= 1 (row i and column j) indicates that school j is
an overlap of school i. For example, in Table 4.1, we show an 6 6× adjacency matrix for
six universities (Brown, Cornell, Harvard, MIT, Penn, and Stanford). The entry in the
Penn row and the Harvard column is 1, that is, Harvard is an overlap of Penn. In the
second step, Condon et al. converted the adjacency matrix S to a directed graph with 100
nodes (one node for each school) and a directed arc for each non-zero s
ij
entry
,
where i is
the start node, j is the end node, and the directed arc connects i and j. In Figure 4.1, we
converted the 6 6× adjacency matrix, which is shown in Table 4.1, into a directed graph
with six nodes. We see that there is a directed arc starting at the Penn node and ending at
the Harvard node.
In the third step, Condon et al. set the distance of an arc in the directed graph to
one and computed the all-pairs shortest path distance matrix T, where each entry t
ij
is
calculated as the shortest distance from node i to node j. In the final step, the authors
modified the distance matrix for disconnected nodes, that is, they set each entry t
ij
for a
disconnected pair to a value V that is greater than the longest distance in the matrix. It is
necessary to carefully choose the value V -- a large value of V will push the connected
points closer together so that it will be difficult to observe the inner relationships. A
small value of V will result in merging disconnected points.
53
School Brown Cornell Harvard MIT Penn Stanford
Brown 0 1 1 0 0 1
Cornell 1 0 1 0 1 0
Harvard 1 0 0 1 0 1
MIT 0 1 1 0 0 1
Penn 1 1 1 0 0 0
Stanford 1 0 1 1 0 0
Table 4.1 Adjacency matrix for six schools.
Figure 4.1 Directed graph generated from the adjacency matrix.
The American college selection data set generated by Condon et al. contains four
groups of schools denoted by A, B, C, and D (see Table 4.2). There are 74 schools in A,
11 schools in B (these are schools from the southern United States), 8 schools in C (six
schools are from the Ivy League), and 7 schools in D (all from California).
Brown Cornell
MIT
Penn Stanford
Harvard
54
Key School State
A2 Arizona State University AZ
A3 Arizona, University of AZ
A5 Barnard College (Columbia University) NY
A6 Bates College ME
A7 Boston College MA
A8 Boston University MA
A9 Bowdoin College ME
A11 Bryn Mawr College PA
A12 Bucknell University PA
A19 Carleton College MN
A20 Carnegie Mellon University PA
A23 Colby College ME
A24 Colgate University NY
A25 Colorado College CO
A26 Colorado, University of—Boulder CO
A27 Connecticut, University of CT
A29 Delaware, University of DE
A30 Denver, University of CO
A31 Emory University GA
A34 George Mason University VA
A35 Georgetown University DC
A38 Grinnell College IA
A40 Illinois, University of—Urbana-Champaign IL
A41 Indiana University IN
A42 Iowa State University IA
A43 Iowa, University of IA
A44 James Madison University VA
A45 Lafayette College PA
A46 Lehigh University PA
A47 Lewis and Clark College OR
A48 Macalester College MN
A49 Marquette University WI
A50 Mary Washington College VA
A51 Maryland, University of—College Park MD
A53 Massachusetts, University of—Amherst MA
A55 Michigan State University MI
A56 Michigan, University of MI
A57 Middlebury College VT
A58 Minnesota, University of—Twin Cities MN
Table 4.2 One hundred schools selected from The Fiske Guide for analysis.
55
A59 Mount Holyoke College MA
A60 New Hampshire, University of NH
A61 New Jersey, The College of NJ
A62 New York University NY
A63 North Carolina State University NC
A64 North Carolina, University of—Chapel Hill NC
A65 Northeastern University MA
A66 Northwestern University IL
A67 Notre Dame, University of IN
A68 Oberlin College OH
A69 Oregon State University OR
A70 Oregon, University of OR
A71 Pennsylvania State University PA
A73 Pittsburgh, University of PA
A75 Puget Sound, University of WA
A76 Purdue University IN
A77 Reed College OR
A78 Richmond, University of VA
A79 Rutgers University NJ
A80 Smith College MA
A85 Tufts University MA
A86 Vanderbilt University TN
A87 Vassar College NY
A88 Vermont, University of VT
A89 Villanova University PA
A90 Virginia Polytechnic Institute and State University VA
A91 Virginia, University of VA
A92 Wake Forest University NC
A93 Washington University in St. Louis MO
A94 Washington, University of WA
A95 Wellsley College MA
A96 Whitman College WA
A97 Willamette University OR
A98 William and Mary, College of VA
A99 Wisconsin, University of--Madison WI
B1 Alabama, University of --Tuscaloosa AL
B4 Auburn University AL
B21 Charleston, College of SC
B22 Clemson University SC
B32 Florida State University FL
Table 4.2 (Continued).
56
B33 Florida, University of FL
B36 Georgia Institute of Technology GA
B37 Georgia, University of GA
B54 Miami, University of FL
B81 South Carolina, University of SC
B84 Tennessee, University of--Knoxville TN
C10 Brown University RI
C28 Cornell University NY
C39 Harvard University MA
C52 Massachusetts Institute of Technology MA
C72 Pennsylvania, University of PA
C74 Princeton University NJ
C83 Stanford University CA
C100 Yale University CT
D13 California, University of--Berkeley CA
D14 California, University of--Davis CA
D15 California, University of--Irvine CA
D16 California, University of--Los Angeles CA
D17 California, University of--San Diego CA
D18 California, University of--Santa Barbara CA
D82 Southern California, University of CA
Table 4.2 (Continued).
Each of the four groups is a strongly connected component in the directed graph
of 300 schools given in The Fiske Guide. If a group is strongly connected, then there
exists at least one directed path from any school in the group to any of the other schools
in the same group. In other words, any one school is considered to be a competitor to all
of the other schools in the group. The distance between each pair of schools measures
the magnitude of the competitiveness. The shorter the distance, the more competitive the
schools are. For example, if the distance between schools S and T is shorter than the
distance between schools S and R, then T is more likely a competitor of S than R.
57
4.2 Experimental Design
We implement our modified Sammon mapping method in C/C++. Our code reads
in an n × n asymmetric distance matrix and generates random starting coordinates for
each data item i (i = 1,…,n) to be plotted in the mapped D-dimensional (usually, two
dimensional) space. We use the error function E
1
and the associated updating rule given
in Section 3.1 to adjust the coordinates iteratively to minimize the value of the error
function. If no improvement is found after a certain number of iterations, the modified
SM method is considered converged and the obtained configuration is the resulting
Sammon map. Currently, we employ the steepest gradient method as the optimization
procedure when applying the modified SM method to the asymmetric distance matrix.
We realize that the resulting configuration is a local minimum, multiple
experiments with different random starts are necessary to look for an approximate
solution. Meanwhile, we tested several different magic factors ranging from 0.1 to 0.8 in
our experiments. Tables 4.3 and 4.4 provide the sensitivity analysis of the magic factor.
The values listed in the two tables are average values of five experiments with different
random starts on each data set. The minimum average error measure(s) and the
maximum average order preservation coefficient(s) of each data set can be found in
boldface in Tables 4.3 and 4.4 respectively. For example, on the data set of 100 schools,
the minimum error measure is 28829.5600 (see Table 4.3), which is associated with a
magic factor of 0.6. The error measure associated with a magic factor of 0.6 is the
second minimum (28829.5600). Their corresponding average order preservation
coefficients are tied at the value of 0.4720, which is the maximum coefficient obtained
58
Magic
Factor 100Schools 30Schools ASchools BSchools CSchools DSchools
0.1 28830.5200 773.6160 5688.3040 18.3131 6.1831 5.5503
0.2 28835.6200 773.3146 5688.1640 18.3131 6.0645 5.5502
0.3 28846.9600 773.9876 5688.6180 18.3131 6.0677 5.5490
0.4 28830.3200 773.0448 5688.3020 18.3131 6.0645 5.5490
0.5 28831.8400 773.0162 5706.6340 18.3131 6.0677 5.5490
0.6 28829.5600 773.9876 5709.3380 18.4980 6.1799 5.5369
0.7 28831.6200 773.9776 5815.3500 18.4980 6.0677 5.5490
0.8 28835.4400 773.0448 5760.8320 18.3131 6.0645 5.5490
Table 4.3 Average error measures obtained from the modified SM method.
Magic
Factor 100Schools 30Schools Aschools Bschools Cschools Dschools
0.1 0.4720 0.6065 0.5943 0.7562 0.5683 0.5552
0.2 0.4718 0.6064 0.5944 0.7562 0.5677 0.5533
0.3 0.4714 0.6065 0.5943 0.7562 0.5661 0.5524
0.4 0.4720 0.6065 0.5944 0.7562 0.5677 0.5524
0.5 0.4720 0.6065 0.5939 0.7562 0.5661 0.5524
0.6 0.4720 0.6065 0.5940 0.7539 0.5661 0.5524
0.7 0.4719 0.6065 0.5916 0.7539 0.5661 0.5524
0.8 0.4718 0.6065 0.5937 0.7565 0.5677 0.5524
Table 4.4 Average order preservation coefficients obtained from the modified SM
method.
59
for the 100 school data. When considering the overall performance on all six data sets in
terms of both error measures and order preservation coefficients, each tested magic factor
turns to give either the best error measure or the best order preservation coefficient. In
other words, the experiments are not sensitive to the magic factors. Therefore, as
suggested by Sammon (1969), we use 0.4 as the step size to adjust the locations of data
items iteratively in the mapped space.
For each data set examined in the following sections, five experiments with
different random starts are performed. We choose the best maps in terms of error
measures and order preservation measures for comparison. Listed in the tables are
average values of five experiments on each data set.
We point out that the Sammon maps generated with different starting
configurations are usually similar, that is, the relative relationships among schools are
roughly the same in each map. In our experiments, we keep the substitute value of
infinity distance as what is used in Condon’s work, which is slightly larger than the
longest distance in the distance matrix.
We apply our modified Sammon mapping method to six data sets which are:
American college selection data set with 100 schools, each of the four strongly connected
groups (A, B, C, D), and 30 schools selected from A. We experiment with five different
starting configurations for each data set and then compare the average performance of our
modified SM method to that of the standard SM method and that of Merino’s method.
We use the typical resulting Sammon map of each data set to illustrate the similarities
and differences of the resulting maps generated by these three methods.
60
In Figure 4.2, we show the map of all 100 colleges and universities that was
generated by the modified SM method. In Figures 4.3 and 4.4, we show the Sammon
maps of all 100 schools that were generated by the standard SM method and by Merino’s
method respectively with the standard error function (1) given in Section 3.1. In Figures
4.5 to 4.19, we show Sammon maps generated by our modified procedure, the standard
procedure and Merino’s procedure for five data sets (A, 30 schools from A, B, C, and D).
We discuss the maps and results in the next section.
4.3 Discussion of the Results
Shown in Figures 4.2, 4.3, and 4.4 are Sammon maps of all 100 schools generated
by the modified, the standard, and Merino’s methods respectively. In Figure 4.2, the
group of B schools and group of D schools are separated from the group of A schools and
the group of C schools. This is consistent with the existing structure of the data set that
consists of four strongly connected groups of schools. The most interesting phenomenon
is that the group of C schools is located in the center of the map and is surrounded by
some schools of group A, e.g., Tufts University (A85), New York University (A62),
Boston College (A7), Columbia University (A5) and Georgetown University (A35).
From our experimental results, it shows that the more chances a university is considered
as a competitor by other universities, the more likely this university will be placed in the
center or near the center of the map. Boston College (A7) and New York University
(A62) have more chances to be considered competitors by many other universities so that
they, we think they are popular, are placed in the center of the maps (see Figures 4.2, 4.3,
61
Figure 4.2 Map of 100 schools generated by the modified SM method.
Figure 4.3 Map of 100 schools generated by the standard SM method.
62
Figure 4.4 Map of 100 schools generated by Merino’s method.
and 4.4). In Figure 4.2, the location of group C tells us that group C and some popular A
schools have some in common to some extents. For example, group C consists of Ivy
League schools which have high education quality and expensive tuition costs, etc., and
some A schools that surround group C (i.e., New York University) provide qualified
education the same as or no worse than group C schools provide. This relationship
between group C and these schools of group A is reasonable in practice. However, this
relationship cannot be detected from the maps generated by the standard and Merino’s
methods, where there is no clue that schools in group C are competitors of schools in
group A.
63
In addition, in Figure 4.2, Group B is closer to group A than to other groups.
Group D is also closer to group A than to other groups. In the maps generated by the
standard and Merino’s methods (Figures 4.3 and 4.4), we see that the four strongly
connected groups (As, Bs, Cs, Ds) of schools are separated from each other. It is hard to
determine which group(s) is (are) closer to another group; in other words, the
relationships between groups are not as clear as they are in the map generated by the
modified SM method.
Besides, in terms of inner group relationship, schools in each group are pushed
close together in Figures 4.3 and 4.4 so that it becomes difficult to ascertain within-group
relationships from the maps. Most of relationships that can be seen in the maps generated
by the standard and Merino’s methods can also be detected in the map generated by our
modified SM method. For example, in Figures 4.3 and 4.4, the University of Maryland
(A51) is close to schools A73 (University of Pittsburgh), A29 (University of Delaware),
and A34 (George Mason University). These relationships are still preserved in Figure
4.2.
However, the map generated by the modified SM method has its limitations. It
seems that the modified SM method might not represent some local structures as
precisely as other two SM methods. For example, although these four schools that are
considered close competitors by each other can still be thought of forming a cluster (see
Figure 4.2), A6, A23, A9 and A57 are not placed together in Figure 4.2 as closely as they
are, while in maps generated by the other two methods the relationship between these
four schools is represented more clearly.
64
Regarding the differences between the maps generated by the standard and
Merino’s methods, there is no significant evidence that one outperforms the other. This
observation is also confirmed in our performance measurements that are discussed later
in this section. The general structures of the maps generated by these two methods are
similar and it is reasonable to see local differences due to different random starting
configurations and other factors such as stopping criterion etc.
Figures 4.5, 4.6 and 4.7 show the Sammon maps of group A generated by the
modified SM, the standard SM method, and Merino’s method, respectively. In Figures
4.8, 4.9 and 4.10, we show the Sammon maps of the subset of 30 schools from group A
generated by these three methods. As compared to groups B, C, and D, group A and the
30 schools from group A have a relatively large number of schools.
65
Figure 4.5 Map of group A generated by the modified SM method.
Figure 4.6 Map of group A generated by the standard SM method.
66
Figure 4.7 Map of group A generated by the Merino’s method.
The Sammon map of group A generated by the modified SM method has roughly
similar structure as has the Sammon maps generated by the standard SM and Merino’s
methods. For example, the locations of most A schools are roughly same in these three
maps (see Figures 4.5, 4.6 and 4.7). Grinnell College (A38), McAlester College (A48),
Carleton College (A19), and Oberlin College (A68) are close to each other and are
located in right middle/bottom of the maps away from the other A schools, so that they
can be considered a cluster. Wellsley College (A95), Smith College (A80), Mount
Holyoke College (A59), and Bryn Mawr College (A11) can be considered another cluster
for the same reason. Bates College (A6), Colby College (A23), Bowdoin College (A9)
67
and Middlebury College (A57) are a third cluster that is placed in left bottom of the
maps.
The within-group relationships of schools are illustrated similarly in the maps
generated by these three methods. For example, the order relationships between
Middlebury College (A57), Bowdoin College (A9), Colby College (A23) and Bates
College (A6) are obvious, i.e., Middlebury College and Bates College have the longest
pairwise distance of all pairwise distances of these four schools.
There are some local differences between the map generated by the modified
method and the other two methods. For example, the distance between Arizona State
University (A2) and University of Arizona (A3) is smaller than the distance between
Arizona State University and Colorado College (A25) because, in the asymmetric
distance matrix, d
3,2
< d
25,2
and d
2,3
< d
2,25
. In Figure 4.5, it is clear that Arizona State
University is closer to University of Arizona than it is to Colorado College, while in the
map generated by Merino’s method (Figure 4.7), it appears that Arizona State University
is closer to Colorado College.
As another example to show local differences between these three maps. The
distance between Minnesota University (A58) and Marquette University (A49) seems
equal to the distance between Minnesota University and Iowa State University (A42) in
the map generated by the modified method, while in the maps generated by the other two
methods Minnesota University is obviously closer to Iowa State University. However, in
the original asymmetric distance matrix, d
58,49
< d
58,42
, i.e. 1< 4, and d
49,58
> d
42,58
, i.e.
5>1, so that the relationships are not as obvious as they are in maps.
