Interactive Visual Summarization for Visualizing Large, Multidimensional Datasets

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Abstract

KOCHERLAKOTA, SARAT MOHAN. Interactive Visual Summarization for Visualizing Large Multidimensional Datasets. (Under the direction of Christopher G. Healey.)

Because of its ability to help users analyze and explore data from a diverse set of domains,

visualization is becoming integral to the knowledge discovery process. However, existing

vi-sualization techniques for displaying large, multidimensional datasets often produce detailed,

cluttered images that overwhelm the user’s ability to effectively absorb the underlying data.

To visualize such datasets effectively we have developed a visual summarization framework

that intelligently summarizes datasets by extracting its important and relevant characteristics

prior to visualization. The summaries are then visualized both in place of the original data, or

along with the original data. Our approach performs this summarization in three broad steps.

First, size and dimensionality of the data are reduced meaningfully. Next, patterns and

de-pendencies in the form of association rules, along with outliers are extracted from the reduced

data. Finally, these summary characteristics are visualized using techniques that are aimed at

enhancing the comprehension of the data. Summary characteristics, as well as summarization

steps are also recorded.

Our framework is designed to harness the benefits of both visual and non-visual methods

to intuitively guide users to produce relevant data summaries. Initial results in applying our

approach to practical datasets suggest that our approach could be used to generate effective

visual summaries of large, multidimensional datasets from a wide variety of domains and

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INTERACTIVE VISUAL SUMMARIZATION

FOR VISUALIZING LARGE

MULTIDIMENSIONAL DATASETS

by

SARAT M. KOCHERLAKOTA

A dissertation submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the Degree of

Doctor of Philosophy

COMPUTER SCIENCE

Raleigh, North Carolina

2006

APPROVED BY:

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BIOGRAPHY

Sarat Mohan Kocherlakota was born to Mohan Kocherlakota and Girija Kocherlakota, in

Bom-bay (now Mumbai), India. He received a Bachelor of Engineering degree in Computer Science

and Engineering from the Mahatma Gandhi Mission’s College of Engineering and Technology,

Kamothe, India after which he worked for a few months with Larsen and Toubro, Bombay,

In-dia, and later also worked for a few months with an animation and special effects studio, The

Fix, Bombay, India. Sarat joined the computer science masters program at North Carolina

State University in the spring of 2000. He received his M.S. in computer science in the fall of

2002 and since the spring of 2003 has been enrolled in the computer science doctoral program

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ACKNOWLEDGMENTS

I have been looking forward to this moment when I can say “thank you” to all those people

who have contributed to my progress thus far. I also realize that it is perhaps impossible to

acknowledge each and every one of those very important people in my life while also ensuring

that I don’t leave anyone out. But I will plow on nevertheless.

I would like to thank, first and foremost, Dr. Christopher Healey without whose guidance

and support none of this would have been possible. Even though I never thought I would be

saying this, I know that I truly will miss working for him. I will never be able to thank him

enough for all that he’s done for me.

Dr. Thomas Honeycutt has been a pillar of support, inspiration and guidance to me during

my stay at NC State. I would like to especially thank him for motivating me to stay on for the

Ph.D. program on completion of my M.S. degree.

I would also like to thank Dr. Xiaosong Ma and Dr. Rada Chirkova for agreeing to be in

my advisory committee. Their queries and valuable comments have been extremely useful in

guiding my Ph.D. research in the right direction.

I would like to thank my parents Mohan and Girija for all the things great, responsible

parents do for thankless children like me. They’ve always supported me selflessly and

uncon-ditionally and have always encouraged me to pursue my dreams. I would also like to thank my

sister Mamata and her husband Prabhakar for having supported me in so many ways during my

entire stay at North Carolina State University. Also, I would like to thank them for bringing

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these past five years.

I cannot thank Prasad and Kumari Vanguri enough for providing me support, and for always

ensuring that I never miss home too much during my time here in Raleigh, North Carolina.

You cannot go too far in life without the support of good friends. I must be truly blessed

because I have several. My lab-mates, especially Brent, Laura, Amit, Thomas, Lloyd and

Ping-Lin, among several others, have helped me immeasurably in getting thus far. Over the years

Subhayu and Ravi have been a great source of support to me, as indeed have Raoul, Micah and

Alex.

Sibin, Dhruba, Meeta, Nikola, Salil, Radha, Saurabh, Ajit, Surendra, and all the other

regulars at Cup-A-Joe, have over the last couple of years given me a reason to look forward to

spending my evenings in the company of great friends. Their support has helped make all of

this possible.

I am also a very lucky person for the friendship and support of old friends, especially that

of Pranav Mehta. I cannot imagine how boring life would have been without him around,

as indeed without the presence of wonderful friends such as Brijesh, Gurmeet, Venky, Lalit,

Shaibal, Meghna, Soumya and Dipti, among several others.

My lovely wife Sally Chopra has made life so much more richer, wonderful, interesting and

meaningful over the past several years that we have been together. She has been my staunch

companion, especially through the toughest of journeys that I have undertaken in these last few

years. Her unflinching love and support has provided me great strength and has made

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my life.

I am also thankful for the opportunity and privilege of studying at North Carolina State

University. I would especially like to thank the tremendous faculty and staff of the department

of computer science at NC State. In particular, I would like to thank Dr. David Thuente and

Margery Page, for their help and guidance over these past several years.

I know there are several others that I have left out here. It is to them and all the others

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List of Figures

1.1 Multidimensional Visualization Example . . . 4

1.2 Multidimensional Visualization Example . . . 5

2.1 Parallel coordinates visualization example . . . 31

2.2 Star Coordinates element layout example . . . 33

2.3 Self Organizing Maps example . . . 34

2.4 Design Gallery layout example . . . 37

2.5 A RuleViz visualization of association rules . . . 46

3.1 Attribute Viz Display . . . 54

3.2 Initial Layout . . . 59

3.3 Final Layout . . . 62

3.4 Attribute Cluster Display . . . 63

4.1 Attribute Viz Display . . . 72

4.2 Outlier Detection Results . . . 79

5.1 Summary View Display . . . 84

6.1 Temperature . . . 97

6.2 Precipitation . . . 97

6.3 frost days . . . 98

6.4 Vapor Pressure . . . 98

6.5 Cloud Cover . . . 99

6.6 wet days . . . 99

6.7 Diurnal Range . . . 100

6.8 Weather Dataset Scaling Viz Display . . . 101

6.9 Weather Dataset Attribute Partitions Display . . . 103

6.10 Cluster 1 RuleView . . . 105

6.11 Cluster 2 and 3 RuleView . . . 106

6.12 Precipitation Outlier Values . . . 107

6.13 Cluster 1 Summary View Display . . . 109

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6.15 Cluster 1 Summary View 3D Display . . . 111

