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 A Non-Archimedean Absolute Value, or Norm, is Less Than or Equal
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
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