68
For the 30 schools from group A, the Sammon maps generated by the modified,
the standard SM and Merino’s methods are visually different (see Figures 4.8, 4.9 and
4.10). However, we can see that A41, A76, A56, A93, A31, etc. are placed in the same
sequence in these three maps but in different direction, i.e., clockwise in Figures 4.8 and
4.10, and counter-clockwise in Figure 4.9. Therefore, the general structures of the maps
generated by these three SM methods are actually similar. As another example, Reed
College (A77), Carleton College (A19), University of Washington (A94), Lafayette
College (A45), and Carnegie Mellon University (A20), which are located as outliers in
the map generated by the modified method, are still outliers and apart from other schools
of group A in the other two maps.
The maps generated by the modified SM method for groups B, C, and D have few
visual differences from the maps generated by the standard SM and Merino’s methods.
In these three types of maps, schools are scattered about and the relative relationships
among schools are roughly the same. For example, in Figures 4.11, 4.12 and 4.13,
Auburn University (B4) is about the same distance from University of Alabama (B1) and
University of Georgia (B37).
69
Figure 4.8 Map of 30 schools from group A generated by the modified SM method.
Figure 4.9 Map of 30 schools from group A generated by the standard SM method.
70
Figure 4.10 Map of 30 schools from group A generated by Merino’s method.
Figure 4.11 Map of group B generated by the modified SM method.
71
Figure 4.12 Map of group B generated by the standard SM method.
Figure 4.13 Map of group B generated by Merino’s method.
72
There are some differences in the relative relationships among schools in the
maps generated by the modified SM method and the maps obtained by the standard SM
and Merino’s methods. For example, in the asymmetric data set of group C, Harvard
(C39) and Yale (C100) are considered competitors by all other schools so that it is
desired to place these two schools in the center of the maps and let other schools scatter
about. Only in the map (Figure 4.14) generated by the modified SM method Harvard and
Yale are located near the center and surrounded by other schools. UPenn (C72), MIT
(C52), and Cornell (C28) that are less likely considered competitors by other schools are
located in the marginal area of the maps in Figures 4.14, 4.15 and 4.16.
The pairwise distances between Berkeley (D13) and Los Angeles (D16) are the
same with the pairwise distances between Berkeley and San Diego (D17) in the
asymmetric distance matrix and we expect that Berkeley is equally away from school Los
Angeles and San Diego. In Figures 4.17, 4.18 and 4.19, the distances from Berkeley to
Los Angeles and to San Diego are approximately equal. For data sets with small size, the
modified SM method yields results visually similar to those generated by the standard
and Merino’s methods.
73
Figure 4.14 Map of group C generated by the modified SM method.
Figure 4.15 Map of group C generated by the standard SM method.
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Figure 4.16 Map of group C generated by Merino’s method.
Figure 4.17 Map of group D generated by the modified SM method.
75
Figure 4.18 Map of group D generated by the standard SM method.
Figure 4.19 Map of group D generated by Merino’s method.
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The standard and Merino’s methods take as inputs symmetrized distance matrices,
which are derived from original asymmetric distance matrices by averaging. The
modified SM method takes original asymmetric distance matrices as inputs directly. Two
performance measurements are employed to assess these three methods’ capability of
representing original asymmetric distance data. We use order preservation coefficient to
show how effectively a SM method preserves order relationships of original data. We
use error measurement to indicate how precisely original distances between data are
represented by a SM method. The represented order relationships among schools and the
resulting error measures are expected to be different in the maps generated by the three
SM methods because their input distances and rationales behind the methods are
different.
As shown in Tables 4.5 and 4.6, the order preservation coefficients of maps
generated by the standard and Merino’s SM methods are very close. The error measures
of these two SM methods do not change dramatically (see Tables 4.7 and 4.8). These can
be explained by their taking the same symmetrized distance matrices. Another
explanation may be the asymmetry coefficients introduced in Merino’s method
(described in section 2.5). In these six data sets most of schools have similar chances to
be considered competitors by others and asymmetry coefficients of schools fall in a
narrow range of values. Introducing similar asymmetry coefficients will not make the
objective function better off and therefore asymmetry coefficients in these cases have
little impact on the optimization results.
By incorporating the upper triangular part and the lower triangular part of the
original asymmetric distance matrix into the objective function and the associated
77
Data Set Standard SM Modified SM
% Improvement of Standard over
Modified
100 Schools
0.482 0.472 2.10
30 Schools
0.619 0.607 1.97
A
0.605 0.594 1.71
B 0.759 0.756 0.34
C 0.580 0.568 2.14
D 0.567 0.552 2.59
Table 4.5 Order preservation coefficients for the six data sets (standard vs. modified).
Data Set Merino’s SM Modified SM
% Improvement of Merino’s over
Modified
100 Schools
0.475 0.472 0.70
30 Schools
0.616 0.607 1.51
A
0.603 0.594 1.45
B 0.759 0.756 0.36
C 0.582 0.568 2.52
D 0.567 0.552 2.59
Table 4.6 Order preservation coefficients for the six data sets (Merino’s vs. modified).
updating rules, the search procedure of the modified SM method takes into account the
entire original distance matrix instead of the symmetrized distance matrix and looks for
an optimal configuration for the entire distance matrix. Therefore, the error measures of
maps generated by the modified SM method are significantly smaller than the error
measures of maps generated by the standard and Merino’s SM methods (see Tables 4.7
and 4.8). It seems that the modified SM method reduces the distance error measures
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Data Set Standard SM Modified SM
% Improvement of Modified over
Standard
100 Schools 54906.620 28830.320 47.49
30 Schools 999.067 773.045 22.62
A 6926.616 5688.302 17.88
B 20.192 18.313 9.31
C 6.547 6.065 7.37
D 6.048 5.549 8.25
Table 4.7 Error measures for the six data sets (standard vs. modified).
Data Set Merino’s SM Modified SM
% Improvement of Modified over
Merino’s
100 Schools 59774.640 28830.320 51.77
30 Schools 1024.385 773.045 24.54
A 7096.426 5688.302 19.84
B 19.927 18.313 8.10
C 6.615 6.065 8.33
D 6.073 5.549 8.62
Table 4.8 Error measures for the six data sets (Merino’s vs. modified).
proportionally to the size of data sets. For example, in 100-school data set, the modified
SM method reduced the error measure over the standard SM method by 47.49% (100 ×
(54906.620 – 28830.320)/ 54906.620). In the remaining data sets, the improvements
range from 8.25% to 22.62%. We point out that the values given in the tables of
performance measures are averages from the five experiments on each data set.
The modified SM method produces slightly smaller order preservation
coefficients than the other two SM methods (large coefficients are better). This may be
79
due to asymmetric distances in original data sets. When the whole asymmetric distance
matrix is taken into account, the search procedure goes through the upper triangular part
and the lower triangular part separately to find an improved result. This adds more
constraints to the optimization process and introduces more difficulty to achieve a better
result because some constraints are conflict due to asymmetry.
Although the modified SM method did not do a better job than the standard SM
and Merino’s SM methods in preserving the order relationships in our American college
selection data, the maps generated by the modified SM method are still considered better
than the maps generated by the standard SM method and Merino’s method. The
modified SM method is capable of preserving order relationships with similar accuracy
and reducing the distance errors with significant improvement as compared to the
standard and Merino’s SM method. The maps generated by the modified SM method
show us the intra-group relationships, which were not reflected in the maps generated by
other two methods. For example, it is interesting to see that schools of group C mixed
with several A schools in the maps generated by the modified SM method. In addition,
the generated maps by the modified method are more readable -- schools were not
squeezed as tightly in Figure 4.2 as they were in Figure 4.3.
In our study of analyzing American college selection data using different
Sammon mapping visualization techniques, currently, the modified SM method seems to
be able to generate visualization maps with higher quality as compared to the standard
and Merino’s methods. The modified method yields results that are with significantly
reduced distance errors and reasonably preserved order relationships compared to the
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results generated by the other two methods. We hope that our modified SM method can
be robust on other applications.
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Chapter 5
Visualizing Canadian Ranked College Data
The modified SM method was shown to be good at recovering the structure of
American college data at least comparable to the standard and Merino’s methods. In
American college selection data, the overlaps of each school are not ordered so that there
is no indication of ranks among the overlaps. The generated visualization maps are
unable to provide students with further details such as which school among the overlaps
is the closest competitor. If ranking information is incorporated into the data set,
decisions by students can be made more easily and effectively.
Data sets with ranking information can be collected in many fields. In marketing,
ranked data can be gathered from customers who give ranks of different brands of
products of the same category, i.e., car brands. Consider another example in which a
survey is sent to apartment managers in a metropolitan area. The survey asks the
managers to identify top 10 rival apartment buildings. The managers respond with a top
10 listing in which the first building is the most competitive rival, the second building is
next most competitive rival, and so on. For a person seeking an apartment, a visual map
of competitors can help narrow the search effectively and naturally.
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Our work in this chapter includes building a visual model of asymmetric data sets
that incorporate ranking information. The Canadian ranked college data set is the one
that has ranking information and will be analyzed in the following sections. The modified
SM method will be applied to this data set to see if ranking information is represented
accurately. For comparison, the standard and Merino’s SM methods will also be applied
to the data set.
5.1 Description of Canadian Ranked College Data
The idea of visualizing universities originates with the work of Yin (2002) and
Condon et al. (2002). Yin proposed ViSOM (visualization-induced SOM) to detect
clusters of universities in the United Kingdom. Condon et al. created visual maps of 100
American universities that can be used to view patterns and clusters and gain insights.
We collected data from surveys that were sent to the admission directors of
undergraduate programs in 52 Canadian universities. The directors were asked to list the
five most competitive rivals in terms of overall education quality. We have received
responses from 44 universities and we list these in Table 5.1. The competitors of each
Canadian university are given in Table 5.2.
83
Key School
1 Acadia University
2 Bishop's University
3 Brandon University
4 Brock University
5 Carleton
6 Concordia University
7 Dalhousie University
8 Lakehead University
9 Laval University
10 McGill University
11 McMaster University
12 Memorial University of Newfoundland
13 Mount Saint Vincent University
14 Nipissing University
15 Queens University
16 Simon Fraser University
17 St. Francis Xavier University
18 Univeristy College of Cape Breton
19 Universite de Moncton
20 Universite de Sherbrooke
21 Universite du Quebec a Rimouski
22 Universite du Quebec en Outaouais
23 University of Alberta
24 University of British Columbia
25 University of Calgary
26 University of Guelph
27 University of Lethbridge
28 University of Manitoba
29 University of Montreal
30 University of New Brunswick
31 University of Ontario Institute of Technology
32 University of Ottowa
33 University of Prince Edward Island
34 University of Regina
35 University of Saskatchewan
36 University of Toronto
37 University of Victoria
38 University of Waterloo
Table 5.1 44 Canadian universities collected from surveys.
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39 University of Western Ontario
40 University of Windsor
41 Wilfrid Laurier
42 Laurentian University
43 University of St. Anne
44 York University
Table 5.1 (continued).
Key School 1
st
2
nd
3
rd
4
th
5
th
1 Acadia University 38 36 10 15 23
2 Bishop's University 1 41 0 0 0
3 Brandon University 28 25 34 35 0
4 Brock University 11 26 41 44 39
5 Carleton 36 32 10 15 38
6 Concordia University 10 29 20 7 0
7 Dalhousie University 10 36 15 24 23
8 Lakehead University 26 4 39 38 0
9 Laval University 29 20 10 6 0
10 McGill University 36 24 15 7 44
11 McMaster University 36 15 38 39 10
12 Memorial University of Newfoundland 7 30 1 17 13
13 Mount Saint Vincent University 1 17 12 0 0
14 Nipissing University 4 41 8 0 0
15 Queens University 36 24 10 26 38
16 Simon Fraser University 24 37 38 26 36
17 St. Francis Xavier University 7 1 0 0 0
18 Univeristy College of Cape Breton 7 17 13 30 0
19 Universite de Moncton 30 9 32 7 43
20 Universite de Sherbrooke 29 9 2 0 0
21 Universite du Quebec a Rimouski 9 20 0 0 0
22 Universite du Quebec en Outaouais 32 20 0 0 0
23 University of Alberta 36 24 10 0 0
24 University of British Columbia 36 10 15 23 39
25 University of Calgary 36 24 15 10 23
Table 5.2 Competitors of 44 Canadian universities.
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26 University of Guelph 11 38 15 36 39
27 University of Lethbridge 23 25 34 35 37
28 University of Manitoba 23 35 36 15 0
29 University of Montreal 36 10 32 24 23
30 University of New Brunswick 7 12 1 0 0
31 University of Ontario Institute of Technology 39 36 15 44 38
32 University of Ottowa 11 39 15 29 7
33 University of Prince Edward Island 1 17 30 0 0
34 University of Regina 23 24 28 0 0
35 University of Saskatchewan 34 23 25 15 24
36 University of Toronto 39 24 23 15 10
37 University of Victoria 24 16 25 23 0
38 University of Waterloo 36 39 15 41 11
39 University of Western Ontario 36 10 15 38 11
40 University of Windsor 39 11 41 36 15
41 Wilfrid Laurier 39 38 26 11 36
42 Laurentian University 4 40 8 32 5
43 University of St. Anne 7 17 30 0 0
44 York University 36 15 39 41 0
Table 5.2 (continued).
5.2 Modeling Steps
In the Canadian university data set that we have collected, some universities
specified their competitors that were not on the list of the 44 universities. We did not
include universities that did not respond and yet were selected as rivals by other
universities. For example, York University chose University of Toronto as its top
competitor, Ryerson University was the second, and Queens, Western Ontario, and
Wilfrid Laurier were third, fourth, and fifth, respectively. Ryerson University did not
respond to our survey, so it is not on the list. Therefore, in the rival list of York
University, we excluded Ryerson and replaced it with Queens as the second most
86
competitive school, and then moved Western Ontario and Wilfrid Laurier up one rank,
respectively.
In order to create a visual map of the Canadian college data set, we need to
generate a distance matrix from the data set and then input the distance matrix into the
modified SM method. Using procedures that are described in Chapter 4, we start by
creating a 44 44× 0-1 adjacency matrix, where entry s
ij
= 1 (row i and column j) if
school j is a competitor of school i. Next, we adjust the entries with the value of 1 to
reflect the ranking information. In terms of adjacency or distance, the more competitive
school j is to school i, the smaller the value of the s
ij
entry is. The entry of the most
competitive school is 1 by default. We set the gap between entries of two consecutive
competitors in ranking is 0.5, in other words, the entry of the second competitor is 1.5,
and the entry of the third competitor is 2, and so forth.
After adjusting the adjacency matrix, we construct the distance matrix using a
shortest path procedure and replace infinity entries with an appropriate value, i.e., 25%
larger than the longest distance. After the distance matrix is constructed, we apply the
modified SM method that is described in Chapter 3 to Canadian ranked college data
trying to detect some interesting relationships of Canadian schools.
87
5.3 Discussion of the Results
The map shown in Figure 5.1 is generated by the modified SM method. The
distance gap, which is used to separate two consecutive competitors, is set to 0.5 and the
substitute value of infinity distance is 10.5, which is 25% larger than the longest finite
distance i.e. 8.5. It provides us with a general view of the structure of Canadian colleges.
Universities such as Toronto (36), Queens (15), and Western Ontario (39) that are
frequently considered competitors by other universities are located in the center of the
map. Other universities such as Waterloo (38) and British Columbia (24) with high
frequencies are also placed near the center. Therefore, the map tells us that universities
placed in or near the center are those that are considered popular. Besides, it shows in the
map roughly five clusters of universities – one in the center and the other four
surrounding the center.
The map generated by the modified SM method also reveals some ranking
information of competitors of a particular school. For example, the top five competitors
of Queens University (15) are Toronto (36), British Columbia (24), McGill (10), Guelph
(26), and Waterloo (38), where Toronto is the most competitive school to Queens. In
Figure 5.1, among these five competitors, Toronto is the closest to Queens. Although
Waterloo is not the school farthest from Queens, it is the second farthest school among
these five competitors. Guelph is the school farthest from Queens, because the top rival
of Guelph is McMaster and Queens is its third top. The asymmetric characteristic of a
data set makes it very difficult to generate a map that is a completely accurate
representation of the data set. A map that shows roughly similar structures of the data set
can be generated by visualization techniques.
88
Figure 5.1 Sammon map of the Canadian ranked college generated by the modified
method. Gap is set to 0.5.
Figures 5.2 and 5.3 show the maps generated by the standard and Merino
methods. Again the maps generated by these two methods are similar to each other due
to the same reasons that have been discussed in previous chapter. Universities located in
the center of Figure 5.1 are still placed in the center of these two maps. Other
universities scatter about surrounding popular universities.