6.16 Auction ID . . . 116

6.17 User ID . . . 116

6.18 Price . . . 117

6.19 Scaling Viz . . . 117

6.20 Rule View 1 . . . 119

6.21 Auction ID Outlier . . . 121

6.22 Cluster 1 Summary View 3D . . . 122

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Chapter 1 Introduction

Visualization is the area of computer science that primarily deals with the representation of

data in the form of images. These images provide users the ability to analyze and explore

information contained in large, complex datasets. An effective visualization aids in the rapid

and accurate identification of important data aspects such as patterns, relationships,

dependen-cies, boundaries, gaps, anomalies and other characteristics embedded within the underlying

data [16]. Studies in interactive data analysis and exploration have supported the notion that

visualization helps in increased cognition and understanding [5, 9, 22, 37]. These

capabili-ties of visualization systems have contributed greatly towards the growing interest in the area.

Visualization is regarded as an integral part of the knowledge discovery process [6].

The widespread use of visualization has made it ubiquitous in our daily lives. Daily weather

reports broadcast by various television stations rely on visualizations to convey weather related

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commu-nity in analyzing and presenting their data. More and more visualization techniques are being

utilized to assist humans analyze and explore datasets from a wide variety of applications

and domains. Some of these include such varied areas as meteorology, medical diagnosis,

e-commerce applications, network monitoring and intrusion detection, among several others.

Visualization has become an invaluable aid in our quest for managing and comprehending the

large volumes of information that are being generated on a daily basis in today’s information

age.

However, the ever-increasing sizes and dimensionalities of real-world datasets are placing

current visualization techniques under more and more duress [16, 37, 41]. Current techniques

largely tackle the size and dimensionality issue by increasing the amount of information

dis-played within the images. The visual detail contained within these images however can often

overwhelm the user’s ability to accurately comprehend the underlying information. These

im-ages also suffer from cluttering, occlusion and visual interference problems, all of which

dimin-ish the user’s ability to meaningfully absorb the underlying information. Furthermore,

process-ing and renderprocess-ing these large, multi-dimensional datasets requires tremendous computational

effort. Current techniques are thus finding it increasingly difficult to produce visualizations

of large, multi-dimensional datasets that remain coherent, meaningful, relevant, and accurate,

while simultaneously ensuring that the viewers’ cognitive abilities are not overwhelmed.

To address this problem we have proposed and developed a visual summarization approach

aimed at handling such large, multi-dimensional datasets more effectively. Our approach

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abstrac-tions of the original data; abstracabstrac-tions that include only its important characteristics while

excluding extraneous information. Visualizing these summaries could then help reduce the

amount of visual detail displayed to users while also highlighting the underlying

characteris-tics of the dataset. Such an approach could potentially lead to more effective comprehension

of the data.

1.1 Motivation

Figure 1.1 illustrates an example of visualizing a multidimensional dataset from the

Intergov-ernmental Panel on Climate Change (IPCC). It comprises of average monthly weather

condi-tions recorded over the years 1961 to 1990. This visualization is an instance of a glyph based

visualization technique in which individual data elements (i.e. weather readings) are

visual-ized using small 2D glyphs. The features of the glyphs such as color, size and texture, and

their individual properties are used to represent different data attributes and their values. In the

image, hue is used to represent temperature; hot temperatures are displayed using red glyphs,

and cold temperatures are shown using blue glyphs. Size represents pressure; large glyphs

rep-resent high pressure areas, and small glyphs show low pressure. Luminance is used to display

cloud coverage; dark glyphs denote low cloud coverage and bright glyphs denote high

cover-age areas. Orientation is used to display precipitation values; vertical glyphs represent little or

no rainfall, and horizontal glyphs denote high rainfall areas. Finally, wind speed is shown using

the density of the glyphs; high numbers of glyphs packed together denote high wind speed, and

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Figure 1.1: A visualization of a weather dataset using perceptual texture elements with cloud coverage→ lumi-nance, temperature→hue, pressure→size, precipitation→orientation, and wind speed→density

Visualizations such as these harness the ability of the low level human visual system to

absorb large quantities of visual information rapidly and effectively thereby aiding in the

ef-fective comprehension of the underlying data. However, exploratory data analysis of large,

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Figure 1.2: A visualization of a weather dataset of the entire earth

1.1.1 Size and Dimensionality Issues

Current multi-dimensional visualization techniques, including glyph-based techniques,

pixel-based techniques, parallel coordinates visualization, scatter-plot matrices among others, are

designed to visualize up to only a few million data elements and a handful of attributes at a

time [41]. Real-world datasets generated from across a wide variety of applications and

do-mains frequently contain millions (if not billions) of elements where each element can have

several hundred attributes. Directly visualizing such large datasets by displaying each and

ev-ery individual element generally results in highly detailed images that can seem vev-ery cluttered.

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can also occlude one another [11]. An example of the problems associated with directly

visu-alizing a large dataset, particularly cluttering and overcrowding, is illustrated in the figure 1.2.

The figure represents a visualization of the weather dataset of the entire globe within a single

image.

In addition to the problem posed by large sizes of datasets, effectively handling

multi-dimensionality can also be a difficult task. Glyph-based techniques such as the one shown in

figure 1.1 for instance display multi-dimensional datasets by assigning data attributes to visual

features such as color, texture, motion and flicker among others. This approach is effective

when visualizing a low-dimensional dataset. However, visualizing a high-dimensional dataset

within a single image would involve assigning additional visual features and their properties

to represent the large number of attributes. This could lead to visual interference which occurs

when one visual feature interferes with the user’s cognition of the properties of another visual

feature and thereby negatively impacts the comprehension of the underlying attribute values.

Cluttering, occlusion and visual interference are all problems that could hamper the

compre-hension of the underlying data. Additionally, the computational effort required in displaying

large, high-dimensional datasets can put a strain on the processing and rendering abilities of

visualization systems. Ideally, visualizations must be free from these problems in order to be

effective.

