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E-raamat: Foundations and Methods in Combinatorial and Statistical Data Analysis and Clustering

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Foundations and Methods in Combinatorial and Statistical Data Analysis and ClusteringSuurem pilt
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This book offers an original and broad exploration of the fundamental methods in Clustering and Combinatorial Data Analysis, presenting new formulations and ideas within this very active field.





With extensive introductions, formal and mathematical developments and real case studies, this book provides readers with a deeper understanding of the mutual relationships between these methods, which are clearly expressed with respect to three facets: logical, combinatorial  and  statistical.





Using relational mathematical representation, all types of data structures can be handled in precise and unified ways which the author highlights in three stages:









Clustering a set of descriptive attributes Clustering a set of objects or a set of object categories Establishing correspondence between these two dual clusterings

Tools for interpreting the reasons of a given cluster or clustering are also included.





Foundations and Methods in Combinatorial and Statistical Data Analysis and Clustering will be a valuable resource for students and researchers who are interested in the areas of Data Analysis, Clustering, Data Mining and Knowledge Discovery.

Arvustused

This book provides a synthetic and systematic presentation of clustering, combinatorial, and statistical data analysis. the presentation is interesting and original. Keeping a smart balance between theoretical concepts and practical issues, the book is addressed to students and researchers interested in data mining, data analysis, and clustering. (Florin Gorunescu, zbMATH 1338.62012, 2016)