Differences between the map generated by the modified method and the maps
generated by the other two methods can be summarized as follows. Universities
surrounding the center are separated more evenly in the maps generated by latter two
methods. This makes it harder to detect clusters of universities. Universities in the
center of the maps generated by the latter two methods squeeze more tightly than they are
in the
89
Figure 5.2 Sammon map of the Canadian ranked college generated by the standard
method. Gap is set to 0.5.
Figure 5.3 Sammon map of the Canadian ranked college generated by Merino’s method.
Gap is set to 0.5.
90
map generated by the modified method. Some interesting insights may be hidden in the
maps of Figures 5.2 and 5.3. For example, all five competitors of Dalhousie (7) are
located in the center of maps and therefore it is reasonable to place Dalhousie near the
center. However, most of the universities that consider Dalhousie a competitor are
located in the top middle area of the maps. It is expected that Dalhousie would be near
these universities while keeping near the center universities. Only in the map generated
by the modified SM map the location of Dalhousie reminds us of the relationships while
in other two maps it is not clear that we can detect the relationships between Dalhousie
and these universities that choose Dalhousie as a competitor.
In terms of insights that possibly will be gained from visualization maps, the
modified SM method provides us with a more reasonable map. In terms of performance
measures, the modified SM method most of the time in our experiments does slightly
better than the standard and Merino’s methods in order relationship preservation (see
Tables 5.3 and 5.4) and always outperforms the other two methods in error measurement
(see Tables 5.5 and 5.6).
In order to see if the gap value affects the performance of these three methods, we
experimented with three different gap values: 0.2, 0.5, and 1. 0.5 is the one that is used in
Figures 5.1, 5.2 and 5.3. Given five random starts, we used each gap value on the
Canadian college data. Tables 5.3 and 5.4 provide average order preservation measures
of these three methods, and Tables 5.5 and 5.6 provide average error measures of these
three methods. In Figures 5.4 and 5.5 are given generated maps by the modified method
with gap values of 0.2 and 1.0 respectively. Comparing these two figures to Figure 5.1,
we can see that although there are some local differences between these three maps, the
91
Gap Value Standard SM Modified SM
% Improvement of Modified over
Standard
Gap-0.2
0.555 0.560 0.99
Gap-0.5
0.546 0.548 0.46
Gap-1.0
0.537 0.542 0.89
Table 5.3 Order preservation measures of different gap values of Canadian ranked
college data (standard vs. modified).
Gap Value Merino’s SM Modified SM
% Improvement of Modified over
Merino’s
Gap-0.2
0.559 0.560 0.14
Gap-0.5
0.543 0.548 0.88
Gap-1.0
0.545 0.542 -0.64
Table 5.4 Order preservation measures of different gap values of Canadian ranked
college data (Merino’s vs. modified).
Gap Value Standard SM Modified SM
% Improvement of Modified over
Standard
Gap-0.2
2413.83 2100.84 12.97
Gap-0.5
3921.53 3336.99 14.91
Gap-1.0
7001.58 5414.63 22.67
Table 5.5 Error measures of different gap values of Canadian ranked college data
(standard vs. modified).
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Gap Value Merino’s SM Modified SM
% Improvement of Modified over
Merino’s
Gap-0.2
2344.01 2100.84 10.37
Gap-0.5
3972.14 3336.99 15.99
Gap-1.0
6675.48 5414.63 18.89
Table 5.6 Error measures of different gap values of Canadian ranked college data
(Merino’s vs. modified).
Figure 5.4 Sammon map of the Canadian ranked college generated by the modified
method. Gap is set to 0.2.
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Figure 5.5 Sammon map of the Canadian ranked college generated by the modified
method. Gap is set to 1.0.
general structure of the data set is kept similar in each of these maps, i.e., popular
universities in the center and surrounded by other clusters of universities. Different gap
values have effect on constructing maps especially on local details however the general
structure of the data set is not changed dramatically.
To analyze Canadian ranked college data, the modified SM method seems better
than the standard and Merino’s methods. The modified method reduces distance errors
significantly and preserves the order relationships reasonably well compared to other two
methods. The modified method also helps gain some interesting insights that can hardly
be detected in maps generated by other two methods. We have done some sensitivity
analysis of gap values and it seems to us that gap values do not dramatically affect the
general structure of the data set.
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So far, we have analyzed two visualization applications using the modified
Sammon mapping method. As compared to the modified SM method, the self-organizing
map, which is a neural-network based method, can also be employed to visualize and
analyze data sets. In the following chapters, we will discuss some applications using
SOMs.
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Chapter 6
Self-Organizing Maps: State Sponsored Murder Data Set
6.1 Introduction
As a neural-network based unsupervised method, self-organizing maps (SOMs)
are mainly used for clustering, which is one of activities of data analysis in data
visualization applications. In our previous applications, we used the Sammon mapping
method to visualize two college data sets and analyze intra-cluster and inter-cluster
relationships among clusters. In this chapter, we will use SOMs to analyze clusters in the
state sponsored murder data set and to analyze the sport records data in the next chapter.
For discovering clustering information hidden in data sets, there are a few
methods that have been proposed in the literature such as hierarchical clustering methods
(single linkage, average linkage, and complete linkage), K-means clustering, and
Kohonen’s self-organizing maps (SOMs) (1995). Among these methods, Kohonen’s
SOM has received increased attention in the literature in recent years. Some recently
proposed clustering methods such as ViSOM (Yin, 2002) are based on Kohonen’s SOM.
Several software packages (i.e., Viscovery SOMine) with SOM-based clustering
procedures have been released. A natural question arises: How well do these software
packages perform? We want to make sure that we choose the right clustering procedure
96
to analyze datasets. Therefore, in the first part of this chapter, we evaluate the
performance of four software implementations of SOM-based clustering methods and
determine the best SOM-based procedure to be used in our clustering analysis. In the
second part of this chapter, we apply the chosen SOM-procedure to the state sponsored
murder data set.
6.2 Evaluating SOM-based Methods
Four clustering implementations are compared based on their performances: Ward
clustering, modified Ward clustering, single linkage clustering, and classic SOM. The
first three clustering methods are implemented in a commercial package, Viscovery
SOMine 4.0 from Eudaptics Software (www.eudaptics.com). These three SOM-based
clustering methods make use of the representation of the data set given by Kohonen’s
SOM scheme. SOM-Ward clustering uses Ward’s classic minimum distance method
(Ward, 1963). In SOM-modified Ward clustering, the classic Ward method is modified
to use a different distance measure (Viscovery, 2002). SOM-single linkage uses an
adaptation of the classic single-linkage clustering algorithm (Viscovery, 2002). Classic
SOM clustering is implemented in a research package, SOM_Pak from the Helsinki
University of Technology (SOM_Pak, 1997).
Mangiameli et al. (1996) conducted a comprehensive evaluation of seven
hierarchical clustering algorithms and the SOM network generated by the commercial
software package NeuralWorks. Our study can be viewed as an extension of their work.
We assess the clustering performance of four procedures in two current SOM software
97
packages. To our knowledge, the performance of these packages has not been reported in
the open literature.
The standard approach to studying the performance of a clustering procedure is to
apply the procedure to a problem for which the clusters are already known. This
approach allows the researcher to measure the method’s success in assigning data points
to their correct clusters. We adopt this approach and evaluate the performance of the
clustering methods on 96 data sets that we construct. The clustering methods that we
evaluate are applied to data sets in which the clusters are well separated. Figure 6.1
presents a two-dimensional plot of a four-cluster data set.
6.2.1 Constructing data sets
Four experimental factors are used to characterize each data set: the number of
clusters (three, four, five, and six), number of dimensions (three and four), number of
data points (50, 100, 150, and 200), and amount of intra-cluster dispersion (low, medium,
and high). Using this design, we construct 4 × 2 × 4 × 3 = 96 data sets, each of which
exhibits both external isolation and internal cohesion (see Cormack (1971), Mangiameli
et al. (1996), and Milligan (1980)). External isolation means that the members of one
cluster are separated from members of another cluster by empty space. Internal cohesion
means that members of the same cluster are similar (close) to each other.
We use a procedure that is similar to the method proposed by Milligan (1980,
1985) to construct data sets. The first step is to determine the cluster lengths and the
cluster boundaries for the first dimension of the variable space. For each cluster, the
98
Figure 6.1 Example of a four-cluster data set.
cluster length is selected randomly from the uniform distribution on the interval (10, 40).
In order to achieve external isolation, the boundaries of adjacent clusters are separated by
an amount selected randomly from the uniform distribution on the interval (0.25, 0.75).
The mean of each cluster is taken to be the midpoint of the cluster’s boundaries, and the
standard deviation of each cluster is set to 0.5.
The second step is to specify the characteristics of the clusters in the remaining
dimensions. We select the cluster lengths randomly from the uniform distribution on the
interval (10, 40) and then select the cluster boundaries randomly. This makes it possible
that cluster boundaries overlap with each other, unlike in the first dimension where
external isolation is guaranteed. The mean of each cluster is taken to be the midpoint of
the cluster’s boundaries. Three levels of intra-cluster dispersion are used here: low,
medium, and high; these are the same levels specified in Mangiameli et al. (1996). At the
Cluster 1
Cluster 2
Cluster 3
Cluster 4
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low, medium, and high levels of dispersion, the standard deviation of the cluster is set
equal to 1/12, 1/6, and 1/3 times the cluster length, respectively. The level of the intra-
cluster dispersion indicates the density of data points around the cluster boundaries. The
higher the intra-cluster dispersion, the higher is the density of data points near the cluster
boundaries. Thus, internal cohesion decreases as the intra-cluster dispersion increases.
We generate data points in each cluster from a multivariate normal distribution
with mean vector given by the midpoints of the cluster boundaries. The diagonal
elements of the variance-covariance matrix are given by the squares of the standard
deviations, and all of the off-diagonal elements are equal to zero. We discard data points
that fall outside the cluster boundaries.
6.2.2 Measuring performance
In order to evaluate the performance of a clustering procedure, we use two
measures: the cluster recovery rate and the Rand statistic. The cluster recovery rate is
defined to be the proportion of times a clustering procedure correctly recovers the cluster
structure, that is, the percentage of times a procedure correctly determines the cluster
membership of each data point. The Rand statistic (Rand, 1971) is a widely used
performance metric (Milligan, 1981). The definition of the Rand statistic can be
illustrated using the notation given in Table 6.1. Cell A is the number of pairs of points in
the data set that are from the same cluster and are correctly assigned by a clustering
procedure to the same cluster. Cell B is the number of pairs of points that are from
different clusters and are correctly assigned by a clustering procedure to different
100
Table 6.1 Pairwise classification notation.
clusters. In Cells C and D, clustering errors are counted. The Rand statistic provides the
proportion of correct pairwise classifications for the data set and is given by (A + B)/(A +
B + C + D). If the solution generated by a clustering procedure is correct, then the Rand
statistic equals one; if the generated solution is incorrect, then the value of the Rand
statistic will be less than one. Clearly, the larger the value of the Rand statistic, the better
the solution.
6.2.3 Comparison results and conclusions
We applied the three SOM-based clustering procedures in Viscovery (Ward,
modified Ward, and single linkage, all with default settings), the classic SOM clustering
procedure in SOM_Pak (there are no default settings; for each run, we had to specify
values for several parameters found in the package), and the K-means algorithm in
Clementine from SPSS to each of our 96 data sets. We specified the number of clusters in
each data set as input to each procedure.
The cluster recovery rates for the five procedures are given in Table 6.2. We see
Correct Solution
Clustering Procedure
Solution
Pair in Same Cluster Pair Not in Same Cluster
Pair in Same Cluster A C
Pair Not in Same Cluster D B
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SOM-Ward
SOM-
Modified Ward
SOM-Single
Linkage SOM-Classic K-Means
92.7 91.7 82.3 14.6 80.2
Table 6.2 Cluster recovery rates (in %).
that two of Viscovery’s procedures, SOM-Ward and SOM-modified Ward, recover the
true clusters more than 90% of the time, while Viscovery’s SOM-single linkage and the
K-means algorithm recover the clusters about 80% of the time. SOM-classic performs
poorly, only recovering the clusters about 15% of the time.
In Table 6.3, we show the effect of intra-cluster dispersion on the cluster recovery
rates of the five procedures. As the intra-cluster dispersion increases, thereby reducing
the internal cohesion of the clusters, we see that the cluster recovery rates decrease. At
all three levels of dispersion, SOM-Ward performs the best, closely followed by SOM-
modified Ward.
In addition to the cluster recovery rate, we also examine the performance of the
five procedures by calculating the Rand statistic. The average value of the Rand statistic
is given in Table 6.4. We apply SOM-Ward to the eight data sets with low dispersion
and three clusters, calculate the Rand statistic for each data set, and then average over the
eight data sets (we see that the entry is 1). The average Rand statistic for SOM-Ward for
all 32 data sets with low dispersion is 1 (this is the row average for SOM-Ward in the
first row of Table 6.4).
In Table 6.4, at the low and medium levels of dispersion, we see that all three of
Viscovery’s procedures perform better than SOM-classic and K-means. At the high level
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Level of Dispersion
Procedure Low Medium High
SOM-Ward 100 94 84
SOM-Modified Ward 100 91 84
SOM-Single Linkage 100 91 56
SOM-Classic 19 16 9
K-Means 88 78 75
Table 6.3 Cluster recovery rates (in %) by level of dispersion.
Number of Clusters Row
Clustering Procedure 3 4 5 6 Average
Low Intra-Cluster Dispersion
SOM-Ward 1.000 1.000 1.000 1.000 1.000
SOM-Modified Ward 1.000 1.000 1.000 1.000 1.000
SOM-Single Linkage 1.000 1.000 1.000 1.000 1.000
SOM-Classic 0.846 0.899 0.911 0.886 0.886
K-Means 1.000 1.000 0.988 0.965 0.988
Medium Intra-Cluster Dispersion
SOM-Ward 0.994 0.989 1.000 1.000 0.996
SOM-Modified Ward 0.995 0.986 1.000 0.997 0.995
SOM-Single linkage 1.000 0.999 1.000 0.999 0.999
SOM-Classic 0.898 0.878 0.893 0.893 0.890
K-Means 0.995 1.000 0.988 0.931 0.979
High Intra-Cluster Dispersion
SOM-Ward 0.914 0.954 0.984 1.000 0.963
SOM-Modified Ward 0.951 0.971 0.996 1.000 0.980
SOM-Single linkage 0.946 0.977 0.992 0.991 0.977
SOM-Classic 0.915 0.931 0.876 0.898 0.905
K-Means 1.000 0.960 0.990 0.950 0.975
Table 6.4 Values of the Rand statistics.
103
of dispersion, the three procedures in Viscovery and K-means perform nearly the same
and SOM-classic is close behind. It appears that as the level of dispersion increases and
when the number of clusters is small (three or four), the performance of each of the
Viscovery procedures deteriorates somewhat (the value of the average Rand statistic
decreases). We note that SOM-classic and K-means seem to be less affected than
Viscovery’s three procedures by increases in intra-cluster dispersion.
Both SOM-classic and K-means require the user to specify the number of clusters
(that is why we input the number of clusters into all five procedures). Viscovery,
however, does not have this requirement. After inputting the data set, Viscovery can
determine the number of clusters and the assignment of points to clusters. This is a
desirable feature of the package since in practice a user usually does not know how many
clusters to specify in advance. We applied Viscovery to each of the 96 data sets and let it
determine the number of clusters. In Table 6.5, we give the cluster recovery rates for the
procedures. For each procedure, the recovery rate drops about 10 percentage points from
the recovery rate generated by Viscovery when cluster size was specified. These results
are still competitive with the recovery rate from K-means (80.2%) when K-means has the
advantage of knowing the true cluster size.
In this study, we assessed the performance of four SOM-based clustering
procedures that are implemented in commercial and research software. The three
procedures in Viscovery SOMine 4.0 performed generally well in clustering. We found
that Viscovery’s procedures performed slightly better than the K-means algorithm and
much better than the procedure in SOM_Pak. In addition, when clusters were well
separated (i.e., exhibited external isolation), the clustering procedures in Viscovery were
104
SOM-Ward SOM-Modified Ward SOM-Single Linkage
83.3 82.3 71.9
Table 6.5 Cluster recovery rates (in %) for Viscovery (number of clusters is not
specified).
fairly effective at determining the appropriate number of clusters in a data set. This
feature may help Viscovery users who are not sure of the number of clusters to
determine.
6.3 Self-Organizing Maps: the State Sponsored Murder Data Set
As we discussed in the previous section, Viscovery has useful features such as
helping determine the number of clusters and recovering cluster structures of data sets.
In this section, we apply Viscovery SOM-Ward procedure to a state sponsored murder
(also called genocide and politicide) data set. Genocides and politicides refer to actions
committed by governing elites or, in the case of civil war, either of the contending
authorities that are intended to destroy a national, ethnical, racial, religious, or political
group (Harff, 2003).