Hence, the basic problem that we attempt to address using our summarization approach can

be stated as: “How do we coherently and effectively display large, high-dimensional datasets

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Visualization studies generally address the problem of large sizes and high dimensionality

in two ways. The first approach is aimed at enhancing the graphical or rendering abilities of

current techniques to try and display more and more data within the images. These approaches

do not adequately focus on effective data preparation prior to visualizing the data. The second

approach focuses more on the data preparation or pre-processing performed prior to displaying

the data. Effective data pre-processing could involve meaningfully transforming the data that

is to be visualized to ease the burden posed by large, multi-dimensional datasets on current

visualization systems. Such an approach has not been adequately addressed in visualization

studies.

1.2 Summarization

In our visual summarization approach, we focus on investigating methods for effective

pre-processing of the data prior to visualization. Specifically, we are interested in generating

sum-maries of the datasets which can then be used to comprehend the original data more effectively

and efficiently.

A summary can be described as a concise representation of a body of information. In our

pre-processing approach we focus on meaningfully abstracting the important aspects of the

data within a summary form prior to generating a final visualization of the dataset. One of

the main objectives of our approach is to produce a summary that contains only the important

characteristics of the original data and excludes extraneous information. Generating a

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data. Once the summaries are generated they can be visualized in place of the original data or

visualized along with the original data.

Generating summaries of a dataset prior to visualizing it has two possible advantages:

1. Reduce the amount of data to be visualized:

In our approach visual summaries are generated by reducing both the size and the

di-mensionality of the original data. The reduction also involves preventing extraneous and

irrelevant details from being incorporated within the summary. Such a reduction could

potentially reduce the number of data elements as well as attributes that are visualized.

This in turn could potentially minimize the effects of cluttering, occlusion and visual

interference.

2. Highlight important data characteristics:

Our summarization involves extracting high level characteristics of the original dataset.

Some of these characteristics include high-relevance attributes, patterns and

relation-ships, and outliers. Visualizing these summary characteristics could help users analyze

the underlying data more effectively and efficiently when compared to direct

visualiza-tion techniques.

1.2.1 Automated Analysis Techniques

In order to generate summaries we are interested in extracting important data characteristics

such as high-relevance elements and attributes subsets, patterns and relationships, and outliers.

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the techniques that we are interested in include multi-dimensional scaling,k-means clustering,

association rule mining and outlier detection. However, directly applying these techniques to

the datasets may not be the best approach for the following reasons:

1. Unsuited to high-dimensional datasets: Automated methods are often better suited to

low-dimensional datasets. The algorithmic complexities associated with handling

high-dimensional datasets can make these techniques computationally inefficient for handling

such datasets.

2. Lack of user involvement: Automated techniques generally function as “black boxes”:

the data analysis performed is not visible and only final results are displayed to users.

Additionally, these techniques do not provide users with the ability to participate in or

interact with the analysis process. The inability to display intermediate steps and their

results could make it difficult for users to understand the analysis process. Also if the

user is unsatisfied with the results, any modifications made to the algorithms are not

apparent until the whole process is repeated and the final results are re-displayed. The

techniques themselves could seem complicated to most users.

Clearly, using these techniques directly for summarization purposes may not be the most

ef-fective approach. A more meaningful approach is required whereby complementary analysis

techniques are chained together as a sequence of analysis steps and augmented by the use of

interactive visualization techniques. One possible approach to making the pre-processing more

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the intermediate steps of the process. This could help users understand the summarization

pro-cess as well as the data itself more clearly. User interaction with the individual steps of the

process could help guide the process towards results that are relevant to users.

1.3 Research Objectives / Goals

Our primary objective in designing a visual summarization approach is to present large,

multi-dimensional datasets to users in a meaningful fashion without overwhelming the user’s ability

to comprehend the information. Specifically we are interested in the following objectives:

1. Effective data pre-processing focused on meaningfully reducing datasets:

For effective data pre-processing purposes, we investigate the use of automated

analy-sis techniques to extract important data characteristics such as high relevance attributes,

patterns, relationships and outliers from the original data. These extracted

characteris-tics can then be used to generate summaries of the data, which can then eventually be

visualized in place of the original data.

2. Integrating automated non-visual algorithms with visualization techniques:

Since automated techniques tend to be non-intuitive and complicated for most users, our

approach is focused more on making the summarization process intuitive by combining

both visual and non-visual techniques in a complementary fashion. Using interactive

visualization techniques, we aim to provide users the ability to interact with and view the

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3. Visualizing the generated summaries:

Once the summary features are extracted, we investigate methods by which the

sum-maries can be visualized coherently. The generated sumsum-maries could be used both in

place of the original dataset or to augment the original data. Visualizing the summaries

in place of the original dataset is aimed at reducing the visual detail presented to the

user while ensuring that the important aspects are presented to the user. Visualizing the

summary along with the original data is aimed at utilizing the summary information to

highlight important characteristics of the underlying data.

4. Managing summaries and summary operators:

We are also interested in storing the summaries, as well as the sequence of summarization

steps applied for possible use with other similar datasets.

In the next section, we briefly discuss our visual summarization approach aimed at achieving

the aforementioned objectives.

1.4 Visual Summarization

Our summarization approach is focused on the pre-processing phase of the visualization

pro-cess. To make the pre-processing more intuitive and interactive, our visual summarization

approach combines the use of both automated analysis techniques and interactive visualization

techniques. The automated techniques are used to extract important data characteristics which

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display intermediate results to users and also to allow users to modify individual parameters

of the summarization steps. Displaying results of the intermediate summarization steps could

provide useful insight into the dataset as well as the summarization process itself without

hav-ing to wait for final results of the analysis. Combinhav-ing both visual and non-visual analysis

techniques is thus aimed at harnessing the advantages of both techniques to produce relevant

summaries.

1.4.1 Data Reduction

We begin our visual summarization by reducing the size and dimensionality of the dataset to

manageable proportions. This reduction is conducted in a serial fashion with size reduction

performed first followed by attribute partitioning.

In the size reduction step attribute values for each attribute are discretized into a

user-defined set of ranges. Ranges are then categorized as dense or sparse based on the number

of data elements that fall into these ranges. User-specified thresholds are used to define

den-sity and sparden-sity of individual ranges. This information is visualized to provide users with a

better understanding of the size reduction steps and the effects of interacting with this step

of the summarization process. Elements belonging to at least one dense range or one sparse

range for any attribute are retained and the remaining elements not satisfying this condition are

discarded. Our purpose for retaining dense elements (identified by dense value ranges) is that

they are more likely to combine with other dense ranges from other attributes to produce strong

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of association mining. Sparse elements (identified by sparse value ranges) are retained as they

are more likely to be identified as outliers with respect to the other elements.

