Foundations and Methods in Combinatorial and Statistical Data Analysis and Clustering 1st Edition by Israel Cesar Lerman ISBN 1447167910 9781447167914

Chapter 1: On Some Facets of the Partition Set of a Finite Set

• 1.1 Lattice of Partition Set of a Finite Set
  • Definition and General Properties
  • Countings
• 1.2 Partitions of an Integer
  • Generalities
  • Representations
• 1.3 Type of a Partition and Cardinality of the Associated Equivalence Binary Relation
• 1.4 Ultrametric Spaces and Partition Chain Representation
  • Definition and Properties of Ultrametric Spaces
  • Partition Lattice Chains of a Finite Set and the Associated Ultrametric Spaces
  • Partition Lattice Chains and the Associated Ultrametric Preordonances
  • Partition Hierarchies and Dendrograms
  • From a Symmetrical Binary Hierarchy to a Directed Binary Hierarchy
• 1.5 Polyhedral Representation of the Partition Set of a Finite Set

Chapter 2: Two Methods of Non-hierarchical Clustering

• 2.1 Preamble
• 2.2 Central Partition Method
  • Data Structure and Clustering Criterion
  • Transfer Algorithm and Central Partition
  • Objects with the Same Representation
  • Statistical Asymptotic Analysis
  • Remarks on the Application of the Central Partition Method and Developments
• 2.3 Dynamic and Adaptative Clustering Method
  • Data Structure and Clustering Criterion
  • The K-Means Algorithm
  • Dynamic Cluster Algorithm
  • Following the Definition of the Algorithm

Chapter 3: Structure and Mathematical Representation of Data

• 3.1 Objects, Categories and Attributes
• 3.2 Representation of the Attributes of Type I
  • The Boolean Attribute
  • The Numerical Attribute
  • Defining a Categorical Attribute from a Numerical One
• 3.3 Representation of the Attributes of Type II
  • The Nominal Categorical Attribute
  • The Ordinal Categorical Attribute
  • The Ranking Attribute
  • The Categorical Attribute Valuated by a Numerical Similarity
  • The Valuated Binary Relation Attribute
• 3.4 Representation of the Attributes of Type III
  • The Preordonance Categorical Attribute
  • The Taxonomic Categorical Attribute
  • The Taxonomic Preordonance Attribute
  • Coding the Different Attributes in Terms of Preordonance or Similarity Categorical Attributes
• 3.5 Attribute Representations When Describing a Set of Categories
  • Introduction
  • Attributes of Type I
  • Nominal or Ordinal Categorical Attributes
  • Ordinal (Preordonance) or Numerical Similarity Categorical Attributes
  • The Data Table: A Tarski System or a Statistical System

Chapter 4: Ordinal and Metrical Analysis of the Resemblance Notion

• 4.1 Introduction
• 4.2 Formal Analysis in the Case of a Description of an Object Set
  • Similarity Index in the Case of Boolean Data
  • Preordonance Associated with a Similarity Index in the Case of Boolean Data
• 4.3 Extension of the Indices Defined in the Boolean Case to Attributes of Type II or III
  • Comparing Nominal Categorical Attributes
  • Comparing Ordinal Categorical Attributes
  • Comparing Preordonance Categorical Attributes

Chapter 5: Comparing Attributes by Probabilistic and Statistical Association I

• 5.1 Introduction
• 5.2 Comparing Attributes of Type I for an Object Set Description
  • The Boolean Case
  • Comparing Numerical Attributes in the LLA Approach
• 5.3 Comparing Attributes for a Description of a Set of Categories
  • Case of a Description by Boolean Attributes
  • Comparing Distributions of Numerical, Ordinal Categorical, and Nominal Categorical Attributes

Chapter 6: Comparing Attributes by Probabilistic and Statistical Association II

• 6.1 Introduction
• 6.2 Comparing Attributes of Type II for an Object Set Description
  • Introduction; Alternatives in Normalizing Association Coefficients
  • Comparing Two Ranking Attributes
  • Comparing Two Nominal Categorical Attributes
  • Comparing Two Ordinal Categorical Attributes
  • Comparing Two Valuated Binary Relation Attributes
  • From the Total Association to the Partial One