The genocide and politicide data set examined in this section includes 28
historical cases of genocide and politicide that began between 1955 and 2002 in
independent countries with populations greater than 500,000. The genocide and
politicide data were originally studied for identifying independent variables (risk factors
or pre-conditions) to distinguish countries that have genocides and politicides from those
that do not (Harff, 2003). It consists of 25 countries that have or have no prior genocides
105
and politicides and six variables that are identified as risk factors contributing to genocide
and politicide. The genocide and politicide data set under consideration is given in Table
6.6.
Among the six risk factors (variables) listed, upheaval in political context is
defined as an abrupt change in the political community resulted from the formation of a
state or regime through violent actions, defeat in international war, or rewriting of state
boundaries. Minority elite is designed to reveal the information about interethnic
disputes over access to political power. A positive value under the column of ‘Minority
Elite’ (i.e., ‘Yes’) indicates that elite ethnicity is a recurring issue of political conflicts,
which possibly leads to genocide or politicide. Exclusionary ideology refers to a belief
system that establishes some cardinal principle that maintains efforts to restrict,
persecute, or eliminate certain categories of people. Elite with ‘exclusionary ideology’ is
more apt to eliminate groups. The type of regime is another risk factor that has vital
intervening effects to cause genocide and politicide. Elite in autocratical regime is likely
to opt for restricting citizens’ participation, especially political opponents’ participation.
The level of trade openness is also an indicator of genocide and politicide. Historical
records have shown that armed conflicts and adverse regime changes are more likely to
occur in poor countries, especially those countries in Africa and Asia.
Countries in this data set are listed according to their number of positive risk
factors. If a country has many positive risk factors (i.e., six or five risk factors), it will be
ranked high. For example, Iraq is the only country with six positive risk factors, which
indicates that Iraq has the greatest potential to have future genocides and politicides and
therefore Iraq is listed in the first place in the data set. By comparison,
106
Country Name
Prior
Genocides
or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of Regime
Trade
Openness
Iraq Yes High Yes Yes Autocracy Very Low
Afghanistan
(2000) Yes Very High Yes Yes Autocracy Very Low
Afghanistan
(2002) Yes Very high No No No effective regime Very Low
Burma Yes High No Yes Autocracy Very Low
Burundi Yes Very High Yes No Autocracy Low
Rwanda Yes High Yes No Autocracy Low
Congo-Kinshasa Yes Very High Yes No No effective regime Medium
Somalia Yes Very High No No No effective regime Very Low
Sierra No Very High Yes No No effective regime Low
Ethiopia Yes High Yes No Autocracy Medium
Uganda Yes High No No Autocracy Low
Algeria Yes Very High No Yes Autocracy Medium
Liberia No High No No Autocracy Low
Pakistan Yes Medium No No Autocracy Low
China Yes Medium No Yes Autocracy Medium
Sri Lanka Yes High No No Partial democracy High
Philippines Yes Very High No No Democracy High
Colombia No Very High No No Partial democracy Low
Turkey No High No Yes Partial democracy Medium
India No High No No Democracy Low
Israel No Very High No Yes Democracy High
Indonesia Yes Medium No No Partial democracy Medium
Russia Yes Low No No Partial democracy Medium
Nigeria No Low No No Partial democracy High
Nepal No Medium No No Partial democracy Medium
Macedonia No None No No Partial democracy High
Table 6.6 Genocide and politicide data set from Harff (2003).
Macedonia is very unlikely to have future genocides and politicides according to
the fact that Macedonia has no positive risk factors.
We apply Viscovery to the genocide and politicide data set to detect clusters of
countries that have similar pre-conditions that may result in future genocides and
politicides. Viscovery needs numerical values as input, so that the categorical values in
107
the genocide and politicide data set need to be transformed to numerical values. We
transformed the categorical values using a commonly used approach that assigns sorted
numerical values to categorical values based on the description of the categorical values
(Ritter & Kohonen, 1989). The transformed data are given in Table 6.7. The notation for
the transformed genocide and politicide data set is given in Table 6.8. For example, in
the second column (prior genocides and politicides), ‘No’ is assigned by 0 and ‘Yes’ is
assigned by 1.
We input the transformed values into Viscovery and used the software’s default
settings. We obtained the map shown in Figure 6.2. This map has five clusters of
countries. These five clusters are formed based on six variables. For example,
Macedonia, Russia, Nigeria, Nepal, Indonesia, and Sri Lanka are grouped into the same
cluster. These six countries share several common characteristics. For example, they all
are partial democratic countries. Half of them (Macedonia, Nigeria, and Sri Lanka) have
a high level of trade openness. Countries in this cluster have relatively infrequent
political upheavals except Sri Lanka. Half of this group (i.e., Russia, Indonesia and Sri
Lanka) has prior genocides or politicides. Overall, this cluster can be viewed as a group
of countries where genocide and politicide is less likely to take place.
India, Colombia, Philippines, Israel and Turkey are clustered together. Countries
in this cluster almost have no prior genocides or politicides except that the Philippines
has one. Members of this cluster have frequent political upheavals, as their levels of
political upheavals are either ‘High or Very High’.
108
Country Name
Prior Genocides or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of
Regime
Trade
Openness
Iraq 1 3 1 1 1 0
Afghanistan (2000) 1 4 1 1 1 0
Afghanistan (2002) 1 4 0 0 0 0
Burma 1 3 0 1 1 0
Burundi 1 4 1 0 1 1
Rwanda 1 3 1 0 1 1
Congo-Kinshasa 1 4 1 0 0 2
Somalia 1 4 0 0 0 0
Sierra 0 4 1 0 0 1
Ethiopia 1 3 1 0 1 2
Uganda 1 3 0 0 1 1
Algeria 1 4 0 1 1 2
Liberia 0 3 0 0 1 1
Pakistan 1 2 0 0 1 1
China 1 2 0 1 1 2
Sri Lanka 1 3 0 0 2 3
Philippines 1 4 0 0 3 3
Colombia 0 4 0 0 2 1
Turkey 0 3 0 1 2 2
India 0 3 0 0 3 1
Israel 0 4 0 1 3 3
Indonesia 1 2 0 0 2 2
Russia 1 1 0 0 2 2
Nigeria 0 1 0 0 2 3
Nepal 0 2 0 0 2 2
Macedonia 0 0 0 0 2 3
Table 6.7 Transformed genocide and politicide data.
Prior
Genocides
or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of Regime
Trade
Openness
0 = No
1 = Yes
0 = None
1 = Low
2 = Medium
3 = High
4 = Very high
0 = No
1 = Yes
0 = No
1 = Yes
0 = No effective regime
1 = Autocracy
2 = Partial democracy
3 = Democracy
0 = Very low
1 = Low
2 = Medium
3 = High
Table 6.8 Notation for the transformed genocide and politicide data set.
109
Figure 6.2 Resulting SOM map of the genocide and politicide data set.
The level of trade openness of this group of countries is relatively high among the
five clusters: more than half of member countries in this group have active trade
openness. In addition, countries in this cluster are either democratic or partially
democratic. Summaries of the remaining clusters are given in Table 6.9.
The genocide and politicide data set we examine in this section includes the six
variables given in Table 6.6. Some researchers suggest including a country’s per capita
income, which they claim is the best predictor of the ethnic insurgencies and civil wars
and which underlies Harff’s work. We consider gross domestic product (GDP) per capita
and number of prior genocides and politicides. GDP per capita is a purchasing power
parity
110
Country Name
Prior
Genocides or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of
Regime
Trade
Openness
Cluster 1
Macedonia 0 0 0 0 2 3
Nigeria 0 1 0 0 2 3
Russia 1 1 0 0 2 2
Nepal 0 2 0 0 2 2
Indonesia 1 2 0 0 2 2
Sri Lanka 1 3 0 0 2 3
Cluster 2
Pakistan 1 2 0 0 1 1
Uganda 1 3 0 0 1 1
Somalia 1 4 0 0 0 0
Afghanistan
(2002) 1 4 0 0 0 0
Liberia 0 3 0 0 1 1
Cluster 3
Sierra 0 4 1 0 0 1
Congo-Kinshasa 1 4 1 0 0 2
Ethiopia 1 3 1 0 1 2
Burundi 1 4 1 0 1 1
Rwanda 1 3 1 0 1 1
Cluster 4
India 0 3 0 0 3 1
Colombia 0 4 0 0 2 1
Philippines 1 4 0 0 3 3
Israel 0 4 0 1 3 3
Turkey 0 3 0 1 2 2
Cluster 5
China 1 2 0 1 1 2
Algeria 1 4 0 1 1 2
Burma 1 3 0 1 1 0
Iraq 1 3 1 1 1 0
Afghanistan
(2000) 1 4 1 1 1 0
Table 6.9 Cluster profiles of the genocide and politicide data set.
111
basis divided by population. In the modified data set M1 given in Table 6.10, we include
GDP per capita. In Table 6.11, we give modified data set M2 that includes GDP per
capita and number of prior genocides and politicides. The GDP per capita of each
country can be found at www.cia.gov. Of these 25 countries, Israel has the highest GDP
per capita ($19000) while Somalia has the lowest GDP per capita ($550). Due to large
differences in GDP per capita among the countries, it is necessary to scale these values
and place them into several categories. Each GDP value is divided by the maximum
GDP value ($19000) and then classified into one of five categories according to its scaled
GDP value. The categories are given in Table 6.12. For example, the scaled GDP value
of Iraq is 0.1263 (2400/19000), which is given 1 in the transformed modified genocide
and politicide data sets shown in Tables 6.13 and 6.14. The corresponding visual maps
generated by Viscovery are shown in Figures 6.3 and 6.4. The associated significance
values (or cluster indicators) are 62, 38, and 55 for data sets O, M1, and M2 respectively.
We point out that significance values are recommended by Viscovery to help determine
the appropriate number of clusters of a data set. The larger the significance values, the
better the choice of a particular number of clusters. The cluster profiles of the data sets
M1 and M2 are given in Tables 6.15 and 6.16.
112
Country
Name
Prior
Genocides
or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of Regime
Trade
Openness
GDP
per
Capita
(US$)
Iraq Yes High Yes Yes Autocracy Very Low 2400
Afghanistan
(2000) Yes Very High Yes Yes Autocracy Very Low 700
Afghanistan
(2002) Yes Very high No No
No effective
regime Very Low 700
Burma Yes High No Yes Autocracy Very Low 1660
Burundi Yes Very High Yes No Autocracy Low 600
Rwanda Yes High Yes No Autocracy Low 1200
Congo-
Kinshasa Yes Very High Yes No
No effective
regime Medium 610
Somalia Yes Very High No No
No effective
regime Very Low 550
Sierra No Very High Yes No
No effective
regime Low 580
Ethiopia Yes High Yes No Autocracy Medium 750
Uganda Yes High No No Autocracy Low 1260
Algeria Yes Very High No Yes Autocracy Medium 5300
Liberia No High No No Autocracy Low 1100
Pakistan Yes Medium No No Autocracy Low 2100
China Yes Medium No Yes Autocracy Medium 4400
Sri Lanka Yes High No No
Partial
democracy High 3700
Philippines Yes Very High No No Democracy High 4200
Colombia No Very High No No
Partial
democracy Low 6500
Turkey No High No Yes
Partial
democracy Medium 7000
India No High No No Democracy Low 2540
Israel No Very High No Yes Democracy High 19000
Indonesia Yes Medium No No
Partial
democracy Medium 3100
Russia Yes Low No No
Partial
democracy Medium 9300
Nigeria No Low No No
Partial
democracy High 875
Nepal No Medium No No
Partial
democracy Medium 1400
Macedonia No None No No
Partial
democracy High 5000
Table 6.10 Modified genocide and politicide data set 1 (M1).
113
Country Name
Number of
Prior
Genocides
or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of
Regime
Trade
Openness
GDP per
Capita
(US$)
Iraq 2 High Yes Yes Autocracy Very Low 2400
Afghanistan
(2000)
1
Very High Yes Yes Autocracy Very Low 700
Afghanistan
(2002)
1
Very high No No
No effective
regime Very Low 700
Burma 1 High No Yes Autocracy Very Low 1660
Burundi 3 Very High Yes No Autocracy Low 600
Rwanda 2 High Yes No Autocracy Low 1200
Congo-
Kinshasa
2
Very High Yes No
No effective
regime Medium 610
Somalia
1
Very High No No
No effective
regime Very Low 550
Sierra
0
Very High Yes No
No effective
regime Low 580
Ethiopia 1 High Yes No Autocracy Medium 750
Uganda 2 High No No Autocracy Low 1260
Algeria 1 Very High No Yes Autocracy Medium 5300
Liberia 0 High No No Autocracy Low 1100
Pakistan 2 Medium No No Autocracy Low 2100
China 3 Medium No Yes Autocracy Medium 4400
Sri Lanka
1
High No No
Partial
democracy High 3700
Philippines 1 Very High No No Democracy High 4200
Colombia
0
Very High No No
Partial
democracy Low 6500
Turkey
0
High No Yes
Partial
democracy Medium 7000
India 0 High No No Democracy Low 2540
Israel 0 Very High No Yes Democracy High 19000
Indonesia
2
Medium No No
Partial
democracy Medium 3100
Russia
2
Low No No
Partial
democracy Medium 9300
Nigeria
0
Low No No
Partial
democracy High 875
Nepal
0
Medium No No
Partial
democracy Medium 1400
Macedonia
0
None No No
Partial
democracy High 5000
Table 6.11 Modified genocide and politicide data set 2 (M2).
114
Prior Genocides or
Politicides
0 = No; 1 = Yes
Number of Prior
Genocides or Politicides
Actual number of genocides or politicides
Upheaval 0 = None; 1 = Low; 2 = Medium; 3 = High; 4 = Very high
Minority Elite 0 = No; 1 = Yes
Exclusionary Ideology 0 = No; 1 = Yes
Type of Regime
0 = No effective regime; 1 = Autocracy;
2 = Partial democracy; 3 = Democracy
Trade Openness 0 = Very low; 1 = Low; 2 = Medium; 3 = High
GDP per Capita
0 = if the scaled value < 0.1;
1 = if the scaled value >0.1 and < 0.2;
2 = if the scaled value >0.2 and < 0.3;
3 = if the scaled value >0.3 and < 0.4;
4 = if the scaled value >0.4
Table 6.12 Notation of the transformed genocide and politicide data with added
variables.
115
Country Name
Prior
Genocides or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of
Regime
Trade
Openness
GDP per
Capita
(US$)
Iraq 1 3 1 1 1 0 1
Afghanistan
(2000) 1 4 1 1 1 0 0
Afghanistan
(2002) 1 4 0 0 0 0 0
Burma 1 3 0 1 1 0 0
Burundi 1 4 1 0 1 1 0
Rwanda 1 3 1 0 1 1 0
Congo-Kinshasa 1 4 1 0 0 2 0
Somalia 1 4 0 0 0 0 0
Sierra 0 4 1 0 0 1 0
Ethiopia 1 3 1 0 1 2 0
Uganda 1 3 0 0 1 1 0
Algeria 1 4 0 1 1 2 2
Liberia 0 3 0 0 1 1 0
Pakistan 1 2 0 0 1 1 1
China 1 2 0 1 1 2 2
Sri Lanka 1 3 0 0 2 3 1
Philippines 1 4 0 0 3 3 2
Colombia 0 4 0 0 2 1 3
Turkey 0 3 0 1 2 2 3
India 0 3 0 0 3 1 1
Israel 0 4 0 1 3 3 4
Indonesia 1 2 0 0 2 2 1
Russia 1 1 0 0 2 2 4
Nigeria 0 1 0 0 2 3 0
Nepal 0 2 0 0 2 2 0
Macedonia 0 0 0 0 2 3 2
Table 6.13 Transformed modified genocide and politicide data set 1 (M1).
116
Country Name
Number of
Prior
Genocides or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of
Regime
Trade
Openness
GDP per
Capita
(US$)
Iraq 2 3 1 1 1 0 1
Afghanistan
(2000)
1 4 1 1 1 0
0
Afghanistan
(2002)
1 4 0 0 0 0
0
Burma 1 3 0 1 1 0 0
Burundi 3 4 1 0 1 1 0
Rwanda 2 3 1 0 1 1 0
Congo-
Kinshasa 2 4 1 0 0 2 0
Somalia 1 4 0 0 0 0 0
Sierra 0 4 1 0 0 1 0
Ethiopia 1 3 1 0 1 2 0
Uganda 2 3 0 0 1 1 0
Algeria 1 4 0 1 1 2 2
Liberia 0 3 0 0 1 1 0
Pakistan 2 2 0 0 1 1 1
China 3 2 0 1 1 2 2
Sri Lanka 1 3 0 0 2 3 1
Philippines 1 4 0 0 3 3 2
Colombia 0 4 0 0 2 1 3
Turkey 0 3 0 1 2 2 3
India 0 3 0 0 3 1 1
Israel 0 4 0 1 3 3 4
Indonesia 2 2 0 0 2 2 1
Russia 2 1 0 0 2 2 4
Nigeria 0 1 0 0 2 3 0
Nepal 0 2 0 0 2 2 0
Macedonia 0 0 0 0 2 3 2
Table 6.14 Transformed modified genocide and politicide data set 2 (M2).