In the attribute partitioning phase, we focus on partitioning the set of attributes into smaller

attribute subsets or clusters in which the attributes belonging to a particular cluster exhibit

sim-ilar behavior. Each of these clusters can be seen as a reduced view of the dataset. To determine

similarities between the attributes they are first compared in a pair-wise fashion. The pair-wise

correlation measures are then used to construct a spatial layout in which individual attributes

are represented by points in a 2D space. The purpose behind generating such a layout is to find

an optimal arrangement of the points such that the proximity of these points in this space

re-sembles, as closely as possible, the degree of correlation between the corresponding attributes.

Multidimensional scaling is utilized for this purpose to generate a layout representing

similari-ties between a set of attributes. Partitions are then generated by applying clustering algorithms

to this layout.

1.4.2 Behavior Characterization

Once the strongly correlated attribute clusters are defined, the clusters are then examined

sep-arately. For each of the clusters, both strong relationships in the form of association rules, as

well as outliers are identified.

For discovering association rules we utilize an Apriori-based association rule mining

algo-rithm. Association rule mining is a powerful technique for extracting relationships and patterns

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both its efficiency and effectiveness are reduced. Our approach attempts to counter this

prob-lem in two ways. First, the discretization of the attribute values in the previous phase helps

reduce the number of individual attribute values that the algorithm needs to track. Second,

ap-plying the association mining technique to the lower-dimensional attribute clusters separately

reduces the complexity of the technique and potentially improves its efficiency as opposed to

applying it to the entire set of attributes.

To detect outliers we use a “densities of local neighborhoods” technique [7]. In our

ap-proach we apply this technique to only those elements that were identified as very sparse in the

first phase. These sparse elements are also referred to as “potential outliers”. We then verify if

the potential outliers are truly outlying with respect to their distances from their neighbors. If

the potential outlier is found to be much farther away from its neighbors when compared to the

distance of its neighbors from one another, the potential outlier is then classified as an actual

outlier.

1.4.3 Summary Visualization

In the summary visualization phase we display each attribute cluster separately to allow users

to assimilate information relevant to each cluster independently. We visualize the summary of

the dataset in two ways. In the first method we only visualize the generated summary instead

of visualizing the entire dataset. For each attribute cluster, both strong patterns represented

by association rules, as well as the outliers that were detected, are displayed using

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with the original data. Here, in order to reduce the amount of visual detail displayed we only

visualize the most important relationships and outliers for each attribute cluster.

1.4.4 Summary Management

The summary management phase essentially involves storing the summary information

ex-tracted from the dataset. This involves storing information relevant to the individual attributes,

as well as information associated with each attribute cluster. Attribute cluster information

in-cludes both association rules as well as outliers. Additionally, we also store the summary steps

or “operators” applied during the summarization process and the order in which they were

applied.

1.5 Contributions of Our Research

Using our visual summarization research, we hope to make the following contributions to the

area of visualization:

1. A new method for summarizing large, multi-dimensional datasets. Our emphasis is on

generating summaries at a greater level of detail compared to existing techniques. The

detailed summaries contain information such as high relevance attribute and attribute

values, patterns and relationships, and outliers.

2. A summarization technique combining both automated, mathematical analysis methods

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fashion.

3. Techniques for visualizing both extracted data characteristics as well as the individual

steps of the summarization process using current visualization techniques.

4. A method to store the summaries generated as well as summary operators applied during

the summarization process.

1.6 Organization

The following chapters in this thesis are organized as follows. The next chapter (Chapter 2)

deals with related work. Chapters 3 to 5 deal with the individual phases of our summarization

approach. Chapter 3 describes our data reduction phase. The behavior characterization phase is

discussed in Chapter 4. The summary visualization phase and the summary management phase

are dealt with in Chapter 5. In Chapter 6, we discuss a few examples of applying our visual

summarization approach to practical datasets. Finally, in Chapter 7 we discuss the conclusions

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Chapter 2 Related Work

As stated earlier our basic premise is that large, high-dimensional datasets could potentially be

visualized more meaningfully and coherently using summaries generated from these datasets.

An important goal of our summarization approach is to ensure that the summaries generated are

relevant to the user. In order to achieve this objective we are interested in the use of both visual

and non-visual techniques for extracting and displaying summary information in an intuitive

and efficient fashion.

Summaries can be described as concise representations of data consisting of their most

important and relevant aspects or features. A quick summary of a dataset could be generated

by computing simple statistical measures such as the mean, median, variance and standard

deviation values. The advantages of using these simple measures to summarize data is that

they are widely understood and can be computed efficiently for small datasets. To summarize

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in-depth description of the dataset and its features is required. Some of the features or aspects

that are vital to summarizing the data could include high relevance attributes, patterns and

relationships, and outliers, among others.

In this chapter we will discuss some of the related techniques that can be used for data

summarization purposes. Specifically we discuss techniques for pattern detection and

visu-alization, clustering, outlier detection and relevant attribute identification. We also discuss

techniques focused on effective data pre-processing.

2.1 Pattern Detection

Patterns play an important part in describing the behavior of a dataset. They represent

rela-tionships between data elements and attributes and provide useful insight to users about their

data.

Decision tree-based techniques, such as ID3 and C4.5 [35] have traditionally been

em-ployed for data classification purposes. The basic purpose behind these techniques involves

learning patterns from a training set using heuristic methods and then applying the patterns (or

classification rules as they are more popularly referred to as) to predict the class of previously

unclassified data elements. The training setT is usually a small subset of the original dataset

Dwith one important difference: inT, the class value for each element is known in advance.

The resulting rules are then used to predict the class of unclassified elements fromD. The

per-formance of such classification algorithms is indicated by their classification accuracy, which

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The process of generating classification rules involves the following broad steps:

1. Generating decision trees: Decision trees are a special kind of tree structures in which

each of the non-leaf nodes (also referred to as the decision nodes) represent an

attribute-value pair and the leaf nodes are represented by classes or categories assigned to the

elements in the training set T. All decision nodes at the same tree-depth belong to the

same attribute. The process of generating a decision tree involves recursively

partition-ingT into smaller and smaller disjoint subsets using a single attribute at a time. At each

step, the attribute Ai ∈ T that has the highest probability of distributing T evenly is

chosen and each of its distinct values or value ranges are used to form a separate node.

