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Core Concepts in Data Analysis: Summarization, Correlation, Visualization September 30, 2010 Core Concepts in Data Analysis: Summarization, Correlation, Visualization Boris Mirkin Department of Computer Science and Information Systems, Birkbeck, University of London, Malet Street, London WC1E 7HX UK Department of Data Analysis and Machine Intelligence, Higher School of Economics, 11 Pokrovski Boulevard, Moscow RF Abstract This book presents an in-depth description of main data analysis methods: 1D summarization, 2D analysis, popular classifiers such as naïve Bayes and linear discriminant analysis, regression and neuron nets, Principal component analysis and its derivates, K-Means clustering and its extensions, hierarchical clustering, and network clustering including additive and spectral methods. These are sys- tematized based on the idea that data analysis is to help enhance concepts and rela- tions between them in the knowledge of the domain. Modern approaches of evolu- tionary optimization and computational validation are utilized. Various relations between criteria and methods are formulated as those underlain by data-driven least-squares frameworks invoked for most of them. The description is organized in three interrelated streams: presentation, formulation and computation, so that the presentation part can be read and studied by students with little mathematical background. A number of self-study tools – worked examples, case studies, pro- jects and questions – are provided to help the reader in mastering the material. ii Acknowledgments Too many people contributed to the material of the book to list all their names. First of all, my gratitude goes to Springer’s editors who were instrumental in bringing forth the very idea of writing such a book and in channeling my efforts by providing good critical reviews. Then, of course, I thank the students at my classes in MS programs in Computer Sci- ence at Birkbeck and, more recently, in BS and MS programs in Applied Mathematics and In- formatics at HSE. Here is a list of people who directly contributed to this book with advice, and sometimes with computation: I. Muchnik (Rutgers University), M. Levin (Higher School of Economics Moscow), T. Fenner (Birkbeck University of London), S. Nascimento (New Univer- sity of Lisbon), T. Krauze (Hofstra University), I. Mandel (Telmar Inc), V. Sulimova (Tula Technical University), and V. Topinsky (Higher School of Economics Moscow). iii Preface This is a textbook in data analysis. Its contents are heavily influenced by the idea that data analysis should help in enhancing and augmenting knowledge of the domain as represented by the concepts and statements of relation between them. According to this view, two main pathways for data analysis are summarization, for developing and augmenting concepts, and correlation, for enhancing and estab- lishing relations. Visualization, in this context, is a way of presenting results in a cognitively comfortable way. The term summarization is understood quite broadly here to embrace not only simple summaries like totals and means, but also more complex summaries such as the principal components of a set of features or clus- ter structures in a set of entities. The material presented in this perspective makes a unique mix of subjects from the fields of statistical data analysis, data mining, and computational intelligence, which follow different systems of presentation. Another feature of the text is that its main thrust is to give an in-depth under- standing of a few basic techniques rather than to cover a broad spectrum of ap- proaches developed so far. Most of the described methods fall under the same least-squares paradigm for mapping an “idealized” structure to the data. This al- lows me to bring forward a number of relations between methods that are usually overlooked. Just one example: a relation between the choice of a scoring function for classification trees and normalization options for dummies representing the target categories. Although the in-depth study approach involves a great deal of technical details, these are encapsulated in specific fragments of the text termed “formulation” parts. The main, “presentation”, part is written in a very different style. The pres- entation involves no mathematical formulas and explains a method by actually ap- plying it to a small real-world dataset – this part can be read and studied with no concern for the formulation at all. There is one more part, “computation”, targeted at a computer-oriented reader. This part describes the computational implementa- tion of the methods, illustrated using the MatLab computing environment. I have arrived at this three-way narrative style as a result of my experiences in teaching data analysis and computational intelligence to students in