Data Science Foundations: Geometry and Topology of Complex Hierarchic Systems and Big Data Analytics

224 Pages·2017·15.3 MB·English

by  Fionn Murtagh

DATA SCIENCE
FOUNDATIONS
Geometry and Topology
of Complex Hierarchic Systems
and Big Data Analytics
Chapman & Hall/CRC
Computer Science and Data Analysis Series
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SERIES EDITORS
David Blei, Princeton University
David Madigan, Rutgers University
Marina Meila, University of Washington
Fionn Murtagh, Royal Holloway, University of London
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Published Titles
Semisupervised Learning for Computational Linguistics
Steven Abney
Visualization and Verbalization of Data
Jörg Blasius and Michael Greenacre
Design and Modeling for Computer Experiments
Kai-Tai Fang, Runze Li, and Agus Sudjianto
Microarray Image Analysis: An Algorithmic Approach
Karl Fraser, Zidong Wang, and Xiaohui Liu
R Programming for Bioinformatics
Robert Gentleman
Exploratory Multivariate Analysis by Example Using R
François Husson, Sébastien Lê, and Jérôme Pagès
Bayesian Artificial Intelligence, Second Edition
Kevin B. Korb and Ann E. Nicholson
Published Titles cont.
Computational Statistics Handbook with MATLAB®, Third Edition
Wendy L. Martinez and Angel R. Martinez
Exploratory Data Analysis with MATLAB®, Third Edition
Wendy L. Martinez, Angel R. Martinez, and Jeffrey L. Solka
Statistics in MATLAB®: A Primer
Wendy L. Martinez and MoonJung Cho
Clustering for Data Mining: A Data Recovery Approach, Second Edition
Boris Mirkin
Introduction to Machine Learning and Bioinformatics
Sushmita Mitra, Sujay Datta, Theodore Perkins, and George Michailidis
Introduction to Data Technologies
Paul Murrell
R Graphics
Paul Murrell
Correspondence Analysis and Data Coding with Java and R
Fionn Murtagh
Data Science Foundations: Geometry and Topology of Complex Hierarchic
Systems and Big Data Analytics
Fionn Murtagh
Pattern Recognition Algorithms for Data Mining
Sankar K. Pal and Pabitra Mitra
Statistical Computing with R
Maria L. Rizzo
Statistical Learning and Data Science
Mireille Gettler Summa, Léon Bottou, Bernard Goldfarb, Fionn Murtagh,
Catherine Pardoux, and Myriam Touati
Music Data Analysis: Foundations and Applications
Claus Weihs, Dietmar Jannach, Igor Vatolkin, and Günter Rudolph
Foundations of Statistical Algorithms: With References to R Packages
Claus Weihs, Olaf Mersmann, and Uwe Ligges

Chapman & Hall/CRC
Computer Science and Data Analysis Series
DATA SCIENCE
FOUNDATIONS
Geometry and Topology
of Complex Hierarchic Systems
and Big Data Analytics
Fionn Murtagh
Boca Raton London New York
CRC Press is an imprint of the
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Contents
Preface xiii
I Narratives from Film and Literature, from Social Media and
Contemporary Life 1
1 The Correspondence Analysis Platform for Mapping Semantics 3
1.1 The Visualization and Verbalization of Data . . . . . . . . . . . . . . . . . 3
1.2 Analysis of Narrative from Film and Drama . . . . . . . . . . . . . . . . . 4
1.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 The Changing Nature of Movie and Drama . . . . . . . . . . . . . . 4
1.2.3 Correspondence Analysis as a Semantic Analysis Platform . . . . . . 5
1.2.4 Casablanca Narrative: Illustrative Analysis . . . . . . . . . . . . . . 5
1.2.5 Modelling Semantics via the Geometry and Topology of Information 6
1.2.6 Casablanca Narrative: Illustrative Analysis Continued . . . . . . . . 8
1.2.7 Platform for Analysis of Semantics . . . . . . . . . . . . . . . . . . . 8
1.2.8 Deeper Look at Semantics of Casablanca: Text Mining . . . . . . . . 10
1.2.9 Analysis of a Pivotal Scene . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Application of Narrative Analysis to Science and Engineering Research . . 11
1.3.1 Assessing Coverage and Completeness . . . . . . . . . . . . . . . . . 12
1.3.2 Change over Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.3 Conclusion on the Policy Case Studies . . . . . . . . . . . . . . . . . 15
1.4 Human Resources Multivariate Performance Grading . . . . . . . . . . . . 19