Each node represents a subset of the data and is connected to the child nodes that

poten-tially partition its subset evenly. In this way, each attribute is utilized only once for the

partitioning. The partitioning is continued until all attributes have been utilized and the

leaf nodes of the decision tree represent data classes or categories.

2. Generating rules:

Rules can be generated by traversing all individual paths within the tree. These rules

are generally of the form X _⇒ Y, whereX denotes a single attribute-value pair or a

combination of such pairs andY denotes a category or class value.

The above method learns patterns existing within a test dataset T, which can then be used

for predicting the class of unclassified elements in the complete dataset D. Increasing the

coverage of the decision tree by using a larger subset of D could potentially produce more

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Decision tree-based methods have their drawbacks however. One major drawback is they

are more suited to low-dimensional datasets. When trying to learn rules from large,

high-dimensional datasets, they can generate very large decision trees which could in turn result

in a very high number of rules being generated. Effectively partitioning attributes with

con-tinuous attribute values could also become difficult. Furthermore, generating and traversing

the decision trees could become computationally intensive thus affecting the efficiency of the

technique. Evaluating the “interestingness” of the many rules derived from large decision trees

can be time consuming. Missing values in the data could also cause the technique to generate

inaccurate patterns or rules [31]. Finally, these techniques may not be widely understood as

they are largely automated and non-intuitive.

2.1.1 Association Rule Mining

Association rule mining, also referred to as frequent pattern mining, is another popularly used

technique in the field of knowledge discovery for extracting patterns in the form of association

rules from a dataset [2]. An association rule is an expression of the form X _⇒ Y, where

X and Y are sets of items. An item could denote a specific attribute value or a range of

values, andXandY could denote combinations of such values. For a multidimensional dataset

D, representing n data attributes A = _{A1, ..., An}, where each data element ei ∈ D is a

combination of n individual data values, one for each attribute Ai, the expression X ⇒ Y

signifies that, ifei containsX thenei probably also contains Y. X is also referred to as the

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Association rule mining algorithms utilize two measures: support and confidence to

gener-ate rules. For a set of itemsX, its support denoted bysupp(X)is given as the fraction of data

elements inDwhich contain X. For a ruleX _⇒ Y, its support denoted bysupp(X _⇒Y)is

given by the fraction of elements inD containing the conjunctionX _∪Y. The confidence of

X _⇒Y denoted byconf(X _⇒Y)is given by the fraction of data elements inDthat contain

X, and which also contain Y [2]. This can be expressed mathematically as: supp(X _⇒ Y)

=supp(X_∪Y), and conf(X _⇒Y) =supp(X_∪Y)/supp(X)[18].

The main task of association rule mining algorithms is to find all association rules withinD

satisfying user defined thresholds for minimum support i.e. min-supp and minimum confidence

min-conf [2]. The approach followed by these techniques generally involves two broad steps.

In the first step, the support of all possible combinations of items (i.e. attribute value pairs)

also called itemsets is calculated. In the second step itemsets found to be frequent, i.e. having

support _≥ min-supp, are then combined into all possible rules of the form X _⇒ Y. Rules

with confidence _≥ min-conf, are referred to as association rules. Modifying min-supp and

min-conf could produce significantly different results depending on the nature of the dataset.

Traditionally, the Apriori technique and Apriori-based techniques have been utilized for mining

association rules [18].

Apriori

Apriori is an association rule mining algorithm introduced by Agrawal et al. [3]. Before we

describe the algorithm, we will discuss the terminologies relevant to the algorithm. A set ofk

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as a largek-itemset. The set of all largek-itemsets is denoted by Lk. A candidatek-itemset

or ak-candidate itemset is used to denote a potentially largek-itemset, and the set of all such

k-candidate itemsets is denoted byCk.

Apriori works in the following manner. It first counts the number of occurrences of each

individual item in D to find all large 1-itemsets L1. Each subsequent pass k involves two

phases. In the first phase Lk−1, the set of (k −1)-itemsets found large in the previous pass

is used to generate a set of k-candidate itemsets Ck. In the second phase, D is scanned to

compute the support of each of thek-candidate itemsets inCk. Those that are found large (i.e.

those that have support greater than or equal to min-supp) are used to formLk, the set of large

k-itemsets.

Thek-candidate itemsets inCkare generated fromLk−1through two steps. In the first step,

referred to as the join step,k-candidate itemsets are generated by merging all(k₋1)-itemsets

X, Y _∈ Lk−1, which share their first(k − 2) items, i.e. X.item1 = Y.item1, X.item2 =

Y.item2, ..., X.itemk−2 = Y.itemk−2, X.itemk−1 6= Y.itemk−1 [3]. In the second step,

referred to as the prune step, all itemsetsc_∈ Ckfor which some(k−1)subset ofcis not in

Lk−1are deleted.

To illustrate the algorithm let us consider an example [3]. Let L1, the set of all large 1

-itemsets in a datasetDgenerated during the first phase of the algorithm, contain the individual

items _{{ {}1_}, _{2_}, _{3_}, _{5_{} }}. To construct C2 from L1, all 2-itemset combinations of items

fromL1 are generated, producingC2 = { {1 2}, {1 3}, {1 5}, {2 3}, {2 5}, {3 5}, }. In the

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they were found to have low support inD. The remaining itemsets defineL2={ {1 2},{1 3},

{2 3_},_{2 5_{} }}. In the first phase of the passk = 3,L2is used to generatedC3. During the join

step, _{1 2_}is merged with _{1 3_}to generate the itemset _{1 2 3_}. Likewise, _{2 3_}is merged

with{2 5}to generate{2 3 5}. C3 now contains { {1 2 3}, {2 3 5} }. During the prune step,

there are threek₋1 = 2-itemsets for candidate_{2 3 5_}: _{2 3_},_{2 5_}and_{3 5_}. Although_{2

3_}and_{2 5_}are contained inLk−1, {3 5}is not. So the candidate itemset{2 3 5}is removed

fromC3. The candidate itemset_{{ {}1 2 3_{} }} is accepted since_{1 2_}, _{2 3_}and_{1 3_} are all

inLk−1. C3 now contains only{ {1 2 3} }. The remaining itemset inC3, if found large in the

second phase of the current pass, becomesL3.