Chapter 7: Comparing Objects or Categories Described by Attributes

• 7.1 Preamble
• 7.2 Comparing Objects or Categories by the LLA Method
  • Similarity Index Between Objects Described by Numerical or Boolean Attributes
  • Similarity Index Between Objects Described by Nominal or Ordinal Categorical Attributes
  • Similarity Index Between Objects Described by Preordonance or Valuated Categorical Attributes
  • Similarity Index Between Objects Described by Taxonomic Attributes
  • Similarity Index Between Objects Described by a Mixed Attribute Types: Heterogenous Descriptions
  • The Goodall Similarity Index
  • Similarity Index Between Rows of a Juxtaposition of Contingency Tables

Chapter 8: The Notion of “Natural” Class, Tools for Its Interpretation, and the Classifiability Concept

• 8.1 Introduction: Monothetic Class and Polythetic Class
  • The Intuitive Approaches of Beckner and Adanson
• 8.2 Discriminating a Cluster of Objects by a Descriptive Attribute
  • Case of Attributes of Type I: Numerical and Boolean
  • Discriminating a Partition by a Categorical Attribute
• 8.3 “Responsibility” Degree of an Object in an Attribute Cluster Formation
  • The Attribute Set is Composed of Attributes of Type I
  • The Attribute Set is Composed of Categorical or Ranking Attributes
• 8.4 Rows or Columns of Contingency Tables
  • Case of a Single Contingency Table
  • Case of a Horizontal Juxtaposition of Contingency Tables
• 8.5 Measuring the “Importance” of a Descriptive Attribute
  • Comparing Clustering “Importance” and Projective “Importance” of a Descriptive Attribute
• 8.6 Crossing Fuzzy Categorical Attributes or Fuzzy Classifications (Clusterings)
  • Crossing Net Classifications
  • Crossing a Net and a Fuzzy Dichotomous Classification
• 8.7 Classifiability
  • Discrepancy Between the Preordonance Structure and the Ultrametric Structure on a Data Set
  • Classifiability Distribution Under a Random Hypothesis of Non-ultrametricity

Chapter 9: Quality Measures in Clustering

• 9.1 Introduction
• 9.2 The Direct Clustering Approach: An Example of a Criterion
  • An Example of Quality Measurement
• 9.3 Quality of a Partition Based on Pairwise Similarities
  • Approximating a Symmetrical Binary Relation by an Equivalence Relation: The Zahn Problem
• 9.4 Measuring the Fitting Quality of a Partition Chain (Classification Tree)
  • Generalization of the Set Theoretic and Metrical Criteria
  • Pure Ordinal Criteria: The Lateral Order and Lexicographic Order Criteria

Chapter 10: Building a Classification Tree

• 10.1 Introduction
• 10.2 “Lexicographic” Ordinal Algorithm
  • Definition of an Ultrametric Preordonance Associated with a Preordonance Data
  • Algorithm for Determining ωu Defined by the H Function
• 10.3 Ascendant Agglomerative Hierarchical Clustering Algorithm
  • “Single Linkage”, “Complete Linkage” and “Average Linkage” Criteria
  • “Inertia Variation” (Ward) Criterion
• 10.4 AAHC Algorithms; Likelihood Linkage Criteria
  • Maximal Likelihood Linkage Family of Criteria
• 10.5 AAHC for Clustering Rows or Columns of a Contingency Table
  • Chi Square Criterion
  • Mutual Information Criterion
• 10.6 Efficient Algorithms in Ascendant Agglomerative Hierarchical Classification
  • Complexity Considerations
  • Reducibility and Monotonic Criterion

Chapter 11: Applying the LLA Method to Real Data

• 11.1 Introduction: CHAVL Software
• 11.2 Real Data Processing
  • Types of Child Characters Through Children’s Literature
  • Profiles Extracted from the Classification Tree
• 11.3 Dayhoff, Henikoff’s Matrices and LLA for Comparing Proteic Sequences
• 11.4 Clustering Categorical Attributes
  • Structuring the Sets of Values
  • Partial Association Between Categorical Attributes

Chapter 12: Conclusion and Thoughts for Future Works

• 12.1 Contribution to Challenges in Cluster Analysis
• 12.2 Developments in the Framework of the LLA Approach
  • Principal Component Analysis
  • Semi-supervised Hierarchical Classification
• 12.3 Future Directions in Big Data