1.5 Data Analytics as the Narrative of the Analysis Processing . . . . . . . . . 21
1.6 Annex: The Correspondence Analysis and Hierarchical Clustering Platform 21
1.6.1 Analysis Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6.2 Correspondence Analysis: Mapping χ2 Distances into Euclidean Dis-
tances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6.3 Input: Cloud of Points Endowed with the Chi-Squared Metric . . . . 22
1.6.4 Output:CloudofPointsEndowedwiththeEuclideanMetricinFactor
Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6.5 Supplementary Elements: Information Space Fusion . . . . . . . . . 23
1.6.6 Hierarchical Clustering: Sequence-Constrained . . . . . . . . . . . . 24
2 Analysis and Synthesis of Narrative: Semantics of Interactivity 25
2.1 Impact and Effect in Narrative: A Shock Occurrence in Social Media . . . 25
2.1.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.2 Two Critical Tweets in Terms of Their Words . . . . . . . . . . . . . 26
2.1.3 Two Critical Tweets in Terms of Twitter Sub-narratives . . . . . . . 26
2.2 Analysis and Synthesis, Episodization and Narrativization . . . . . . . . . 32
2.3 Storytelling as Narrative Synthesis and Generation . . . . . . . . . . . . . 33
vii
viii Contents
2.4 Machine Learning and Data Mining in Film Script Analysis . . . . . . . . . 35
2.5 Style Analytics: Statistical Significance of Style Features . . . . . . . . . . 36
2.6 Typicality and Atypicality for Narrative Summarization and Transcoding . 37
2.7 Integration and Assembling of Narrative . . . . . . . . . . . . . . . . . . . 40
II Foundations of Analytics through the Geometry and Topol-
ogy of Complex Systems 43
3 Symmetry in Data Mining and Analysis through Hierarchy 45
3.1 Analytics as the Discovery of Hierarchical Symmetries in Data . . . . . . . 45
3.2 Introduction to Hierarchical Clustering, p-Adic and m-Adic Numbers . . . 45
3.2.1 Structure in Observed or Measured Data . . . . . . . . . . . . . . . 46
3.2.2 Brief Look Again at Hierarchical Clustering . . . . . . . . . . . . . . 46
3.2.3 Brief Introduction to p-Adic Numbers . . . . . . . . . . . . . . . . . 47
3.2.4 Brief Discussion of p-Adic and m-Adic Numbers . . . . . . . . . . . 47
3.3 Ultrametric Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.1 Ultrametric Space for Representing Hierarchy . . . . . . . . . . . . . 48
3.3.2 Geometrical Properties of Ultrametric Spaces . . . . . . . . . . . . . 48
3.3.3 Ultrametric Matrices and Their Properties . . . . . . . . . . . . . . 48
3.3.4 Clustering through Matrix Row and Column Permutation . . . . . . 50
3.3.5 Other Data Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Generalized Ultrametric and Formal Concept Analysis . . . . . . . . . . . . 52
3.4.1 Link with Formal Concept Analysis . . . . . . . . . . . . . . . . . . 52
3.4.2 Applications of Generalized Ultrametrics. . . . . . . . . . . . . . . . 54
3.5 Hierarchy in a p-Adic Number System . . . . . . . . . . . . . . . . . . . . . 54
3.5.1 p-Adic Encoding of a Dendrogram . . . . . . . . . . . . . . . . . . . 54
3.5.2 p-Adic Distance on a Dendrogram . . . . . . . . . . . . . . . . . . . 57
3.5.3 Scale-Related Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.6 Tree Symmetries through the Wreath Product Group . . . . . . . . . . . . 58
3.6.1 Wreath Product Group for Hierarchical Clustering . . . . . . . . . . 58
3.6.2 Wreath Product Invariance . . . . . . . . . . . . . . . . . . . . . . . 59
3.6.3 Wreath Product Invariance: Haar Wavelet Transform of Dendrogram 60
3.7 Tree and Data Stream Symmetries from Permutation Groups . . . . . . . . 62
3.7.1 Permutation Representation of a Data Stream . . . . . . . . . . . . 62
3.7.2 Permutation Representation of a Hierarchy . . . . . . . . . . . . . . 63
3.8 Remarkable Symmetries in Very High-Dimensional Spaces . . . . . . . . . 64
3.9 Short Commentary on This Chapter . . . . . . . . . . . . . . . . . . . . . . 65
4 Geometry and Topology of Data Analysis: in p-Adic Terms 69
4.1 Numbers and Their Representations . . . . . . . . . . . . . . . . . . . . . . 69