Once combinations of items that satisfy min-supp in D are identified, they are used to

generate association rules. For each large itemsetZ, all rules having the formX _⇒ (Z ₋X)

are generated, whereX is any subset of Z andsupp(Z)/supp(X)_≥min-conf . Initially, for

every large itemsetZ, all rules having only one item on the right hand side of the expression

X _⇒(Z₋X)are generated and tested. Rules that satisfy the minimum confidence requirement

are further expanded to build additional rules.

Another algorithm introduced in [3] called AprioriTID extends the Apriori-based approach.

In this algorithm the entire datasetDis not used for counting support at the end of each pass.

Instead, AprioriTID uses only those items that were found large from the previous pass in an

effort to improve the efficiency.

Research on association rule mining was initially targeted at transaction databases.

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applications and datasets. Visualizing the patterns, rules and dependencies discovered using

as-sociation rule mining could simultaneously highlight the important characteristics of the data,

as well as help reduce the amount of information to be displayed.

2.1.2 Drawbacks

Association rule mining algorithms, in particular Apriori-based techniques, however have

cer-tain drawbacks, such as:

1. Algorithmic complexity: Increase in size and dimensionality of the data reduces the

ef-ficiency of the algorithm. This means that the algorithm may be unsuited to very large

multidimensional datasets [18].

2. Determining usefulness of generated rules: Association rule mining of large

multidimen-sional datasets could result in the generation of thousands of association rules.

Evaluat-ing the relevance and utility of these rules can be a complex task by itself. In some cases,

the rules could also correspond to knowledge that was already known previously, or to

redundant knowledge present in other rules [25]

3. Counting infrequent items: Association rule mining algorithms scan the entire dataset to

discover itemsets that satisfy minimum (user-specified) support and confidence

thresh-olds. This process also includes counting support for itemsets that have low support

within the dataset. This could limit the efficiency of association rule mining techniques

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4. Handling continuous datasets: Techniques for association rule mining were designed

to discover relationships in transaction data in which attributes assume binary values,

i.e. either 0 or 1. In general however data attributes assume values that cover a broad,

continuous range of real numbers [39]. Counting the support of each individual attribute

value can pose serious problems to association rule mining algorithms.

Some of the problems associated with association mining algorithms are addressed in

re-lated techniques such as the template based mining and FP-growth techniques.

2.1.3 Template based mining

The template based approach proposed by Klemittinen et al. is focused towards intelligently

pruning the number of rules generated by association rule mining to retain only interesting

and relevant rules [25]. This is made possible by encoding user knowledge within rule

tem-plates which are of the form A1, A2, ..., Ak ⇒ Ak+1. Each Ai in the template represents an

individual data attribute, or a user-defined attribute class. Templates can be used to explicitly

define classes or rules that are interesting (referred to as inclusive templates) and those that are

irrelevant (referred to as restrictive templates). Both kinds of templates are then used to prune

the rules that are generated; rules matching inclusive templates are retained while those

match-ing restrictive templates are discarded. While such an approach may be useful in guidmatch-ing the

mining process in a more relevant fashion, it may not prove to be of any significant advantage

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2.1.4 Frequent Pattern Growth

The Frequent Pattern growth (FP-growth) technique [14] is targeted towards improving the

efficiency of association rule mining. The basic approach behind this technique involves first

generating a compact structure called the frequent pattern tree, or FP-tree, to store all frequent

items present within D. These frequent items are represented by nodes in the tree, whereby

each node represents a unique item, i.e. an attribute-value pair present in D. A path in the

tree linking the nodes together represents a combination of items that occur together within an

ei ∈D. Multiple occurrences of the same combination of items is represented using the same

path, which helps reduce redundancy or repetition of nodes or paths. Once the tree is ready,

rules are generated from the FP-tree by traversing individual paths within the tree.

While FP-growth reduces Dto a compact tree structure containing all important patterns

present withinD, it may not be useful for very large datasets. For such datasets, constructing

the FP-tree could mean a significant computational overhead, since it requires two complete

scans ofD. Also, traversing such trees could become inefficient.

So far we discussed some automated techniques that could be used for generating

associa-tion rules from a dataset. In the next secassocia-tion we discuss some of the significant techniques used

to display or visualize these rules or patterns.

2.2 Pattern Visualization

Pattern visualization techniques are used to display patterns or rules generated by pattern

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visu-alization techniques include Mineset [8], mosaic plots [19], and parallel coordinates technique

[12]. Mineset, introduced by Brunk et al., supports the use of various display techniques to

vi-sualize the results generated by association mining [8]. However, this technique only vivi-sualizes

a very limited number of rules.

In mosaic plots, data attributes and their values are used to partition a rectangle into

sub-regions of various sizes that are proportional to the support value of combinations of various

attributes and their value ranges [19]. The resulting rectangular mosaic helps distinguish

im-portant relationships between the attributes and properties of these relationships. However, this

technique is limited to allowing at most three attributes to be displayed at a time. Also, as the

number of individual attribute values to be displayed increases, the number of partitions also

increases, which could make it difficult for users to perceive individual sub-regions accurately.

Additionally, both Mineset and mosaic plots do not provide means to visualize or guide the

rule generation process itself.

The parallel coordinates visualization technique has been utilized by Jian et al. [12] as well

as by Yang et al. [42] to visualize association rules. In the parallel coordinates technique,

par-allel axes in the visualization space are used to represent individual data attributes. Rules are

represented by polylines such that the axes connected by the polylines represent combinations

of items present within the same association rule. However, parallel coordinate techniques are

not suitable for visualizing lengthy rules, i.e. rules containing a large number of items.

Over-lapping polylines is another issue when trying to interpret parallel-coordinates visualization.

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from the consequent of a rule and vice-versa could become more and more difficult.

Paral-lel coordinates can also be used to visualize individual data elements in a multidimensional

dataset. We discuss this aspect in the next section.

2.3 Clustering

A cluster is a collection of data elements in which elements belonging to the same cluster are

more similar to each other when compared to elements from different clusters. Clusters are

useful as they provide insight into the behavior of datasets, including characteristics such as

the distribution of the data, as well as, similarities and differences existing within a dataset.

Clusters are also useful for generalizing data elements for classification purposes [21]. They

can be identified using both automated and interactive visual clustering techniques.

Given a datasetDcontainingmdata elements, clustering algorithms divideDinto groups

or clusters where data elements belonging to the same cluster are similar to one another

rel-ative to elements in different clusters [10]. Automated clustering techniques can broadly be

classified as either hierarchical clustering or partitional clustering techniques.