1 On Some Facets of the Partition Set of a Finite Set
1(60)
1.1 Lattice of Partition Set of a Finite Set
1(17)
1.1.1 Definition and General Properties
1(8)
1.1.2 Countings
9(9)
1.2 Partitions of an Integer
18(3)
1.2.1 Generalities
18(2)
1.2.2 Representations
20(1)
1.3 Type of a Partition and Cardinality of the Associated Equivalence Binary Relation
21(9)
1.4 Ultrametric Spaces and Partition Chain Representation
30(22)
1.4.1 Definition and Properties of Ultrametric Spaces
30(3)
1.4.2 Partition Lattice Chains of a Finite Set and the Associated Ultrametric Spaces
33(4)
1.4.3 Partition Lattice Chains and the Associated Ultrametric Preordonances
37(2)
1.4.4 Partition Hierarchies and Dendrograms
39(6)
1.4.5 From a Symmetrical Binary Hierarchy to a Directed Binary Hierarchy
45(7)
1.5 Polyhedral Representation of the Partition Set of a Finite Set
52(9)
References
58(3)
2 Two Methods of Non-hierarchical Clustering
61(40)
2.1 Preamble
61(1)
2.2 Central Partition Method
62(18)
2.2.1 Data Structure and Clustering Criterion
62(7)
2.2.2 Transfer Algorithm and Central Partition
69(3)
2.2.3 Objects with the Same Representation
72(2)
2.2.4 Statistical Asymptotic Analysis
74(4)
2.2.5 Remarks on the Application of the Central Partition Method and Developments
78(2)
2.3 Dynamic and Adaptative Clustering Method
80(21)
2.3.1 Data Structure and Clustering Criterion
80(4)
2.3.2 The AT-Means Algorithm
84(2)
2.3.3 Dynamic Cluster Algorithm
86(5)
2.3.4 Following the Definition of the Algorithm
91(7)
References
98(3)
3 Structure and Mathematical Representation of Data
101(48)
3.1 Objects, Categories and Attributes
101(2)
3.2 Representation of the Attributes of Type I
103(6)
3.2.1 The Boolean Attribute
104(1)
3.2.2 The Numerical Attribute
105(2)
3.2.3 Defining a Categorical Attribute from a Numerical One
107(2)
3.3 Representation of the Attributes of Type II
109(12)
3.3.1 The Nominal Categorical Attribute
110(3)
3.3.2 The Ordinal Categorical Attribute
113(3)
3.3.3 The Ranking Attribute
116(2)
3.3.4 The Categorical Attribute Valuated by a Numerical Similarity
118(2)
3.3.5 The Valuated Binary Relation Attribute
120(1)
3.4 Representation of the Attributes of Type III
121(16)
3.4.1 The Preordonance Categorical Attribute
121(3)
3.4.2 The Taxonomic Categorical Attribute
124(5)
3.4.3 The Taxonomic Preordonance Attribute
129(3)
3.4.4 Coding the Different Attributes in Terms of Preordonance or Similarity Categorical Attributes
132(5)
3.5 Attribute Representations When Describing a Set C of Categories
137(12)
3.5.1 Introduction
137(1)
3.5.2 Attributes of Type I
138(1)
3.5.3 Nominal or Ordinal Categorical Attributes
138(4)
3.5.4 Ordinal (preordonance) or Numerical Similarity Categorical Attributes
142(1)
3.5.5 The Data Table: A Tarski System Tor a Statistical System S
143(3)
References
146(3)
4 Ordinal and Metrical Analysis of the Resemblance Notion
149(50)
4.1 Introduction
149(3)
4.2 Formal Analysis in the Case of a Description of an Object Set O by Attributes of Type I; Extensions
152(26)
4.2.1 Similarity Index in the Case of Boolean Data
152(13)
4.2.2 Preordonance Associated with a Similarity Index in the Case of Boolean Data
165(13)
4.3 Extension of the Indices Defined in the Boolean Case to Attributes of Type II or III
178(21)
4.3.1 Introduction
178(2)
4.3.2 Comparing Nominal Categorical Attributes
180(3)
4.3.3 Comparing Ordinal Categorical Attributes
183(8)
4.3.4 Comparing Preordonance Categorical Attributes
191(5)
References
196(3)
5 Comparing Attributes by Probabilistic and Statistical Association I
199(52)
5.1 Introduction
199(2)
5.2 Comparing Attributes of Type I for an Object Set Description by the Likelihood Linkage Analysis Approach
201(32)
5.2.1 The Boolean Case
201(20)
5.2.2 Comparing Numerical Attributes in the LLA approach
221(12)
5.3 Comparing Attributes for a Description of a Set of Categories
233(18)
5.3.1 Introduction
233(1)
5.3.2 Case of a Description by Boolean Attributes
234(8)
5.3.3 Comparing Distributions of Numerical, Ordinal Categorical and Nominal Categorical Attributes
242(5)
