Mathematical Tools for Data Mining

Cover -- TOC$Contents -- Preface -- Part I: Set Theory -- CH$1 Sets, Relations, and Functions -- 1.1 Introduction -- 1.2 Sets and Collections -- 1.3 Relations and Functions -- 1.4 The Axiom of Choice -- 1.5 Countable Sets -- 1.6 Elementary Combinatorics -- 1.7 Multisets -- 1.8 Relational Databases -- Exercises and Supplements -- Bibliographical Comments -- CH$2 Algebras -- 2.1 Introduction -- 2.2 Operations and Algebras -- 2.3 Morphisms, Congruences, and Subalgebras -- 2.4 Linear Spaces -- 2.5 Matrices -- Exercises and Supplements -- Bibliographical Comments -- CH$3 Graphs and Hypergraphs -- 3.1 Introduction -- 3.2 Basic Notions of Graph Theory -- 3.3 Trees -- 3.4 Flows in Digraphs -- 3.5 Hypergraphs -- Exercises and Supplements -- Bibliographical Comments -- Part II: Partial Orders -- CH$4 Partially Ordered Sets -- 4.1 Introduction -- 4.2 Partial Orders -- 4.3 Special Elements of Partially Ordered Sets -- 4.4 The Poset of Real Numbers -- 4.5 Closure and Interior Systems -- 4.6 The Poset of Partitions of a Set -- 4.7 Chains and Antichains -- 4.8 Poset Product -- 4.9 Functions and Posets -- 4.10 Posets and the Axiom of Choice -- 4.11 Locally Finite Posets and M??bius Functions -- Exercises and Supplements -- Bibliographical Comments -- CH$5 Lattices and Boolean Algebras -- 5.1 Introduction -- 5.2 Lattices as Partially Ordered Sets and Algebras -- 5.3 Special Classes of Lattices -- 5.4 Complete Lattices -- 5.5 Boolean Algebras and Boolean Functions -- 5.6 Logical Data Analysis -- Exercises and Supplements -- Bibliographical Comments -- CH$6 Topologies and Measures -- 6.1 Introduction -- 6.2 Topologies -- 6.3 Closure and Interior Operators in Topological Spaces -- 6.4 Bases -- 6.5 Compactness -- 6.6 Continuous Functions -- 6.7 Connected Topological Spaces -- 6.8 Separation Hierarchy of Topological Spaces -- 6.9 Products of Topological Spaces -- 6.10 Fields of Sets -- 6.11 Measures -- Exercises and Supplements -- Bibliographical Comments -- CH$7 Frequent Item Sets and Association Rules -- 7.1 Introduction -- 7.2 Frequent Item Sets -- 7.3 Borders of Collections of Sets -- 7.4 Association Rules -- 7.5 Levelwise Algorithms and Posets -- 7.6 Lattices and Frequent Item Sets -- Exercises and Supplements -- Bibliographical Comments -- CH$8 Applications to Databases and Data Mining -- 8.1 Introduction -- 8.2 Tables and Indiscernibility Relations -- 8.3 Partitions and Functional Dependencies -- 8.4 Partition Entropy -- 8.5 Generalized Measures and Data Mining -- 8.6 Differential Constraints -- Exercises and Supplements -- Bibliographical Comments -- CH$9 Rough Sets -- 9.1 Introduction -- 9.2 Approximation Spaces -- 9.3 Decision Systems and Decision Trees -- 9.4 Closure Operators and Rough Sets -- Exercises and Supplements -- Bibliographical Comments -- Part III: Metric Spaces -- CH$10 Dissimilarities, Metrics, and Ultrametrics -- 10.1 Introduction -- 10.2 Classes of Dissimilarities -- 10.3 Tree Metrics -- 10.4 Ultrametric Spaces -- 10.5 Metrics on Rn -- 10.6 Metrics on Collections of Sets -- 10.7 Metrics on Partitions -- 10.8 Metrics on Sequences -- 10.9 Searches in Metric Spaces -- Exercises and Sup.