Hierarchical clustering techniques subdivide datasets into clusters, by either subdividing

large clusters into smaller clusters, or merging small clusters of elements into larger clusters

wherever possible. The subdivision hierarchy is represented using a tree structure or

den-drogram [20]. These denden-drograms can be visualized using tools, such as for instance, the

hierarchical clustering explorer (HCE) [36, 37].

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without producing a nested hierarchy (as in hierarchical clustering) [20]. The most popular

partitional clustering algorithm is thek-means technique [33].

2.3.1 k

-means Clustering

In k-means clustering (KMC), D is subdivided into k clusters. The process involves first

choosingkcluster centersg1, ..., gkfrom among the elements inD, and then assigning each of

the remainingei ∈Dto their closest cluster centergc(ei), wherec(ei)is the index of the center

closest to ei. The goal of the k-means algorithm is to minimize the mean-squared distance

between eachei and its nearest cluster centergc(ei). This distance is also sometimes referred to

as distortion error [23], and is computed as [24]:

Ek =

X

i∈D

kei−gc(ei)k

2

Once alleiare assigned to their nearest cluster centers, the centers for each of thekclusters

are recomputed. This is done by finding the element within each cluster that is closest to the

other elements of the same cluster. Once the new cluster centers are determined, the remaining

eiare once again reassigned to the new centers. This process is repeated several times, with the

cluster centers determined at the end of each iteration converging to the actual cluster centers

of D. The algorithm terminates when convergence is achieved, i.e. the distortion error does

not improve significantly. This technique is also sometimes referred to as the centroid-based

clustering technique or thek-mediods based technique.

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the cluster centers can affect the efficiency of convergence to the true cluster centers. Selecting

an appropriate value forkis also critical to the performance of the algorithm. Ideally,kshould

be as close as possible to the actual number of clusters present within the dataset. Recomputing

centers during each iteration of the algorithm affects the efficiency of the technique, especially

for large datasets of high dimensionality. Also, before clustering algorithms can be applied to

large multidimensional data, analyzing whether the data exhibits a tendency to cluster in the

first place is also important.

While hierarchical clustering algorithms are more informative than partitional algorithms

in describing the underlying structure within the datasets, constructing dendrograms for very

large, multidimensional datasets can be computationally expensive. By contrast, partitional

algorithms, such ask-means, inspite of their drawbacks have proved to be much more efficient

in generating clusters for very large collections of samples, and are hence more useful when

dealing with large datasets [20].

2.3.2 Visual Cluster Detection

Visual clustering techniques allow users the ability to directly display their datasets using

mul-tidimensional visualization techniques and visually detect clusters and groupings in the

infor-mation space. For this purpose, the multidimensional data elements are projected onto a

low-dimensional display space. Parallel coordinates, star coordinates technique, self-organizing

maps and multidimensional scaling are some of the techniques that generate layouts that can

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Figure 2.1: A visualization of financial data using parallel coordinates visualization. The visualization shows attributes names and the ranges of values of each attribute.

Parallel Coordinates Visualization

In parallel coordinates, data attributes are represented using parallel axes such that each data

attribute is assigned to a single axis, and each data element is represented by a polyline

con-necting the axes at locations that correspond to the value of the element for each attribute. The

axes could be organized vertically (as shown in figure 2.1) or horizontally. Areas in which the

polylines seem to be concentrated very close to one another could represent clusters within the

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As the number of elements increase, the degree to which the polylines begin to overlap one

another also increases. For very large datasets, this overlapping could make it very difficult

for users to distinguish individual polylines and the corresponding data elements and their

in-dividual attribute values. Also, for a high-dimensional dataset, fitting a very large number of

attribute axes (either horizontally or vertically) within a limited screen space could produce

ineffective visualizations.

Star Coordinates Visualization

In star coordinates data attributes are mapped to axes arranged within a circle on a

two-dimensional plane such that the axes share a common origin [22]. Data elements from a

mul-tidimensional dataset are projected onto the star-coordinates space by computing summations

of the products of individual attribute values and the direction vectors of their corresponding

axes. The computation of the position of an individual point in the star-coordinates space is

shown in figure 2.2. The resulting layout of elements helps users visually detect clusters, if

any, that are present in the original datasetD. Additionally, users can also manipulate the

visu-alization by scaling and rotating individual axes, thereby altering the contribution of the axes

to the position of the data elements in the layout. This helps in effectively separating clusters

in multidimensional space.

However, while this technique may be useful in producing visualizations that help users

visually identify clusters in a multidimensional space more easily, the visualizations produced

could make it difficult to identify individual properties (for e.g., attribute values) associated

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Figure 2.2: An example of how the location of an element in 8-dimensional star coordinates space is calculated.

incorporated within the circular star-coordinates space would be very high. This could cause

the visualizations to become less meaningful to users.

Self Organizing Maps

The self organizing map (SOM) technique first introduced by Kohonen, is used to organize

unstructured data similar to the k-means clustering technique by projecting elements from a

multidimensional space to a two-dimensional space [27]. The technique utilizes the principle

of competitive learning, which is an adaptive process by which neurons in a neural network

become more and more sensitive to different input elements or categories.

A two-dimensional grid of neurons is used for organizing the data elements inD. Initially

the neurons or reference vectors on the grid are assigned random values. When an elementei

is introduced to this grid, it is assigned to the reference vector that is closest in value to theei.

This vector is also referred to as the “winning vector” and its value is then updated to represent

the value ofeimore closely. The values of the reference vectors around the winning vector also

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vector. The closer the reference vectors are to the winning value the more they resemble the

winning vector. These steps are applied to allei ∈ D. The process is repeated several times

until the positions of the reference vectors on the grid do not change significantly. The resulting

layout is a topographical arrangement of the elements wherein similar elements tend to cluster

close to one another [28]. Such a technique not only helps identify similar elements within

a multidimensional dataset, but also helps in reducing dimensionality since data is projected

from a multidimensional space onto a two-dimensional plane.

However, SOM techniques tend to produce different maps each time they are applied to a

dataset. This happens due to the initial random assignment of values to reference vectors. To

select a map that is most appropriate, users would have to generate several different maps.

Ad-ditionally, very large, high-dimensional datasets also increase the computational costs involved

in generating these maps.

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Multidimensional Scaling

Multidimensional scaling (MDS) [29] is another technique used for projecting elements

be-longing to a high-dimensional dataset onto a low-dimensional visualization space. For this

purpose pair-wise similarity or correlation measures between all pairs of elements are utilized.