References
247(4)
6 Comparing Attributes by a Probabilistic and Statistical Association II
251(74)
6.1 Introduction
251(1)
6.2 Comparing Attributes of Type II for an Object Set Description; the LLA Approach
252(73)
6.2.1 Introduction; Alternatives in Normalizing Association Coefficients
252(4)
6.2.2 Comparing Two Ranking Attributes
256(5)
6.2.3 Comparing Two Nominal Categorical Attributes
261(15)
6.2.4 Comparing Two Ordinal Categorical Attributes
276(10)
6.2.5 Comparing Two Valuated Binary Relation Attributes
286(23)
6.2.6 From the Total Association to the Partial One
309(12)
References
321(4)
7 Comparing Objects or Categories Described by Attributes
325(32)
7.1 Preamble
325(3)
7.2 Comparing Objects or Categories by the LLA Method
328(29)
7.2.1 The Outline of the LLA Method for Comparing Objects or Categories
328(3)
7.2.2 Similarity Index Between Objects Described by Numerical or Boolean Attributes
331(3)
7.2.3 Similarity Index Between Objects Described by Nominal or Ordinal Categorical Attributes
334(4)
7.2.4 Similarity Index Between Objects Described by Preordonance or Valuated Categorical Attributes
338(3)
7.2.5 Similarity Index Between Objects Described by Taxonomic Attributes. A Solution for the Classification Consensus Problem
341(3)
7.2.6 Similarity Index Between Objects Described by a Mixed Attribute Types: Heterogenous Description
344(1)
7.2.7 The Goodall Similarity Index
345(4)
7.2.8 Similarity Index Between Rows of a Juxtaposition of Contingency Tables
349(4)
7.2.9 Other Similarity Indices on the Row Set I of a Contingency Table
353(2)
References
355(2)
8 The Notion of "Natural" Class, Tools for Its Interpretation. The Classifiability Concept
357(78)
8.1 Introduction; Monothetic Class and Polythetic Class
357(6)
8.1.1 The Intuitive Approaches of Beckner and Adanson; from Beckner to Adanson
360(3)
8.2 Discriminating a Cluster of Objects by a Descriptive Attribute
363(6)
8.2.1 Introduction
363(1)
8.2.2 Case of Attributes of Type I: Numerical and Boolean
364(2)
8.2.3 Discrimination a Partition by a Categorical Attribute
366(3)
8.3 "Responsibility" Degree of an Object in an Attribute Cluster Formation
369(8)
8.3.1 A is Composed of Attributes of Type I
370(5)
8.3.2 The Attribute Set A is Composed of Categorical or Ranking Attributes
375(2)
8.4 Rows or Columns of Contingency Tables
377(5)
8.4.1 Case of a Single Contingency Table
377(4)
8.4.2 Case of an Horizontal Juxtaposition of Contingency Tables
381(1)
8.5 On Two Ways of Measuring the "Importance" of a Descriptive Attribute
382(9)
8.5.1 Introduction
382(4)
8.5.2 Comparing Clustering "Importance" and Projective "Importance" of a Descriptive Attribute
386(5)
8.6 Crossing Fuzzy Categorical Attributes or Fuzzy Classifications (Clusterings)
391(28)
8.6.1 General Introduction
391(3)
8.6.2 Crossing Net Classifications; Introduction to Other Crossings
394(6)
8.6.3 Crossing a Net and a Fuzzy Dichotomous Classifications
400(4)
8.6.4 Crossing Two Fuzzy Dichotomous Classifications
404(4)
8.6.5 Crossing Two Typologies
408(3)
8.6.6 Extension to Crossing Fuzzy Relational Categorical Attributes
411(8)
8.7 Classifiability
419(16)
8.7.1 Introduction
419(1)
8.7.2 Discrepancy Between the Preordonance Structure and that Ultrametric, on a Data Set
420(6)
8.7.3 Classifiability Distribution Under a Random Hypothesis of Non-ultrametricity
426(5)
8.7.4 The Murtagh Contribution
431(1)
References
432(3)
9 Quality Measures in Clustering
435(78)
9.1 Introduction
435(3)
9.2 The Direct Clustering Approach: An Example of a Criterion
438(5)
9.2.1 General Presentation
438(2)
9.2.2 An Example
440(3)
9.3 Quality of a Partition Based on the Pairwise Similarities
443(46)
9.3.1 Criteria Based on a Data Preordonance
444(7)
9.3.2 Approximating a Symmetrical Binary Relation by an Equivalence Relation: The Zahn Problem
451(5)
9.3.3 Comparing Two Basic Criteria
456(12)
9.3.4 Distribution of the Intersection Criterion on the Partition Set with a Fixed Type
468(6)
9.3.5 Extensions of the Previous Criterion
474(9)