Computer Science. Some students might be mainly interested in just one of the parts, whereas others might try to get to grips with two or even all three of them. One more device to stimulate the reader’s interest is a multi-layer system of pro-active learning materials for class- and self-study: - Worked examples provided to show how specific methods apply to par- ticular datasets; iv - More complex problems solved, case studies, possibly involving a rule for data generation, rather than a pre-specified dataset, or an informal way of analyzing results; - Even more complex problems, projects, possibly involving uncharted terrain and a small-scale investigation; - A number of computational or theoretical problems, questions, formu- lated as self-study exercises; answers are provided for most of them. The text is based on my courses for full-time and part-time students in the MS program in Computer Science at Birkbeck, University of London (2003-2010), in the BS and MS programs in Applied Mathematics and Informatics at Higher School of Economics, Moscow (2008-2010), and post-graduate School of Data Analysis at Yandex, a popular Russian search engine, Moscow (2009-2010). The material covers lectures and labs for about 35-40 lecture hours in advanced BS programs or MS programs in Computer Science or Engineering. It can also be used in application-oriented courses such as Bioinformatics or Methods in Market- ing Research. No prerequisite beyond a conventional school background for reading through the presentation part is required, yet some training in reading academic material is expected. The reader interested in studying the formulation part should have some background in: (a) basic calculus including the concepts of function, derivative and the first-order optimality conditions, (b) basic linear algebra including vectors, inner products, Euclidean distances and matrices (these are reviewed in the Ap- pendix), and (c) basic set theory notation such as the symbols for inclusion and membership. The computation part is oriented towards those interested in coding for computer implementation, specifically focusing on working with MatLab as a user-friendly environment. v Table of contents Acknowledgments ii Preface iii Table of contents v 0 Introduction: What is Core 1 0.1 Summarization and correlation – two main goals of Data Analysis 2 0.2 Case study problems 11 0.3 An account of data visualization 0.3.1 General 25 0.3.2 Highlighting 25 0.3.3 Integrating different aspects 29 0.3.4 Narrating a story 32 0.4 Summary 33 References 33 1 1D analysis: Summarization and Visualization of a Single Feature37 1.1 Quantitative feature: Distribution and histogram 38 P1.1 Presentation 38 F1.1 Formulation 40 C1.1 Computation 42 1.2 Further summarization: centers and spreads 43 P1.2 Centers and spreads: Presentation 43 F1.2 Centers and spreads: Formulation 46 F1.2.1 Data analysis perspective 46 F1.2.2 Probabilistic statistics perspective 49 C1.2 Centers and spreads: Computation 51 1.3 Binary and categorical features 52 P1.3 Presentation 52 F1.3 Formulation 55 C1.3 Computation 58 1.4 Modeling uncertainty: Intervals and fuzzy sets 59 1.4.1 Individual membership functions 59 1.4.2 Central fuzzy set 61 Project 1.1. Computing Minkowski metric’s center 62 Project 1.2 Analysis of a multimodal distribution 65 Project 1.3 Computational validation of the mean by bootstrapping 67 Project 1.4 K-fold cross-validation 71 1.5 Summary 75 References 76 2 2D analysis: Correlation and Visualization of Two Features 77 2.0 General 78 vi 2.1. Two quantitative features case 79 P2.1.1. Scatter-plot, linear regression 79 and correlation coefficient P2.1.2 Validity of the regression 82 F2.1 Linear regression: Formulation 85 F2.1.1 Fitting linear regression 85 F2.1.2. Correlation coefficient and its properties 87 F2.1.3 Linearization of non-linear regression 88 C2.1 Linear regression: Computation. 89 Project 2.1. 2D analysis, linear regression and bootstrapping 90 Project 2.2 Non-linear and linearized regression: a nature-inspired approach 96 2.2 Mixed scale case: Nominal feature versus a quantitative one 102 P2.2.1 Box plot, tabular regression and correlation ratio 102 F2.2.1. Tabular regression: Formulation 106 2.2.2. Nominal target 108 2.2.2.1. Nearest neighbor classifier 108 2.2.2.2. Interval predicate classifier 110 2.3 Two nominal features case 112 P2.3 Analysis of contingency tables: Presentation 112 P2.3.1. Deriving conceptual relations from statistics 113 P2.3.2. Capturing relationship with Quetélet indexes 115 P2.3.3 Chi-squared contingency coefficient as a summary correlation index 118 F2.3 Analysis of contingency tables: Formulation 121 2.4 Summary 125 References 126 3 Learning Multivariate Correlations in Data 127 3.1 General: Decision rules, fitting criteria, and learning protocols 128
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