4.1.1 Series Representations of Numbers . . . . . . . . . . . . . . . . . . . 69
4.1.2 Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 p-Adic Valuation, p-Adic Absolute Value, p-Adic Norm . . . . . . . . . . . 71
4.3 p-Adic Numbers as Series Expansions . . . . . . . . . . . . . . . . . . . . . 72
4.4 Canonical p-Adic Expansion; p-Adic Integer or Unit Ball . . . . . . . . . . 73
4.5 Non-Archimedean Norms as p-Adic Integer Norms in the Unit Ball . . . . 74
4.5.1 Archimedean and Non-Archimedean Absolute Value Properties . . . 74
4.5.2 ANon-ArchimedeanAbsoluteValue,orNorm,isLessThanorEqual
to One, and an Archimedean Absolute Value, or Norm, is Unbounded 74
4.6 Going Further: Negative p-Adic Numbers, and p-Adic Fractions . . . . . . 75
Contents ix
4.7 Number Systems in the Physical and Natural Sciences . . . . . . . . . . . . 76
4.8 p-Adic Numbers in Computational Biology and Computer Hardware . . . . 77
4.9 Measurement Requires a Norm, Implying Distance and Topology . . . . . . 78
4.10 Ultrametric Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.11 Short Review of p-Adic Cosmology . . . . . . . . . . . . . . . . . . . . . . . 80
4.12 Unbounded Increase in Mass or Other Measured Quantity . . . . . . . . . 81
4.13 Scale-Free Partial Order or Hierarchical Systems . . . . . . . . . . . . . . . 81
4.14 p-Adic Indexing of the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.15 Diffusion and Other Dynamic Processes in Ultrametric Spaces . . . . . . . 83
III New Challenges and New Solutions for Information Search
and Discovery 85
5 Fast, Linear Time, m-Adic Hierarchical Clustering 87
5.1 Pervasive Ultrametricity: Computational Consequences . . . . . . . . . . . 87
5.1.1 Ultrametrics in Data Analytics . . . . . . . . . . . . . . . . . . . . . 87
5.1.2 Quantifying Ultrametricity . . . . . . . . . . . . . . . . . . . . . . . 88
5.1.3 Pervasive Ultrametricity . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.1.4 Computational Implications . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Applications in Search and Discovery using the Baire Metric . . . . . . . . 89
5.2.1 Baire Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.2 Large Numbers of Observables . . . . . . . . . . . . . . . . . . . . . 89
5.2.3 High-Dimensional Data . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2.4 First Approach Based on Reduced Precision of Measurement . . . . 90
5.2.5 Random Projections in High-Dimensional Spaces, Followed by the
Baire Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2.6 Summary Comments on Search and Discovery . . . . . . . . . . . . 91
5.3 m-Adic Hierarchy and Construction . . . . . . . . . . . . . . . . . . . . . . 91
5.4 The Baire Metric, the Baire Ultrametric . . . . . . . . . . . . . . . . . . . 92
5.4.1 Metric and Ultrametric Spaces . . . . . . . . . . . . . . . . . . . . . 92
5.4.2 Ultrametric Baire Space and Distance . . . . . . . . . . . . . . . . . 93
5.5 Multidimensional Use of the Baire Metric through Random Projections . . 94
5.6 Hierarchical Tree Defined from m-Adic Encoding . . . . . . . . . . . . . . . 95
5.7 Longest Common Prefix and Hashing . . . . . . . . . . . . . . . . . . . . . 96
5.7.1 From Random Projection to Hashing . . . . . . . . . . . . . . . . . . 96
5.8 Enhancing Ultrametricity through Precision of Measurement . . . . . . . . 97
5.8.1 Quantifying Ultrametricity . . . . . . . . . . . . . . . . . . . . . . . 97
5.8.2 Pervasiveness of Ultrametricity . . . . . . . . . . . . . . . . . . . . . 98
5.9 Generalized Ultrametric and Formal Concept Analysis . . . . . . . . . . . . 99
5.9.1 Generalized Ultrametric . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.9.2 Formal Concept Analysis . . . . . . . . . . . . . . . . . . . . . . . . 99
5.10 Linear Time and Direct Reading Hierarchical Clustering . . . . . . . . . . 100
5.10.1 LinearTime,orO(N)ComputationalComplexity,HierarchicalClus-
tering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.10.2 Grid-Based Clustering Algorithms . . . . . . . . . . . . . . . . . . . 100
5.11 Summary: Many Viewpoints, Various Implementations . . . . . . . . . . . 101