117
Figure 6.3 Resulting SOM map of the first modified data set (M1).
Figure 6.4 Resulting SOM amp of the second modified data set (M2).
118
Country Name
Prior
Genocides
or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of
Regime
Trade
Openness
GDP per
Capita
US$
Cluster 1
Israel 0 4 0 1 3 3 4
Turkey 0 3 0 1 2 2 3
Colombia 0 4 0 0 2 1 3
Cluster 2
Indonesia 1 2 0 0 2 2 1
Russia 1 1 0 0 2 2 4
Sri Lanka 1 3 0 0 2 3 1
Philippines 1 4 0 0 3 3 2
Cluster 3
Nigeria 0 1 0 0 2 3 0
Nepal 0 2 0 0 2 2 0
Macedonia 0 0 0 0 2 3 2
India 0 3 0 0 3 1 1
Liberia 0 3 0 0 1 1 0
Cluster 4
Pakistan 1 2 0 0 1 1 1
Uganda 1 3 0 0 1 1 0
Afghanistan (2002) 1 4 0 0 0 0 0
Somalia 1 4 0 0 0 0 0
Cluster 5
Algeria 1 4 0 1 1 2 2
China 1 2 0 1 1 2 2
Burma 1 3 0 1 1 0 0
Iraq 1 3 1 1 1 0 1
Afghanistan (2000) 1 4 1 1 1 0 0
Cluster 6
Rwanda 1 3 1 0 1 1 0
Burundi 1 4 1 0 1 1 0
Ethiopia 1 3 1 0 1 2 0
Congo-Kinshasa 1 4 1 0 0 2 0
Sierra 0 4 1 0 0 1 0
Table 6.15 Cluster profiles of the first modified data set (M1).
119
Country Name
Number of Prior
Genocides or
Politicides
Upheaval
Minority
Elite
Exclusionary
Ideology
Type of
Regime
Trade
Openness
GDP per
Capita
US$
Cluster 1
Russia 2 1 0 0 2 2 4
Nigeria 0 1 0 0 2 3 0
Nepal 0 2 0 0 2 2 0
Macedonia 0 0 0 0 2 3 2
Cluster 2
Colombia 0 4 0 0 2 1 3
Israel 0 4 0 1 3 3 4
Turkey 0 3 0 1 2 2 3
India 0 3 0 0 3 1 1
Sri Lanka 1 3 0 0 2 3 1
Philippines 1 4 0 0 3 3 2
Cluster 3
Algeria 1 4 0 1 1 2 2
Iraq 2 3 1 1 1 0 1
Afghanistan
(2000) 1 4 1 1 1 0 0
Burma 1 3 0 1 1 0 0
China 3 2 0 1 1 2 2
Cluster 4
Burundi 3 4 1 0 1 1 0
Rwanda 2 3 1 0 1 1 0
Congo-Kinshasa 2 4 1 0 0 2 0
Sierra 0 4 1 0 0 1 0
Ethiopia 1 3 1 0 1 2 0
Cluster 5
Indonesia 2 2 0 0 2 2 1
Pakistan 2 2 0 0 1 1 1
Uganda 2 3 0 0 1 1 0
Afghanistan
(2002) 1 4 0 0 0 0 0
Liberia 0 3 0 0 1 1 0
Somalia 1 4 0 0 0 0 0
Table 6.16 Cluster profiles of the second modified data set (M2).
120
The number of clusters in each map is either five or six, and the corresponding
significance values are large. This suggests that five or six clusters may be good enough
to represent the actual cluster structure in the data sets. Although visual maps (Figures
6.3 and 6.4) of the two modified data sets (M1, and M2) are different from the map in
Figure 6.2 of the original data set (O), they still have some elements in common.
The cluster made up of China, Algeria, Burma, Iraq, and Afghanistan (2002)
appears in all three maps, as does the cluster of Sierra, Congo, Ethiopia, Burundi, and
Rwanda. Cluster membership of other countries is to some extent similar in these
clustering results. It indicates that the two new variables do not significantly affect the
clustering structure of observed countries. This is possibly due to the small size of the
genocide and politicide data set. The six variables listed in Harff’s work may be enough
to determine the clustering structure of the data set and may also be sufficient to forecast
the future genocides or politicides.
However, in Harff’s listing, Iraq was the country most likely to have future
genocides, or politicides, and followed by Afghanistan, Burma, Burundi, Rwanda, Congo
and Somalia. Algeria and China were not close to Iraq in the list, while in our clustering
results these two countries are always in the same cluster with Iraq and countries such as
Afghanistan and Burma. If from the perspective of clustering, Algeria and China would
be ranked close to Iraq, Burma, and Afghanistan, rather than several ranks down from
them in the list.
There might be several factors causing differences between our clustering results
and Harff’s results. One could be the weight assigned to each variable during
121
preprocessing, where all variables are equally weighted. Another reason might be lack of
further domain knowledge.
Comparing our clustering results with Harff’s work, we conclude that countries
in the same cluster are equally likely to have future genocides or politicides. This is
different from what we see in Tables 6.6 and 6.7, which has the forecasted ordering of
countries for possible future genocides or politicides (Harff, 2003).
Our current result might be helpful to future genocide and politicide research.
The variables ‘GDP per capital’ and ‘number of prior genocides and politicides’ have
been shown to be of little influence in determining the clustering structures of the data
sets, though possibly due to the small size of the data sets. This is not in favor of other
scholars’ suggestion that per capita income is the best indicator of genocides and
politicides, and therefore somehow reinforces Harff’s conclusion regarding identification
of risks factors. Moreover, the listing differences may raise research questions such as if
clustering offers a reasonable alternative view of the genocide and politicide data set, and
to what extent will clustering contribute to forecasting countries’ genocides and
politicides in the coming years.
122
Chapter 7
Self-Organizing Maps: Best Values in Colleges
The prospect of graduating with a lot of debt may be one of the many worries that
concern new college students. It becomes necessary for those college applicants and their
parents to critically screen colleges to see which colleges give students the best value for
their money. Kiplinger’s Personal Finance gave a best value list of 100 public colleges
(Kiplinger, 2003) and a best value list of 100 private colleges (Kiplinger, 2004) that
combine great academics and reasonable costs. In each list, the colleges judged to be the
best value is ranked number one with lower rank values preferred.
As we discussed in previous chapters, SOMs are capable of discovering the
hidden structures to help a decision maker understand a data set. SOMs have been
widely used to cluster data and gain insight into data sets using visual maps. In this
chapter we will apply Viscovery SOMine (2002) to analyze public and private colleges to
determine which colleges are the best values and compare our rankings to those given by
Kiplinger.
123
7.1 Data Description and Preprocessing
We used two data sets from Kiplinger’s Personal Finance (2003 and 2004): a
public college data set and a private college data set. The public college data set contains
the top 100 public colleges considered to be the best values. Similarly, the private
college data set contains the top 100 private colleges considered to be the best values. A
total of 11 variables are included in each data set and seven of them are common to both
data sets. The seven common variables are listed in Table 7.1. The four additional
variables are listed in Table 7.2.
Except for the variable Enrollment, other variables in the two data sets indicate
either the academic quality or the financial cost of a school. Typically, college applicants
look for suitable colleges in terms of both academic quality and financial cost. The
variable Enrollment tells us the number of students who were enrolled in the current
academic year. College applicants can get a rough idea of how many students may be
admitted by each college. However, the variable Enrollment does not provide us the
percentage of applicants who may be admitted – which is known as Admission Rate. The
variable Admission Rate reflects a school’s academic quality and it affects applicants’
decisions. Typically, a low admission rate implies a high academic quality. Compared to
the variable Enrollment, the variable Admission Rate speaks about schools’ academic
quality more directly. Therefore, we included the variable Admission Rate in our
datasets and didn’t include the variable Enrollment. In addition, we did not include the
variable Aid from Grants and Cost after Non-need-based Aid in the private college data.
Values in the Aid from Grants variable tell us that at least 70% students in 83 out of 100
private colleges were awarded aid from grants, and at least 50% students were awarded in
124
Variable Description
Enrollment Number of full-time and part-time undergraduates enrolled at the
college during 2002-2003 academic year
Admission Rate Percentage of applicants who were offered admission
SAT or ACT
Percentage of the 2002-03 freshman class that scored above 600
on the verbal and math parts, separated by slash, or the
percentage that scored above 24 on the ACT
Student/Faculty
Ratio
Average number of students for each faculty member
4-yr. Graduate Rate
Percentage of 1996-97 freshmen who earned a bachelor’s degree
in four years or fewer
6-yr. Graduate Rate
Percentage of 1996-97 freshmen who earned a bachelor’s degree
within six years
Average Debt at
Graduation
Average debt a student accumulates before graduation
Table 7.1 Seven common variables in Kiplinger’s public and private data sets.
Data Set Variable Description
In-State Total
Costs
Overall cost for residents
In-State Costs
After Aid
Overall cost for residents after subtracting
average need-based award
Total Out-of-State
Costs
Overall cost for out-of-state students
Public
College
Out-of-State Costs
after Aid
Overall cost for out-of-state students after
subtracting average need-based award
Total Cost Overall cost for college students
Cost After Need-
based Aid
Total cost in 2003-04 academic year after
subtracting the average need-based award
Aid from Grants
Percentage of the average aid package that came
from grants or scholarships
Private
College
Cost after Non-
need-based Aid
2003-04 cost for a student after subtracting non-
need-based award
Table 7.2 Four additional variables in Kiplinger’s public and private data sets.
125
98% private colleges. These statistics indicate that the majority of private schools are
able to support many of their students. Therefore, the variable Aid from Grants may not
help discover private colleges that are the best values. We did not include the variable
Cost after Non-need-based Aid in the private college dataset because the Cost after Non-
need-based Aid is not available in more than 25% of 100 schools in the data set. Other
variables such as Total Cost and Cost after Need-based Aid provide applicants financial
cost information.
After selecting appropriate variables for each data set, we show the public and
private college data in Tables 7.3 and 7.4, respectively.
126
Rank School Name
SAT or
ACT
(%)
Admis.
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-
State
Total
Costs
($)
In-State
Costs
After
Aid ($)
Total
Out-of-
State
Costs ($)
Out-of-
State
Costs
After
Aid ($)
Avg.
Debt at
Grad.
($)
1
University of North
Carolina at Chapel Hill 65/76 35 14 65 79 11,290 6,673 23,138 16,465 11,156
2 University of Virginia 77/84 39 16 81 91 12,640 8,737 28,610 19,873 13,536
3
College of William
and Mary 83/85 35 12 80 89 13,024 6,668 27,724 21,056 19,762
4 University of Georgia 54/59 65 13 46 66 10,534 5,245 21,310 16,065 12,906
5 University of Florida 58/67 58 21 49 77 10,611 6,125 21,639 15,514 14,449
6
New College of
Florida 93/74 65 11 47 72 10,947 5,613 24,185 18,572 16,645
7
Georgia Institute of
Technology 75/95 59 14 18 69 11,340 6,022 23,266 17,244 17,221
8
University of Illinois
at Urbana, Champaign 79 60 13 52 76 14,410 8,045 25,446 17,401 14,791
9
Truman State
University 82 79 15 39 62 10,609 7,604 14,409 6,805 14,382
10
Virginia Polytechnic
Institute and State
University 39/52 65 15 36 72 10,122 5,613 20,006 14,393 16,229
11
North Carolina State
University 39/60 59 15 25 60 10,688 5,508 22,536 17,028 15,476
12
University of
Delaware 41/53 48 12 54 72 13,416 7,666 22,946 15,280 13,610
13
University of
Wisconsin, Madison 86 71 14 41 77 13,391 7,842 27,401 19,559 15,904
14
University of
Michigan, Ann Arbor 83 49 15 61 82 16,671 8,661 33,473 25,463 16,825
15
University of
California, San Diego 47/73 41 19 43 78 16,000 9,043 24,105 17,148 13,275
16
University of
California, Berkeley 65/78 25 17 48 83 17,265 8,982 25,584 17,301 14,990
17
University of
Washington 42/54 68 11 40 70 13,835 6,926 24,991 18,082 14,500
18
New Mexico Institute
of Mining and
Technology 75 63 13 12 40 9,714 2,708 16,151 9,145 9,500
19
University of
Wisconsin, La Crosse 60 65 21 23 58 10,425 7,327 20,102 12,775 14,306
20
University of Texas at
Austin 51/65 61 19 39 71 14,391 9,111 20,999 11,888 16,400
21
University of
Oklahoma 69 89 21 19 51 10,139 6,618 16,652 10,034 16,886
22 University of Kansas 55 67 19 26 55 9,673 6,266 17,149 10,883 17,347
23
University of North
Carolina at Asheville 48/40 67 14 31 48 8,929 6,308 17,754 11,446 14,547
24
State University of
New York at
Binghamton 51/73 42 19 69 80 13,587 9,214 19,537 10,323 13,915
25
Colorado School of
Mines 89 67 12 31 61 13,780 8,532 26,970 18,438 17,500
26 Auburn University 51 83 16 40 68 10,276 7,308 18,736 11,428 18,585
Table 7.3 Kiplinger’s (2003) public college data.
127
27
Colorado State
University 56 77 17 29 62 10,689 6,700 21,161 14,461 16,042
28 College of New Jersey 61/71 48 12 59 80 16,686 13,915 21,261 7,346 5,490
29
Michigan State
University 57 67 18 31 66 12,743 8,670 22,703 14,033 18,663
30
Appalachian State
University 26/30 64 19 31 60 7,913 4,785 16,834 12,049 13,000
31
Iowa State University
of Science and
Technology 58 89 16 24 62 11,588 8,736 20,930 12,194 17,119
32
State University of
New York College at
Geneseo 65/73 49 19 67 79 12,840 10,840 18,790 7,950 15,000
33
Texas A&M
University 36/49 68 21 27 69 11,899 7,057 18,979 11,922 15,670
34
University of Texas at
Dallas 45/59 53 20 30 53 12,075 7,787 19,155 11,368 NA
35
University of North
Carolina at
Wilmington 18/26 55 16 34 60 9,715 6,375 19,290 12,915 13,583
36
University of
Maryland, College
Park 64/77 43 13 33 63 16,304 12,223 22,881 10,658 15,566
37
University of
California, Los
Angeles 62/75 24 17 40 81 17,616 9,975 26,006 16,031 12,775
38
Louisiana State
University and
Agricultural and
Mechanical College 53 77 21 23 58 10,126 7,406 15,426 8,020 17,569
39
University of
Tennessee 49 58 18 24 56 11,681 6,706 20,763 14,057 21,689
40 University of Iowa 59 84 15 34 64 11,763 9,460 22,055 12,595 15,335
41
Rutgers, The State
University of New
Jersey, New
Brunswick 39/55 55 14 44 72 16,519 10,044 23,033 12,989 15,270
42 Clemson University 44/61 52 16 35 69 14,618 11,121 22,216 11,095 14,347
43
University of Colorado
at Boulder 65 80 16 38 64 11,937 7,877 28,253 20,376 16,737
44
University of
Kentucky 54 82 17 27 58 9,432 5,594 16,112 10,518 NA
45 University of Arkansas 59 86 17 20 45 10,706 7,144 17,456 13,894 14,029
46
Mary Washington
College 62/49 60 17 65 75 12,287 9,033 20,035 11,002 13,100
47
Oklahoma State
University 49 92 19 22 56 10,677 7,478 16,623 9,145 15,580
48
Kansas State
University 49 58 20 18 45 9,626 7,104 16,990 9,886 17,000
49
University of Northern
Iowa 39 80 16 30 64 10,634 7,632 17,592 9,960 15,786
50
University of
Mississippi 46 80 19 29 48 11,257 6,607 16,167 9,560 14,459
51
James Madison
University 33/40 58 17 59 78 13,145 9,320 21,367 12,047 11,786
52
University of
California, Davis 42/62 63 19 28 75 16,521 10,334 23,814 13,480 13,507
Table 7.3 (Continued).