9.3.6 "Significant Levels" and "Significant Nodes" of a Classification Tree
483(6)
9.4 Measuring the Fitting Quality of a Partition Chain (Classification Tree)
489(24)
9.4.1 Introduction
489(1)
9.4.2 Generalization of the Set Theoretic and Metrical Criteria
490(3)
9.4.3 Distribution of the Cardinality of the Graph Intersection Criterion
493(9)
9.4.4 Pure Ordinal Criteria: The Lateral Order and the Lexicographic Order Criteria
502(2)
9.4.5 Lexicographic Ranking and Inversion Number Criteria
504(7)
References
511(2)
10 Building a Classification Tree
513(70)
10.1 Introduction
513(6)
10.2 "Lexicographic" Ordinal Algorithm
519(8)
10.2.1 Definition of an Ultrametric Preordonance Associated with a Preordonance Data
519(2)
10.2.2 Algorithm for Determining ωu Defined by the H Function
521(2)
10.2.3 Property of Optimality
523(1)
10.2.4 Case Where ω Is a Total Ordonance
524(3)
10.3 Ascendant Agglomerative Hierarchical Clustering Algorithm; Classical Aggregation Criteria
527(8)
10.3.1 Preamble
527(1)
10.3.2 "Single Linkage", "Complete Linkage" and "Average Linkage" Criteria
528(2)
10.3.3 "Inertia Variation (or Ward) Criterion"
530(4)
10.3.4 From "Lexicographic" Ordinal Algorithm to "Single Linkage" or "Maximal Link" Algorithm
534(1)
10.4 AAHC Algorithms; Likelihood Linkage Criteria
535(14)
10.4.1 Family of Criteria of the Maximal Likelihood Linkage
535(10)
10.4.2 Minimal Likelihood Linkage and Average Likelihood Linkage in the LLA Analysis
545(4)
10.5 AAHC for Clustering Rows or Columns of a Contingency Table
549(6)
10.5.1 Introduction
549(1)
10.5.2 Chi Square Criterion: A Transposition of the Ward Criterion
550(2)
10.5.3 Mutual Information Criterion
552(3)
10.6 Efficient Algorithms in Ascendant Agglomerative Hierarchical Classification (Clustering)
555(28)
10.6.1 Introduction
555(3)
10.6.2 Complexity Considerations of the Basic AAHC Algorithm
558(2)
10.6.3 Reactualization Formulas in the Cases of Binary and Multiple Aggregations
560(6)
10.6.4 Reducibility, Monotonic Criterion, Reducible Neighborhoods and Reciprocal Nearest Neighborhoods
566(6)
10.6.5 Ascendant Agglomerative Hierarchical Clustering (AAHC) Under a Contiguity Constraint
572(4)
10.6.6 Ascendant Agglomerative Parallel Hierarchical Clustering
576(4)
References
580(3)
11 Applying the LLA Method to Real Data
583(56)
11.1 Introduction: the CHAVL Software (Classification Hierarchique par Analyse de la Vraisemblance des Liens)
583(3)
11.2 Real Data: Outline Presentation of Some Processings
586(4)
11.3 Types of Child Characters Through Children's Literature
590(10)
11.3.1 Preamble: Technical Data Sheet
590(1)
11.3.2 General Objective and Data Description
591(2)
11.3.3 Profiles Extracted from the Classification Tree on A
593(3)
11.3.4 Developments
596(1)
11.3.5 Standardized Association Coefficient with Respect to the Hypergeometric Model
596(1)
11.3.6 Return to Individuals
597(3)
11.4 Dayhoff, Henikoffs and LLA Matrices for Comparing Proteic Sequences
600(29)
11.4.1 Preamble: Technical Data Sheet
600(1)
11.4.2 Introduction
601(2)
11.4.3 Construction of the Dayhoff Matrix
603(9)
11.4.4 The Henikoffs Matrix: Comparison with the Dayhoff Matrix
612(4)
11.4.5 The LLA Matrices
616(3)
11.4.6 LLA Similarity Index on a Set of Proteic Aligned Sequences
619(6)
11.4.7 Some Results
625(4)
11.5 Specific Results in Clustering Categorical Attributes by LLA Methodology
629(10)
11.5.1 Structuring the Sets of Values of Categorical Attributes
629(3)
11.5.2 From Total Associations Between Categorical Attributes to Partial Ones
632(5)
References
637(2)
12 Conclusion and Thoughts for Future Works
639
12.1 Contribution to Challenges in Cluster Analysis
639(2)
12.2 Around Two Books Concerning Relational Aspects
641(2)
12.3 Developments in the Framework of the LLA Approach
643(3)
12.3.1 Principal Component Analysis
643(1)
12.3.2 Multidimensional Scaling
644(1)
12.3.3 In What LLA Hierarchical Clustering Method Is a Probabilistic Method?
645(1)
12.3.4 Semi-supervised Hierarchical Classification
645(1)
12.4 Big Data
646
References
646
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