These pair-wise correlation measures are then converted into a distance or dissimilarity

mea-sure, which in turn is used to create a distance matrixC. An entryCi,jin this matrix represents

the distance or dissimilarity between an element pairi, j. Elements are represented as points

in the visualization space.

The basic purpose of the technique is to position points in this space such that their distances

in the visualization space resemble as closely as possible their actual dissimilarity measures in

C. This is done in the following manner:

1. Initially, points are randomly positioned in the visualization space.

2. Then for each pointi, its new position is computed based on the pair-wise distances

be-tweeniand every other elementj. To compute this new position the Euclidean distance

betweeni andj, i.e. d(i, j)is compared to the dissimilarity between the two elements

given byCi,j. The contribution ofj to the computation of a new position foridepends

on the difference between the two measures:

d0(i, j) = d(i, j)₋Ci,j (2.1)

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dissimilarity between the elements, theniis positioned closer toj along the line passing

through both points. Alternately, if d0(i, j) < 0, i.e. the actual dissimilarity between

the two points is greater than the layout distance,iis positioned away fromj along the

line that passes both points. The distanceiis moved towards or away fromj is a small

fraction of the_kd0(i, j)_kvalue.

3. In this fashion, all otherj are compared withiand new positions foriare computed after

each comparison.

4. The process of computing a new position for i(i.e. steps 2 and 3) is repeated with all

other elementsjone after the other. At the end of one iteration all points are assigned new

positions that better resemble the actual similarities between them. The whole process is

repeated several times until the points do not move significantly closer or farther away

from one another i.e. the configuration stabilizes. In the resulting layout similar elements

tend to cluster closer to one another and those that are dissimilar tend to be placed farther

away from each other.

One application of the MDS technique is the Galaxies Visualization system by Wise et al.,

in which MDS is used to cluster documents from a database based on their similarity to one

another [40]. Visualizing the layout generated by MDS in which documents are represented

by two-dimensional glyphs helps users understand the similarities and relationships that exist

between documents in the database. For very large datasets, however, calculating inter-element

similarity measures for all elements and using these measures to determine the layout of

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Figure 2.4: A visualization of image similarities in the Design Galleries technique. The glyphs represent thumb-nails of output images and the visual properties of the glyphs represent properties of the images.

Another application utilizing MDS techniques to visualize similarities contained within

large information sets is the Design Galleries browser developed by Marks, et al [32]. In

this application various output images generated from multidimensional datasets supplied as

input are compared at the pixel level to generate correlation or similarity measures between the

images. The MDS algorithm is then applied to display the global similarities existing between

the various images. Figure 2.4 shows a typical layout generated by the application. Rectangular

glyphs in the central area of the visualization space represent thumbnails of the various output

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consist of the detailed images for each of the thumbnails. In the display, lines are used to link

thumbnails to their corresponding detailed images in the gallery view. Thumbnails that are

closely packed represent images that are relatively similar to one another allowing users to

visually identify similar image clusters.

The MDS technique can also be used to visualize inter-attribute similarities as in the Value

and Relation (VaR) display technique [41]. We will discuss this technique in a later section

that deals with techniques used to identify relevant attributes within a dataset.

One of the main drawbacks associated with MDS algorithms is that, since the algorithms

are heuristic in nature, they may not always produce an optimal layout that accurately

repre-sents the actual similarities existing within the underlying dataset. Also, the algorithms may

not always produce similar layouts every time. In such instances multiple layouts would have

to be generated to verify the technique.

2.4 Outlier Detection

Outliers are primarily data elements which deviate from other elements by so much that they

arouse suspicion of being generated by a mechanism other than that which generated the other

elements [15]. The can also be described as data elements which differ from other elements by

some measure [1].

In the distance based outlier detection approach by Knorr and Ng, outliers are detected

using a distance metric between data elements in a multidimensional information space [26].

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λfromei[1, 26]. Bothkandλare user-defined parameters and the technique heavily depends

on these parameters in order to be accurate and effective. Choosing appropriate values for these

parameters is hence critical to the performance of the distance-based technique. Additionally,

this technique assumes that all elements in D cluster with similar clustering densities, and

would fail to detect outliers in datasets which contain clusters of varying densities [7].

2.4.1 Densities of Local Neighborhoods

The densities of local neighborhoods of elements approach proposed by Breunig et al. is

designed to find outliers in datasets with varying densities [7]. The basic premise behind this

approach is: an elementei is classified as an outlier, if it lies farther away from its neighbors,

when compared to its set of neighborsNei. Such elements, which are found to be “outlying”

relative to their local neighborhoods are also referred to as local outliers [7].

The process of finding local outliers begins with identifying the set ofk nearest neighbors

for each element ei denoted by Nk(ei). If an element ej ∈ Nk(ei) is within a user-defined

distanceλ fromei, thenej is said to lie in the local neighborhood of ei, and the reachability

distance ofei with respect toej, denoted byd(ei, ej), is set toλ. Ifej lies at a distance greater

than λ, then d(ei, ej) is set to the actual distance betweenei and ej. The local reachability

density ofei, denoted byδlocal(ei)is calculated as the inverse of the average ofd(ei, ej)for all

ej ∈Nk(ei), that is:

δlocal(ei) =

1

k

X

∀ej∈Nk(ei)

1

d(ei, ej)

(2.2)

Interactive Visual Summarization for Visualizing Large, Multidimensional Datasets

INTERACTIVE VISUAL SUMMARIZATION

FOR VISUALIZING LARGE

MULTIDIMENSIONAL DATASETS

SARAT M. KOCHERLAKOTA

COMPUTER SCIENCE

APPROVED BY:

Contents

List of Figures

Chapter 1

Introduction

1.1

Motivation

1.1.1

Size and Dimensionality Issues

1.2

Summarization

1.2.1

Automated Analysis Techniques

1.3

Research Objectives / Goals

1.4

Visual Summarization

1.4.1

Data Reduction

1.4.2

Behavior Characterization

1.4.3

Summary Visualization

1.4.4

Summary Management

1.5

Contributions of Our Research

1.6

Organization

Chapter 2

Related Work

2.1

Pattern Detection

2.1.1

Association Rule Mining

2.1.2

Drawbacks

2.1.3

Template based mining

2.1.4

Frequent Pattern Growth

2.2

Pattern Visualization

2.3

Clustering

2.3.1

k

-means Clustering

2.3.2

Visual Cluster Detection

2.4

Outlier Detection

2.4.1

Densities of Local Neighborhoods