128
53 Miami University 84 77 17 61 80 15,833 12,467 25,603 13,136 17,579
54 Purdue University 29/46 76 16 28 64 14,691 8,723 26,311 17,588 15,677
55
Mississippi State
University 50 74 16 19 48 11,130 8,174 16,036 7,862 15,081
56
University of
Nebraska, Lincoln 55 90 19 15 51 10,687 7,277 18,269 10,992 15,682
57
Florida State
University 34/39 70 22 38 64 10,994 7,035 22,022 14,987 16,372
58
University of
California, Irvine 24/55 57 18 34 72 15,635 8,507 22,450 13,944 12,513
59
University of
Missouri, Columbia 68 88 18 32 65 13,208 8,138 22,655 14,517 17,137
60
University of
Minnesota, Morris 61 82 14 50 76 13,477 8,427 13,477 5,050 9,208
61 University of Alabama 46 85 18 31 61 11,197 8,136 18,357 10,221 18,978
62
University of South
Carolina 29/36 70 17 31 58 11,795 8,794 21,133 12,339 15,260
63
St. Mary's College of
Maryland 67/60 59 12 58 67 16,908 12,908 23,228 10,320 17,125
64
Michigan
Technological
University 68 92 11 22 63 14,135 9,339 25,025 15,686 15,711
65
University of
California, Santa
Barbara 38/55 51 19 44 73 16,154 10,231 24,246 14,015 NA
66
University of
Minnesota, Twin
Cities Campus 64 74 15 17 53 13,910 7,902 25,540 17,638 NA
67
Mississippi University
for Women 56 65 13 21 43 8,649 8,649 13,316 4,667 13,500
68
California Polytechnic
State University, San
Luis Obispo 39/64 39 19 17 66 11,781 10,429 15,268 4,839 12,842
69
University of
Wyoming 42 95 15 22 54 10,863 6,833 16,713 9,880 18,311
70
George Mason
University 26/29 66 16 25 48 11,602 7,597 21,442 13,845 14,143
71
University of Central
Florida 30/36 62 24 25 49 10,839 8,353 21,867 13,514 14,927
72 Ohio State University 69 74 14 25 59 15,249 11,315 25,113 13,798 15,011
73
Illinois State
University 47 81 19 28 55 12,971 7,199 17,441 10,242 13,921
74
University at Buffalo,
The State University
of New York 28/41 61 14 32 56 13,422 9,956 19,372 9,416 16,255
75 Salisbury University 26/35 50 17 50 68 12,895 9,506 19,783 10,277 14,773
76
University of
Massachusetts
Amherst 32/38 58 19 41 61 14,480 9,714 23,333 13,619 15,321
77 University of Vermont 35/38 71 14 48 67 17,116 8,258 30,168 21,910 22,425
78 College of Charleston 47/47 60 14 32 52 14,008 11,214 21,270 10,056 15,135
79
Indiana University
Bloomington 27/33 81 20 40 65 13,129 8,704 24,164 15,460 16,930
80
Pennsylvania State
University University
Park Campus 40/60 57 17 43 80 17,017 12,872 26,639 13,767 17,900
Table 7.3 (Continued).
129
81
State University of
New York at Albany 26/35 56 21 52 66 13,574 9,599 19,524 9,925 15,108
82 University of Arizona 27/33 86 19 29 55 11,163 8,506 19,933 11,427 17,340
83 Towson University 18/26 58 19 30 56 12,776 8,651 20,422 11,771 15,530
84
Rutgers, The State
University of New
Jersey, Camden 19/24 54 11 21 60 16,189 10,111 22,703 12,592 15,223
85
University of
Maryland, Baltimore
County 46/63 63 17 28 53 16,516 12,946 23,368 10,422 14,500
86
University of
Connecticut 33/42 62 17 23 70 14,413 9,141 25,197 16,056 16,093
87
University of
Pittsburgh 48/56 55 17 35 60 17,025 12,723 26,337 13,614 20,154
88
State University of
New York College at
Fredonia 22/25 53 18 47 66 11,782 8,961 17,732 8,771 12,430
89
Stony Brook
University, State
University of New
York 25/50 54 18 30 51 13,645 9,704 19,595 9,891 15,747
90
State University of
New York at New
Paltz 30/29 40 17 21 52 11,276 8,276 16,176 7,900 15,000
91 University of Maine 22/29 79 15 29 56 12,780 8,162 21,480 13,318 17,917
92
University of New
Hampshire 25/32 77 14 48 71 15,779 13,693 26,139 12,446 20,700
93
University of
Missouri, Rolla 83 92 14 10 52 14,326 9,416 23,144 13,728 17,991
94
University of
California, Santa Cruz 37/39 80 19 40 64 16,877 9,309 24,250 14,941 13,282
95 Rowan University 22/33 44 14 37 63 15,416 10,632 20,812 10,180 NA
96
University of Illinois
at Chicago 43 63 15 9 37 14,299 6,399 23,995 17,596 17,000
97 University of Oregon 30/31 86 18 36 59 12,795 9,586 24,231 14,645 22,783
98 Texas Tech University 23/29 69 20 22 51 12,968 9,878 20,048 10,170 13,805
99 Ohio University 49 75 20 43 70 16,514 13,102 24,737 11,635 15,285
100 UC Riverside 16/34 86 19 39 64 16,751 9,932 24,530 17,711 13,226
Table 7.3 (Continued).
130
Rank School Name
Admission
Rate (%)
SAT or
ACT
(%)
Student/
faculty
Ratio
4-Year
Grad.
Rate (%)
6-Year
Grad.
Rate (%)
Total
Costs ($)
Cost After
Need-
Based Aid
($)
Average
Debt at
Graduation
($)
1
California Institute of
Technology 21 99/100 3 71 85 32,682 10,981 10,244
2 Rice University 24 89/92 5 68 89 28,350 14,779 12,705
3 Williams College 23 93/93 8 89 94 36,550 14,737 12,316
4 Swarthmore College 24 94/98 8 86 92 38,676 17,386 12,759
5 Amherst College 18 94/92 9 84 94 38,492 14,453 11,544
6 Webb Institute 42 100/100 7 79 83 8,079 5,579 5,700
7 Yale University 8 96/97 7 88 95 38,432 15,729 19,228
8
Washington and Lee
University 31 89/89 11 86 89 30,225 15,452 15,634
9 Harvard University 11 90*/90* 8 86 97 38,831 17,456 10,465
10 Stanford University 13 93/95 7 77 93 38,875 17,746 15,782
11 Princeton University 11 95/97 5 91 97 40,169 18,325 12,000
12
Massachusetts
Institute of
Technology 16 95/100 6 82 91 39,213 19,609 22,855
13 Pomona College 23 98/97 9 83 88 38,130 17,411 15,600
14 Emory University 42 89/94 7 82 87 37,272 19,657 17,675
15 Columbia University 12 91/93 7 83 93 39,493 17,778 15,331
16 Duke University 25 91/94 11 88 93 40,080 19,996 20,025
17 Davidson College 34 86/89 10 89 91 34,706 21,455 13,697
18 Wellesley College 47 88/89 9 84 88 37,419 17,526 15,697
19 Vassar College 31 93/89 9 81 87 37,870 19,404 17,170
20 Haverford College 32 89/90 8 89 92 38,928 17,826 15,253
21
Northwestern
University 33 88/92 7 83 92 38,817 20,376 14,551
22 Bowdoin College 25 87/92 10 83 90 38,663 17,773 15,307
23
University of
Pennsylvania 21 91/96 6 83 91 39,040 20,596 20,247
24
Johns Hopkins
University 35 85/93 8 81 88 39,188 19,142 13,600
25 Cooper Union 14 81/83 7 57 78 14,652 11,167 9,250
26
Washington
University 24 93/98 7 75 86 39,253 20,700 NA
27 Dartmouth College 23 92/96 9 87 95 38,898 19,546 NA
28
Claremont McKenna
College 28 89/95 7 82 86 37,730 17,988 16,914
29
University of Notre
Dame 34 83/91 12 88 95 35,392 18,011 25,595
30 Colgate University 34 80/86 10 85 89 38,820 18,856 12,984
31 The Colorado College 53 66/70 9 72 79 35,275 16,516 13,500
32
University of
Richmond 41 74/83 10 79 84 31,679 17,588 16,115
33
Georgetown
University 21 87/89 11 86 91 39,182 24,382 20,000
34 Brown University 17 86/90 8 79 94 40,248 20,838 21,700
35 Carleton College 35 88/89 9 82 86 35,288 21,677 14,543
36 Lafayette College 36 63/78 11 79 84 35,713 15,147 17,380
Table 7.4 Kiplingers’ (2004) private college data.
131
37 Middlebury College 27 93/96 11 81 87 39,532 18,288 21,751
38 Grinnell College 65 87/85 10 78 84 31,460 16,585 13,854
39
Illinois Wesleyan
University 48 96 12 76 81 30,780 18,858 17,722
40 Bates College 28 90/91 10 82 87 38,932 18,258 17,045
41 Cornell University 29 85/92 9 82 90 38,974 23,122 15,587
42 Wesleyan University 28 89/92 9 76 81 39,127 21,401 23,753
43 Colby College 33 84/89 11 85 88 38,699 18,168 17,270
44 Bucknell University 39 72/84 12 83 87 36,165 19,165 16,000
45 Kenyon College 52 87/81 9 80 84 36,273 17,905 20,850
46 Centre College 78 89 11 71 73 28,529 15,842 14,300
47 Rhodes College 70 95 11 71 73 30,080 18,899 15,100
48 Macalester College 44 87/88 10 71 77 32,847 16,394 NA
49 Barnard College 34 88/88 10 72 84 37,940 17,826 14,030
50 Brandeis University 42 88/88 8 79 85 39,101 22,257 NA
51
College of the Holy
Cross 43 71/76 11 88 90 36,851 23,846 16,063
52 Harvey Mudd College 37 97/100 9 75 83 38,880 22,041 20,219
53
Wake Forest
University 41 79/86 10 77 87 36,079 21,196 24,769
54 Bryn Mawr College 50 86/75 9 76 80 37,890 18,609 NA
54 Wheaton College 54 84/83 11 70 84 27,076 17,341 15,864
55 Tufts University 27 81/90 9 81 88 39,173 20,115 15,499
56 Oberlin College 33 87/80 10 63 76 37,688 21,081 13,926
57
Mount Holyoke
College 52 80/70 10 75 79 38,668 19,268 14,200
58 Furman University 58 65/70 11 74 81 29,430 16,296 17,741
59 St. Olaf College 73 84 13 71 75 29,879 17,458 18,806
60
Brigham Young
University 73 86 18 31 73 9,663 7,621 11,000
61 Lehigh University 44 59/85 11 70 84 35,670 19,123 16,972
62 Smith College 53 72/66 9 76 80 37,937 18,466 19,911
63 Beloit College 70 82 11 60 72 30,264 17,452 14,942
64 Taylor University 78 74 15 71 75 24,723 15,678 15,117
65 Union College 45 56/71 11 75 80 36,455 18,431 15,725
66 Hamilton College 35 77/82 10 79 84 38,463 19,474 16,856
67 DePauw University 61 55/60 11 75 79 32,150 15,531 14,481
68 Hillsdale College 82 74 11 53 71 23,353 13,853 14,500
69 Knox College 72 72 12 67 74 30,894 15,494 16,920
70
University of Southern
California 30 79/91 10 51 73 37,968 21,606 20,619
71 Trinity College 36 73/79 9 77 83 38,890 19,667 17,000
72 Trinity University 69 71/80 11 65 75 27,086 16,706 NA
73
Gustavus Adolphus
College 77 70 13 72 75 27,820 17,609 17,400
74 Vanderbilt University 46 83/90 9 78 84 38,847 20,971 24,023
75 Whitman College 50 80/81 10 60 71 33,776 21,176 15,000
76 Scripps College 58 85/78 12 63 68 36,500 17,984 12,941
77
Franklin and Marshall
College 62 62/71 11 78 83 36,580 20,925 19,656
Table 7.4 (Continued).
132
78 Saint Louis University 72 72 12 52 67 29,983 16,902 14,989
79
Carnegie Mellon
University 38 75/95 11 61 77 38,460 24,689 19,195
80 Lawrence University 68 83 11 58 68 32,875 17,882 18,311
81 Connecticut College 35 83/84 11 75 81 37,057 16,930 17,250
82
Case Western Reserve
University 78 74/85 8 49 75 32,802 18,323 21,830
84 Dickinson College 51 65/64 13 74 78 36,600 19,753 17,586
85 Kalamazoo College 73 91 12 60 69 30,917 17,947 20,000
86
Saint John's
University 87 66 13 67 74 27,272 19,544 20,680
87 Boston College 34 78/85 13 0 86 37,745 24,470 16,732
88 Reed College 55 95/86 10 45 67 37,900 18,804 16,758
89 Bard College 36 85/67 9 59 71 38,282 20,558 15,400
90
University of
Rochester 50 79/87 12 65 76 37,246 20,297 NA
91 New York University 28 87/86 11 65 74 40,105 28,282 21,495
92 Villanova University 47 57/74 13 79 84 36,560 26,463 28,217
93 Skidmore College 46 70/69 11 71 75 38,838 21,023 15,560
94
Rose-Hulman Institute
of Technology 65 62/91 13 58 71 32,625 28,677 27,000
95 St. John's College 71 95/75 8 63 71 36,635 21,940 20,753
96 Babson College 48 50/77 13 77 81 38,443 21,316 NA
97
Rhode Island School
of Design 32 49/55 11 0 87 34,472 26,447 21,125
98
Rensselaer
Polytechnic Institute 70 68/92 17 48 75 39,200 22,360 24,590
99
Sarah Lawrence
College 40 79/44 6 51 66 42,121 22,847 14,864
100
The George
Washington
University 40 68/72 14 62 73 40,240 25,866 NA
Table 7.4 (Continued).
As shown in Tables 7.3 and 7.4, there are some missing values, denoted by NA or
“0”, in the Average Debt columns. For example, there are five NAs in the Average Debt
column in the public college data and nine “0”s in the same column in the private college
data. Values in the SAT or ACT fields are inconsistent with values in other fields
because of their value format. For example, the SAT or ACT value for George
Washington University in the private college data (see Table 7.4) is 68/72, which is not a
single value needed for a SOM method.
133
To meet the numerical input requirements of an SOM method, the data sets have
to be preprocessed. In the Average Debt column, we used the average value of the
column to replace the missing values. For example, we replaced the missing values with
the average value of 15480 in the public college data. The average value of 16957 was
used to replace the missing values in the private college data. In the SAT or ACT
column, many entries that are not in a single value format (i.e., percentages above 600 on
the verbal and math parts of SAT are separated by a slash). We used the average
percentage instead of the two individual percentages as input. For example, for George
Washington University, the average percentage value of 70 for the two individual
percentages, that is, 68 and 72, in the SAT or ACT column was used as input. After
preprocessing data in the two data sets, we applied Viscovery on the public and private
college data sets to see which groups of colleges are real bargains.
7.2 Discussion of Results
The Viscovery SOM map of the public college data is given in Figure 7.1. There
are three clusters separated by solid lines. The summary of the clusters is given in Table
7.5. Cluster membership of each public college is given in Table 7.6.
Cluster A has 18 schools. The average SAT or ACT percentage of schools in
cluster A is 70%, which is the highest among three clusters, indicating that these schools
have a good academic atmosphere. Other academic quality variables provide the same
insight. For example, schools in cluster A have the lowest average admission rate (53%),
the highest average student/faculty ratio (15), the highest average 4-year and 6-year
134
Figure 7.1 Map for the public college data set.
Cluster
SAT
or
ACT
(%)
Admis.
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-State
Total
Costs ($)
In-State
Costs
After Aid
($)
Total Out-
of-State
Costs ($)
Out-of-State
Costs After
Aid ($)
Avg. Debt
at Grad.
($)
A 70 53 15 49 75 13,657 7,605 25,845 18,432 15,500
B 48 62 16 43 69 15,238 10,682 22,737 12,158 14,915
C 47 72 17 27 57 11,469 7,737 19,384 11,634 15,798
Table 7.5 Summary of clusters of the public colleges.
135
Rank
School
Name
SAT
or
ACT
(%)
Admis..
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-
State
Total
Costs
($)
In-State
Costs
After Aid
($)
Total
Out-of-
State
Costs ($)
Out-of-
State
Costs
After
Aid ($)
Avg.
Debt at
Grad.
($)
Cluster A
1
University of
North Carolina
at Chapel Hill 71 35 14 65 79 11,290 6,673 23,138 16,465 11,156
2
University of
Virginia 81 39 16 81 91 12,640 8,737 28,610 19,873 13,536
3
College of
William and
Mary 84 35 12 80 89 13,024 6,668 27,724 21,056 19,762
4
University of
Georgia 57 65 13 46 66 10,534 5,245 21,310 16,065 12,906
5
University of
Florida 63 58 21 49 77 10,611 6,125 21,639 15,514 14,449
6
New College
of Florida 84 65 11 47 72 10,947 5,613 24,185 18,572 16,645
7
Georgia
Institute of
Technology 85 59 14 18 69 11,340 6,022 23,266 17,244 17,221
8
University of
Illinois at
Urbana,
Champaign 79 60 13 52 76 14,410 8,045 25,446 17,401 14,791
12
University of
Delaware 47 48 12 54 72 13,416 7,666 22,946 15,280 13,610
13
University of
Wisconsin,
Madison 86 71 14 41 77 13,391 7,842 27,401 19,559 15,904
14
University of
Michigan, Ann
Arbor 83 49 15 61 82 16,671 8,661 33,473 25,463 16,825
15
University of
California, San
Diego 60 41 19 43 78 16,000 9,043 24,105 17,148 13,275
16
University of
California,
Berkeley 72 25 17 48 83 17,265 8,982 25,584 17,301 14,990
17
University of
Washington 48 68 11 40 70 13,835 6,926 24,991 18,082 14,500
25
Colorado
School of
Mines 89 67 12 31 61 13,780 8,532 26,970 18,438 17,500
37
University of
California, Los
Angeles 69 24 17 40 81 17,616 9,975 26,006 16,031 12,775
43
University of
Colorado at
Boulder 65 80 16 38 64 11,937 7,877 28,253 20,376 16,737
77
University of
Vermont 37 71 14 48 67 17,116 8,258 30,168 21,910 22,425
Average 70 53 15 49 75 13,657 7,605 25,845 18,432 15,500
Table 7.6 Cluster profiles of the public colleges.
136
Rank
School
Name
SAT or
ACT
(%)
Admis.
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-
State
Total
Costs
($)
In-State
Costs
After Aid
($)
Total
Out-of-
State
Costs ($)
Out-of-
State
Costs
After
Aid ($)
Avg.
Debt at
Grad.
($)
Cluster B
20
University of
Texas at
Austin 58 61 19 39 71 14,391 9,111 20,999 11,888 16,400
24
State
University of
New York at
Binghamton 62 42 19 69 80 13,587 9,214 19,537 10,323 13,915
28
College of
New Jersey 66 48 12 59 80 16,686 13,915 21,261 7,346 5,490
32
State
University of
New York
College at
Geneseo 69 49 19 67 79 12,840 10,840 18,790 7,950 15,000
36
University of
Maryland,
College Park 71 43 13 33 63 16,304 12,223 22,881 10,658 15,566
41
Rutgers, The
State
University of
New Jersey,
New
Brunswick 47 55 14 44 72 16,519 10,044 23,033 12,989 15,270
42
Clemson
University 53 52 16 35 69 14,618 11,121 22,216 11,095 14,347
46
Mary
Washington
College 56 60 17 65 75 12,287 9,033 20,035 11,002 13,100
51
James
Madison
University 37 58 17 59 78 13,145 9,320 21,367 12,047 11,786
52
University of
California,
Davis 52 63 19 28 75 16,521 10,334 23,814 13,480 13,507
53
Miami
University 84 77 17 61 80 15,833 12,467 25,603 13,136 17,579
54
Purdue
University 38 76 16 28 64 14,691 8,723 26,311 17,588 15,677
58
University of
California,
Irvine 40 57 18 34 72 15,635 8,507 22,450 13,944 12,513
60
University of
Minnesota,
Morris 61 82 14 50 76 13,477 8,427 13,477 5,050 9,208
63
St. Mary's
College of
Maryland 64 59 12 58 67 16,908 12,908 23,228 10,320 17,125
65
University of
California,
Santa Barbara 47 51 19 44 73 16,154 10,231 24,246 14,015 15,480
74
University at
Buffalo, The
State
University of
New York 35 61 14 32 56 13,422 9,956 19,372 9,416 16,255
Table 7.6 (Continued).
137
Rank
School
Name
SAT or
ACT
(%)
Admis.
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-
State
Total
Costs
($)
In-State
Costs
After Aid
($)
Total
Out-of-
State
Costs ($)
Out-of-
State
Costs
After
Aid ($)
Avg.
Debt at
Grad.
($)
Cluster B (Continued)
76
University of
Massachusetts
Amherst 35 58 19 41 61 14,480 9,714 23,333 13,619 15,321
78
College of
Charleston 47 60 14 32 52 14,008 11,214 21,270 10,056 15,135
79
Indiana
University
Bloomington 30 81 20 40 65 13,129 8,704 24,164 15,460 16,930
80
Pennsylvania
State
University
University
Park Campus 50 57 17 43 80 17,017 12,872 26,639 13,767 17,900
84
Rutgers, The
State
University of
New Jersey,
Camden 22 54 11 21 60 16,189 10,111 22,703 12,592 15,223
85
University of
Maryland,
Baltimore
County 55 63 17 28 53 16,516 12,946 23,368 10,422 14,500
86
University of
Connecticut 38 62 17 23 70 14,413 9,141 25,197 16,056 16,093
87
University of
Pittsburgh 52 55 17 35 60 17,025 12,723 26,337 13,614 20,154
92
University of
New
Hampshire 29 77 14 48 71 15,779 13,693 26,139 12,446 20,700
94
University of
California,
Santa Cruz 38 80 19 40 64 16,877 9,309 24,250 14,941 13,282
95
Rowan
University 28 44 14 37 63 15,416 10,632 20,812 10,180 15,480
99
Ohio
University 49 75 20 43 70 16,514 13,102 24,737 11,635 15,285
100 UC Riverside 25 86 19 39 64 16,751 9,932 24,530 17,711 13,226
Average 48 62 16 43 69 15,238 10,682 22,737 12,158 14,915
Table 7.6 (Continued).
138
Rank
School
Name
SAT or
ACT
(%)
Admis.
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-
State
Total
Costs
($)
In-State
Costs
After Aid
($)
Total
Out-of-
State
Costs ($)
Out-of-
State
Costs
After
Aid ($)
Avg.
Debt at
Grad.
($)
Cluster C
9
Truman State
University 82 79 15 39 62 10,609 7,604 14,409 6,805 14,382
10
Virginia
Polytechnic
Institute and
State
University 46 65 15 36 72 10,122 5,613 20,006 14,393 16,229
11
North
Carolina State
University 50 59 15 25 60 10,688 5,508 22,536 17,028 15,476
18
New Mexico
Institute of
Mining and
Technology 75 63 13 12 40 9,714 2,708 16,151 9,145 9,500
19
University of
Wisconsin, La
Crosse 60 65 21 23 58 10,425 7,327 20,102 12,775 14,306
21
University of
Oklahoma 69 89 21 19 51 10,139 6,618 16,652 10,034 16,886
22
University of
Kansas 55 67 19 26 55 9,673 6,266 17,149 10,883 17,347
23
University of
North
Carolina at
Asheville 44 67 14 31 48 8,929 6,308 17,754 11,446 14,547
26
Auburn
University 51 83 16 40 68 10,276 7,308 18,736 11,428 18,585
27
Colorado
State
University 56 77 17 29 62 10,689 6,700 21,161 14,461 16,042
29
Michigan
State
University 57 67 18 31 66 12,743 8,670 22,703 14,033 18,663
30
Appalachian
State
University 28 64 19 31 60 7,913 4,785 16,834 12,049 13,000
31
Iowa State
University of
Science and
Technology 58 89 16 24 62 11,588 8,736 20,930 12,194 17,119
33
Texas A&M
University 43 68 21 27 69 11,899 7,057 18,979 11,922 15,670
34
University of
Texas at
Dallas 52 53 20 30 53 12,075 7,787 19,155 11,368 15,480
35
University of
North
Carolina at
Wilmington 22 55 16 34 60 9,715 6,375 19,290 12,915 13,583
Table 7.6 (Continued).
139
Rank
School
Name
SAT or
ACT
(%)
Admis.
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-
State
Total
Costs
($)
In-State
Costs
After Aid
($)
Total
Out-of-
State
Costs ($)
Out-of-
State
Costs
After
Aid ($)
Avg.
Debt at
Grad.
($)
Cluster C (Continued)
38
Louisiana
State
University and
Agricultural
and
Mechanical
College 53 77 21 23 58 10,126 7,406 15,426 8,020 17,569
39
University of
Tennessee 49 58 18 24 56 11,681 6,706 20,763 14,057 21,689
40
University of
Iowa 59 84 15 34 64 11,763 9,460 22,055 12,595 15,335
44
University of
Kentucky 54 82 17 27 58 9,432 5,594 16,112 10,518 15,480
45
University of
Arkansas 59 86 17 20 45 10,706 7,144 17,456 13,894 14,029
47
Oklahoma
State
University 49 92 19 22 56 10,677 7,478 16,623 9,145 15,580
48
Kansas State
University 49 58 20 18 45 9,626 7,104 16,990 9,886 17,000
49
University of
Northern Iowa 39 80 16 30 64 10,634 7,632 17,592 9,960 15,786
50
University of
Mississippi 46 80 19 29 48 11,257 6,607 16,167 9,560 14,459
55
Mississippi
State
University 50 74 16 19 48 11,130 8,174 16,036 7,862 15,081
56
University of
Nebraska,
Lincoln 55 90 19 15 51 10,687 7,277 18,269 10,992 15,682
57
Florida State
University 37 70 22 38 64 10,994 7,035 22,022 14,987 16,372
59
University of
Missouri,
Columbia 68 88 18 32 65 13,208 8,138 22,655 14,517 17,137
61
University of
Alabama 46 85 18 31 61 11,197 8,136 18,357 10,221 18,978
62
University of
South
Carolina 33 70 17 31 58 11,795 8,794 21,133 12,339 15,260
64
Michigan
Technological
University 68 92 11 22 63 14,135 9,339 25,025 15,686 15,711
66
University of
Minnesota,
Twin Cities
Campus 64 74 15 17 53 13,910 7,902 25,540 17,638 15,480
67
Mississippi
University for
Women 56 65 13 21 43 8,649 8,649 13,316 4,667 13,500
68
California
Polytechnic
State
University,
San Luis
Obispo 52 39 19 17 66 11,781 10,429 15,268 4,839 12,842
Table 7.6 (Continued).
140
Rank
School
Name
SAT or
ACT
(%)
Admis.
Rate
Student/
faculty
Ratio
4-Year
Grad.
Rate
(%)
6-Year
Grad.
Rate
(%)
In-
State
Total
Costs
($)
In-State
Costs
After Aid
($)
Total
Out-of-
State
Costs ($)
Out-of-
State
Costs
After
Aid ($)
Avg.
Debt at
Grad.
($)
Cluster C (Continued)
69
University of
Wyoming 42 95 15 22 54 10,863 6,833 16,713 9,880 18,311
70
George
Mason
University 28 66 16 25 48 11,602 7,597 21,442 13,845 14,143
71
University of
Central
Florida 33 62 24 25 49 10,839 8,353 21,867 13,514 14,927
72
Ohio State
University 69 74 14 25 59 15,249 11,315 25,113 13,798 15,011
73
Illinois State
University 47 81 19 28 55 12,971 7,199 17,441 10,242 13,921
75
Salisbury
University 31 50 17 50 68 12,895 9,506 19,783 10,277 14,773
81
State
University of
New York at
Albany 31 56 21 52 66 13,574 9,599 19,524 9,925 15,108
82
University of
Arizona 30 86 19 29 55 11,163 8,506 19,933 11,427 17,340
83
Towson
University 22 58 19 30 56 12,776 8,651 20,422 11,771 15,530
88
State
University of
New York
College at
Fredonia 24 53 18 47 66 11,782 8,961 17,732 8,771 12,430
89
Stony Brook
University,
State
University of
New York 38 54 18 30 51 13,645 9,704 19,595 9,891 15,747
90
State
University of
New York at
New Paltz 30 40 17 21 52 11,276 8,276 16,176 7,900 15,000
91
University of
Maine 26 79 15 29 56 12,780 8,162 21,480 13,318 17,917
93
University of
Missouri,
Rolla 83 92 14 10 52 14,326 9,416 23,144 13,728 17,991
96
University of
Illinois at
Chicago 43 63 15 9 37 14,299 6,399 23,995 17,596 17,000
97
University of
Oregon 31 86 18 36 59 12,795 9,586 24,231 14,645 22,783
98
Texas Tech
University 26 69 20 22 51 12,968 9,878 20,048 10,170 13,805
Average 47 72 17 27 57 11,469 7,737 19,384 11,634 15,798
Table 7.6 (Continued).
141
graduation rates (49% and 75%) among three clusters. The financial cost is another
important concern for students as to choosing schools of the best values. Although the
average total costs for out-of-state students of cluster A schools (i.e., Total Out-of-State
Cost and Out-of-State Costs after Aid) are the highest, when cluster A schools are
compared to cluster B schools and cluster C schools, the average In-State Total Costs
($13657) is the second highest and the average In-State Costs after Aid ($7605) is the
lowest. For cluster A, average of the variable Average Debt at Graduation ($15500) is
the second highest and it is only about $600 greater than the lowest average debt
($14915) of cluster B schools. Therefore, cluster A schools are the group of schools that
not only have excellent education quality but also are financially comparable for students.
Perhaps those in-state students who have excellent academic performance are more
willing to consider cluster A schools because of the lowest average In-State Costs after
Aid.
Cluster C has 52 schools. The overall academic quality of schools in this cluster
is worse than schools in the other two clusters. For example, the average SAT or ACT
percentage of 47% is the lowest, the 4-year and 6-year graduation rates are the lowest,
i.e., 27% and 57% respectively, and the Student/faculty Ratio of 17 is the highest.
However, the average total costs for both residents and non-residents are the lowest
among the three groups. The average In-State Total Costs is $11469, which is about
$2200 less than cluster A schools.
The average Out-of-State Total Costs for cluster B schools is $19384, which is
about $6500 less than that of schools in cluster A. The average total costs after aid for in-
142
state and out-of-state students are also low. It tells that cluster C schools may be good
choices for those students who care more about financial costs.
Cluster B has 30 colleges that have good academic quality, i.e., the second highest
average values in SAT/ACT percentage, Admission Rate, Student/faculty Ratio, 4-year
Graduation Rate, and 6-year Graduation Rate. This cluster of schools has the highest
average costs ($15238) for in-state students, the second highest average costs ($22737)
for non-resident students, and the lowest Average Debt at Graduation ($14915). Schools
in the cluster B may be considered as alternatives to schools in cluster A and schools in
cluster C because they are comparable to cluster A schools in terms of good education
quality and similar to cluster C schools in terms of less expensive financial costs.
We did not include the Kiplinger’s rank variable to generate our SOM maps. The
Rank variable is used as school labels in the maps. Fifteen schools in cluster A are from
the top 25 public schools in Kiplinger’s list, such as University of North Carolina at
Chapel Hill, University of Virginia, College of William & Mary, University of Georgia
(Table 7.6). Two (i.e., UCLA and University of Colorado-Boulder) of the rest schools
come from the middle range, which is between No.26 and No.75, and the last one
(University of Vermont) is ranked No.77. Cluster A schools have the highest education
quality and require reasonable financial expenditures. Cluster B schools have better
academic quality and higher financial expenditures for students than cluster C schools.
When looking at individual schools, we found some schools belonging to the
same cluster have different ranks in Kiplinger’s list. For example, University of
Colorado at Boulder and Colorado School of Mines are in the same cluster A. They have
comparable education quality and the financial costs in these two schools are close. In
143
Kiplinger’s ranking, University of Colorado ranks No.43, which is 18 places lower than
Colorado school of Mines whose rank is No.25. Additional examples can be found in
cluster B schools and cluster C schools. For example, University of California at
Riverside and University of Clemson are members of cluster B. However, UC Riverside
is ranked No.100 and Clemson is ranked No.42. Although there are some differences
between them, most of their educational and financial measures are close. For example,
their 4-year and 6-year graduation rates and Average Debt at Graduation are close.
Therefore University of Clemson and UC Riverside should have close rankings. In
cluster C, University of Maine and Iowa State University have the same situation. Maine
is ranked No.31 while Iowa State is ranked No.91 despite their close educational and
financial qualities as shown in Table 7.6.
The Viscovery SOM map for the private colleges is given in Figure 7.2. There
are five clusters and the summary of the clusters is given in Table 7.7. Cluster
memberships of colleges are provided in Table 7.8.
Cluster A has two member colleges: Webb Institute (6) and Copper Union (25).
Both colleges have excellent educational qualities: the lowest average Student/faculty
Ratio (7), the highest average SAT or ACT percentage (91%) among the five clusters,
and the highest average 4-year and 6-year graduation rates (68% and 81% respectively)
as well. In addition, the financial costs in both colleges are very low. The average total
cost of both schools is $11366, less than half of the second lowest average total cost
($31335) of cluster E schools. The average total cost after need-based aid is the lowest
($8373) among five clusters. The average debt at graduation is the lowest ($7475), which
144
Figure 7.2 Map for the private college data set.
Cluster
Admission
Rate (%)
SAT or
ACT
(%)
Student/
faculty
Ratio
4-Year
Grad. Rate
(%)
6-Year
Grad. Rate
(%)
Total Costs
($)
Total Costs after
Need-based Aid
($)
Avg. Debt at
Grad. ($)
A 28 91 7 68 81 11,366 8,373 7,475
B 27 91 8 82 90 38,051 18,949 16,914
C 46 77 11 77 82 35,574 18,567 16,608
D 43 76 13 47 78 37,486 25,429 21,770
E 65 78 11 61 73 31,115 17,749 16,387
Table 7.7 Summary of clusters of the public colleges.
145
Rank School Name
Admiss.
Rate (%)
SAT or
ACT %
Student/
faculty
Ratio
4-Year
Grad.
Rate %
6-Year
Grad.
Rate %
Total
Costs
Cost After
Need- Based
Aid
Average
Debt
Cluster A
6 Webb Institute 42 100 7 79 83 8,079 5,579 5,700
25 Cooper Union 14 82 7 57 78 14,652 11,167 9,250
Average 28 91 7 68 81 11366 8373 7475
Cluster B
1
California Institute of
Technology 21 100 3 71 85 32,682 10,981 10,244
2 Rice University 24 91 5 68 89 28,350 14,779 12,705
3 Williams College 23 93 8 89 94 36,550 14,737 12,316
4 Swarthmore College 24 96 8 86 92 38,676 17,386 12,759
5 Amherst College 18 93 9 84 94 38,492 14,453 11,544
7 Yale University 8 97 7 88 95 38,432 15,729 19,228
9 Harvard University 11 90 8 86 97 38,831 17,456 10,465
10 Stanford University 13 94 7 77 93 38,875 17,746 15,782
11 Princeton University 11 96 5 91 97 40,169 18,325 12,000
12
Massachusetts Institute
of Technology 16 98 6 82 91 39,213 19,609 22,855
13 Pomona College 23 98 9 83 88 38,130 17,411 15,600
14 Emory University 42 92 7 82 87 37,272 19,657 17,675
15 Columbia University 12 92 7 83 93 39,493 17,778 15,331
16 Duke University 25 93 11 88 93 40,080 19,996 20,025
17 Davidson College 34 88 10 89 91 34,706 21,455 13,697
18 Wellesley College 47 89 9 84 88 37,419 17,526 15,697
19 Vassar College 31 91 9 81 87 37,870 19,404 17,170
20 Haverford College 32 90 8 89 92 38,928 17,826 15,253
21 Northwestern University 33 90 7 83 92 38,817 20,376 14,551
22 Bowdoin College 25 90 10 83 90 38,663 17,773 15,307
23
University of
Pennsylvania 21 94 6 83 91 39,040 20,596 20,247
24
Johns Hopkins
University 35 89 8 81 88 39,188 19,142 13,600
26 Washington University 24 96 7 75 86 39,253 20,700 16,957
27 Dartmouth College 23 94 9 87 95 38,898 19,546 16,957
28
Claremont McKenna
College 28 92 7 82 86 37,730 17,988 16,914
29
University of Notre
Dame 34 87 12 88 95 35,392 18,011 25,595
30 Colgate University 34 83 10 85 89 38,820 18,856 12,984
33 Georgetown University 21 88 11 86 91 39,182 24,382 20,000
34 Brown University 17 88 8 79 94 40,248 20,838 21,700
35 Carleton College 35 89 9 82 86 35,288 21,677 14,543
Table 7.8 Cluster profiles of the private college data.
146
Rank School Name
Admiss.
Rate (%)
SAT or
ACT %
Student/
faculty
Ratio
4-Year
Grad.
Rate %
6-Year
Grad.
Rate %
Total
Costs
Cost After
Need- Based
Aid
Average
Debt
Cluster B (Continued)
37 Middlebury College 27 95 11 81 87 39,532 18,288 21,751
40 Bates College 28 91 10 82 87 38,932 18,258 17,045
41 Cornell University 29 89 9 82 90 38,974 23,122 15,587
42 Wesleyan University 28 91 9 76 81 39,127 21,401 23,753
43 Colby College 33 87 11 85 88 38,699 18,168 17,270
50 Brandeis University 42 88 8 79 85 39,101 22,257 16,957
52 Harvey Mudd College 37 99 9 75 83 38,880 22,041 20,219
53 Wake Forest University 41 83 10 77 87 36,079 21,196 24,769
56 Tufts University 27 86 9 81 88 39,173 20,115 15,499
75 Vanderbilt University 46 87 9 78 84 38,847 20,971 24,023
Average 27 91 8 82 90 38,051 18,949 16,914
Cluster C
8
Washington and Lee
University 31 89 11 86 89 30,225 15,452 15,634
31 The Colorado College 53 68 9 72 79 35,275 16,516 13,500
32 University of Richmond 41 79 10 79 84 31,679 17,588 16,115
36 Lafayette College 36 71 11 79 84 35,713 15,147 17,380
38 Grinnell College 65 86 10 78 84 31,460 16,585 13,854
39
Illinois Wesleyan
University 48 96 12 76 81 30,780 18,858 17,722
44 Bucknell University 39 78 12 83 87 36,165 19,165 16,000
45 Kenyon College 52 84 9 80 84 36,273 17,905 20,850
48 Macalester College 44 88 10 71 77 32,847 16,394 16,957
49 Barnard College 34 88 10 72 84 37,940 17,826 14,030
51
College of the Holy
Cross 43 74 11 88 90 36,851 23,846 16,063
54 Bryn Mawr College 50 81 9 76 80 37,890 18,609 16,957
55 Wheaton College 54 84 11 70 84 27,076 17,341 15,864
58 Mount Holyoke College 52 75 10 75 79 38,668 19,268 14,200
62 Lehigh University 44 72 11 70 84 35,670 19,123 16,972
63 Smith College 53 69 9 76 80 37,937 18,466 19,911
66 Union College 45 64 11 75 80 36,455 18,431 15,725
67 Hamilton College 35 80 10 79 84 38,463 19,474 16,856
72 Trinity College 36 76 9 77 83 38,890 19,667 17,000
78
Franklin and Marshall
College 62 67 11 78 83 36,580 20,925 19,656
82 Connecticut College 35 84 11 75 81 37,057 16,930 17,250
84 Dickinson College 51 65 13 74 78 36,600 19,753 17,586
93 Skidmore College 46 70 11 71 75 38,838 21,023 15,560
96 Babson College 48 64 13 77 81 38,443 21,316 16,957
Average 46 77 11 77 82 35,574 18,567 16,608
Table 7.8 (Continued).
147
Rank School Name
Admiss.
Rate (%)
SAT or
ACT %
Student/
faculty
Ratio
4-Year
Grad.
Rate %
6-Year
Grad.
Rate %
Total
Costs
Cost After
Need- Based
Aid
Average
Debt
Cluster D
71
University of Southern
California 30 85 10 51 73 37,968 21,606 20,619
80
Carnegie Mellon
University 38 85 11 61 77 38,460 24,689 19,195
87 Boston College 34 82 13 0 86 37,745 24,470 16,732
91 New York University 28 87 11 65 74 40,105 28,282 21,495
92 Villanova University 47 66 13 79 84 36,560 26,463 28,217
94
Rose-Hulman Institute of
Technology 65 77 13 58 71 32,625 28,677 27,000
97
Rhode Island School of
Design 32 52 11 0 87 34,472 26,447 21,125
98
Rensselaer Polytechnic
Institute 70 80 17 48 75 39,200 22,360 24,590
100
The George Washington
University 40 70 14 62 73 40,240 25,866 16,957
Average 43 76 13 47 78 37,486 25,429 21,770
Table 7.8 (Continued).
148
Rank School Name
Admiss.
Rate (%)
SAT or
ACT %
Student/
faculty
Ratio
4-Year
Grad.
Rate %
6-Year
Grad.
Rate %
Total
Costs
Cost After
Need- Based
Aid
Average
Debt
Cluster E
46 Centre College 78 89 11 71 73 28,529 15,842 14,300
47 Rhodes College 70 95 11 71 73 30,080 18,899 15,100
57 Oberlin College 33 84 10 63 76 37,688 21,081 13,926
59 Furman University 58 68 11 74 81 29,430 16,296 17,741
60 St. Olaf College 73 84 13 71 75 29,879 17,458 18,806
61
Brigham Young
University 73 86 18 31 73 9,663 7,621 11,000
64 Beloit College 70 82 11 60 72 30,264 17,452 14,942
65 Taylor University 78 74 15 71 75 24,723 15,678 15,117
68 DePauw University 61 58 11 75 79 32,150 15,531 14,481
69 Hillsdale College 82 74 11 53 71 23,353 13,853 14,500
70 Knox College 72 72 12 67 74 30,894 15,494 16,920
73 Trinity University 69 76 11 65 75 27,086 16,706 16,957
74
Gustavus Adolphus
College 77 70 13 72 75 27,820 17,609 17,400
76 Whitman College 50 81 10 60 71 33,776 21,176 15,000
77 Scripps College 58 82 12 63 68 36,500 17,984 12,941
79 Saint Louis University 72 72 12 52 67 29,983 16,902 14,989
81 Lawrence University 68 83 11 58 68 32,875 17,882 18,311
83
Case Western Reserve
University 78 80 8 49 75 32,802 18,323 21,830
85 Kalamazoo College 73 91 12 60 69 30,917 17,947 20,000
86 Saint John's University 87 66 13 67 74 27,272 19,544 20,680
88 Reed College 55 91 10 45 67 37,900 18,804 16,758
89 Bard College 36 76 9 59 71 38,282 20,558 15,400
90 University of Rochester 50 83 12 65 76 37,246 20,297 16,957
95 St. John's College 71 85 8 63 71 36,635 21,940 20,753
99 Sarah Lawrence College 40 62 6 51 66 42,121 22,847 14,864
Average 65 78 11 61 73 31,115 17,749 16,387
Table 7.8 (Continued).
149
is nearly $9000 less than the second lowest average debt at graduation ($16387) of the
cluster E. Therefore, cluster A schools can be considered as schools of great value.
Cluster B has 40 schools. From the average Admission Rate, the average SAT or
ACT, and other academic-related variables (i.e., SAT/ACT, Student/Faculty Ratio, 4-year
and 6-year Graduation Rates), we see that most of schools in cluster B have excellent
education quality. All of Ivy League schools are included. Most of them are financially
expensive, which is shown by the highest average Total Costs ($38051) (see Table 7.7)
among the five clusters. However, if the need-based aid is taken into account, the
average total cost is reduced to $18949 and the average Average Debt at Graduation is
$16914, which makes cluster B schools comparable to schools in clusters C, D, and E. If
students care more about educational quality, then schools in cluster B are worthwhile.
There are 24 colleges in cluster C. Compared to cluster B schools, schools in
cluster C are financially less expensive. For example, the average Total Cost ($35574) is
about $2400 less than that of cluster B schools. However, the education quality of
schools in this cluster is not as exceptional as schools in cluster B. For example, in
cluster C, the average SAT or ACT percentage is less than the average SAT or ACT
percentage of cluster B by 14 points. In addition, the Student/faculty Ratio and, the 4-
year and 6-year graduation rates, are all lower than those of cluster B. Therefore, cluster
C schools provide relatively good academic quality and ask for reasonable financial
sacrifice.
There are nine schools in cluster D. The overall academic quality of these schools
is not as good as the overall academic quality of schools in Cluster C, while the financial
costs of cluster D schools are much higher than the financial costs of cluster C. As
150
shown in Table 7.7, the average academic-related measures are slightly lower than those
of cluster C schools. For example, the average 6-year graduation rate of cluster D (78%)
is a little worse than the average 6-year graduation of cluster C (82%). The average
financial costs in cluster D are the highest among the five clusters. For example, the
average Cost after Need-based Aid of this cluster is $25429 which is about $6500 higher
than the second highest of cluster B ($18949). In terms of educational quality and
financial consideration, schools in cluster D might be considered less competitive than
schools in clusters A, B, and C.
Cluster E has 25 colleges. Compared to the other four clusters, the academic
quality of schools in this cluster is not as good as schools in other four clusters. The
average 4-year and 6-year graduation rates are the lowest among five clusters. However,
the average financial cost of schools in cluster E is the second lowest. Since the
academic quality of cluster E is close to or slightly worse than that of cluster D and the
financial cost of cluster E is much lower than that of cluster D, schools in cluster E might
be considered alternatives of schools in cluster D for students who have financial
concerns.
Although our analysis of cluster structures agrees with Kiplinger’s list in most
cases, there are some discrepancies, especially when examining individual schools within
each cluster. For example, in cluster B, Vanderbilt University has similar academic and
financial measures as Wake Forest (53) and Wesleyan (42). Vanderbilt is ranked No. 75
in Kiplinger’s list, where Vanderbilt’s Kiplinger rank is 23 places and 32 places lower
than Wake Forest and Wesleyan, respectively. Barnard College (49) and Connecticut
College (82) in cluster C is another example. These two schools have comparable
151
academic measures and close financial measures (see Table 7.8). However, the
difference between their Kiplinger’s ranks is 47. This kind of information can not be
discovered on Kiplinger’s list. With the help of the SOM visual maps, alternatives can
easily be found by examining a school’s neighbors.
Our results of the public and private college data sets have shown that
Viscovery’s SOM map helps identify alternatives of a particular school, which may not
be detected from Kiplingers’ rankings. For example, Figure 7.1 gives us a visual map of
the hidden cluster structure of the public college data. On this map, three clusters are
clearly visible, where alternatives of a school can be easily recognized. Colorado State
(27) has six neighbors: Virginia Polytechnic Institute and State University (10), North
Carolina State University (11), Auburn University (26), Michigan State University (29),
Iowa State University (31), and University of Missouri at Columbia (59). If we look for
Colorado State’s alternatives solely on the Kiplinger’s list, we are very likely to think
about schools with close ranks to Colorado State as its alternatives. However, not every
alternative has a close rank to Colorado State, such as North Caroline State. Therefore,
Viscovery can help college students identify alternatives, gain more insights from the
data, and facilitate them to make a better decision.
The differences between our results and Kiplinger’s list do not mean that the
Kiplinger’s list is of no value, although our analysis shows that the Kiplinger’s list can be
misleading. If we could combine the results obtained from our maps and Kiplinger’s
rankings, we might have better way to analyze and explain the data set to help students.
152
Chapter 8
Summary and Future Work
In order to visualize a data set with an asymmetric distance matrix, the standard
SM method takes as inputs the symmetrized distance matrix by averaging entries in the
asymmetric distance matrix. This approach is the simplest way to represent asymmetric
data onto a 2-dimentional map. However, some interesting information hidden in
asymmetric data may be ignored due to averaging. Merino et al.’s method introduces
into the standard SM method an asymmetry coefficient that is expected to reflect
asymmetric information. However, when the data set under consideration is not very
asymmetric, the asymmetry coefficient defined by Merino et al. has little influence on the
resulting map. The map obtained from Merino’s method is very similar to the one
obtained from the standard SM method. Our modified SM method takes into account the
upper triangular part and the lower triangular part of an asymmetric distance matrix
simultaneously. It is reasonable to expect that the modified SM method may outperform
the standard and Merino’s method to some extents.
We applied the modified method to two asymmetric data sets: American college
selection data and Canadian ranked college data. From the results obtained on these two
data sets, we found that the modified SM method always did a fairly good job at reducing
153
distance errors and performed reasonably well at preserving order relationships at least
comparable to the standard and Merino’s methods. Since asymmetric proximity data
arise in other research areas such as marketing, psychology, sociology, etc, one research
problem could be how to use our modified SM method to visualize these data sets. If
such maps could be generated with reasonable interpretability, they might be used to
discover relationships between data items that may hardly be detected by other methods,
and therefore assist the analysis of the asymmetric data sets in different research and
business disciplines.
In terms of helping detect hidden structures, clustering has been widely used in
the applications of data visualization. We have found that the clustering procedures in
Viscovery outperformed the K-means clustering method and the classic SOM method.
We have applied Viscovery to the state-sponsored murder data set and got some
interesting results. Meanwhile, through analyzing 200 public and private colleges with
Viscovery, we have generated several clusters for the public college data set and the
private college data set, respectively. These college clustering results are not quite
similar to Kiplinger’s results. In practice, there are lots of ranking lists trying to give
readers the idea of which is the best/worst or which is most likely to happen. However,
the ranking lists such as Kiplinger’s list do not include alternative information that
readers want to look for. Maybe in addition to their original ranking lists, SOM maps
showing the clustering information should be included as well to better deliver useful
information to readers to help make their decisions.
154
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