Main Data
Author: Dan Simovici, Chabane Djeraba
Title: Mathematical Tools for Data Mining Set Theory, Partial Orders, Combinatorics
Publisher: Springer-Verlag
ISBN/ISSN: 9781848002012
Edition: 1
Price: CHF 154.70
Publication date: 01/01/2008
Category: Informatik, EDV Buch
Language: English
Technical Data
Pages: 615
Kopierschutz: DRM
Geräte: PC/MAC/eReader/Tablet
Formate: PDF
Table of contents
This volume was born from the experience of the authors as researchers and educators,whichsuggeststhatmanystudentsofdataminingarehandicapped in their research by the lack of a formal, systematic education in its mat- matics. The data mining literature contains many excellent titles that address the needs of users with a variety of interests ranging from decision making to p- tern investigation in biological data. However, these books do not deal with the mathematical tools that are currently needed by data mining researchers and doctoral students. We felt it timely to produce a book that integrates the mathematics of data mining with its applications. We emphasize that this book is about mathematical tools for data mining and not about data mining itself; despite this, a substantial amount of applications of mathematical c- cepts in data mining are presented. The book is intended as a reference for the working data miner. In our opinion, three areas of mathematics are vital for data mining: set theory,includingpartially orderedsetsandcombinatorics;linear algebra,with its many applications in principal component analysis and neural networks; and probability theory, which plays a foundational role in statistics, machine learning and data mining. Thisvolumeisdedicatedtothestudyofset-theoreticalfoundationsofdata mining. Two further volumes are contemplated that will cover linear algebra and probability theory. The ?rst part of this book, dedicated to set theory, begins with a study of functionsandrelations.Applicationsofthesefundamentalconceptstosuch- sues as equivalences and partitions are discussed. Also, we prepare the ground for the following volumes by discussing indicator functions, ?elds and?-?elds, and other concepts.
Table of contents
Part I Set Theory14
1 Sets, Relations, and Functions15
1.1 Introduction15
1.2 Sets and Collections15
1.3 Relations and Functions21
1.4 The Axiom of Choice46
1.5 Countable Sets47
1.6 Elementary Combinatorics50
1.7 Multisets56
1.8 Relational Databases58
Exercises and Supplements61
Bibliographical Comments67
2 Algebras69
2.1 Introduction69
2.2 Operations and Algebras69
2.3 Morphisms, Congruences, and Subalgebras73
2.4 Linear Spaces76
2.5 Matrices80
Exercises and Supplements86
Bibliographical Comments89
3 Graphs and Hypergraphs91
3.1 Introduction91
3.2 Basic Notions of Graph Theory91
3.3 Trees104
3.4 Flows in Digraphs123
3.5 Hypergraphs130
Exercises and Supplements133
Bibliographical Comments136
Part II Partial Orders139
4 Partially Ordered Sets141
4.1 Introduction141
4.2 Partial Orders141
4.3 Special Elements of Partially Ordered Sets145
4.4 The Poset of Real Numbers149
4.5 Closure and Interior Systems151
4.6 The Poset of Partitions of a Set156
4.7 Chains and Antichains160
4.8 Poset Product167
4.9 Functions and Posets170
4.10 Posets and the Axiom of Choice172
4.11 Locally Finite Posets and M¨ obius Functions174
Exercises and Supplements180
Bibliographical Comments184
5 Lattices and Boolean Algebras185
5.1 Introduction185
5.2 Lattices as Partially Ordered Sets and Algebras185
5.3 Special Classes of Lattices192
5.4 Complete Lattices200
5.5 Boolean Algebras and Boolean Functions204
5.6 Logical Data Analysis223
Exercises and Supplements231
Bibliographical Comments236
6 Topologies and Measures237
6.1 Introduction237
6.2 Topologies237
6.3 Closure and Interior Operators in Topological Spaces238
6.4 Bases247
6.5 Compactness251
6.6 Continuous Functions253
6.7 Connected Topological Spaces256
6.8 Separation Hierarchy of Topological Spaces259
6.9 Products of Topological Spaces261
6.10 Fields of Sets263
6.11 Measures268
Exercises and Supplements277
Bibliographical Comments284
7 Frequent Item Sets and Association Rules285
7.1 Introduction285
7.2 Frequent Item Sets285
7.3 Borders of Collections of Sets291
7.4 Association Rules293
7.5 Levelwise Algorithms and Posets295
7.6 Lattices and Frequent Item Sets300
Exercises and Supplements302
Bibliographical Comments304
8 Applications to Databases and Data Mining307
8.1 Introduction307
8.2 Tables and Indiscernibility Relations307
8.3 Partitions and Functional Dependencies310
8.4 Partition Entropy317
8.5 Generalized Measures and Data Mining333
8.6 Di.erential Constraints337
Exercises and Supplements342
Bibliographical Comments344
9 Rough Sets345
9.1 Introduction345
9.2 Approximation Spaces345
9.3 Decision Systems and Decision Trees349
9.4 Closure Operators and Rough Sets357
Exercises and Supplements359
Bibliographical Comments360
Part III Metric Spaces361
10 Dissimilarities, Metrics, and Ultrametrics363
10.1 Introduction363
10.2 Classes of Dissimilarities363
10.3 Tree Metrics369
10.4 Ultrametric Spaces378
10.5 Metrics on389
10.6 Metrics on Collections of Sets400
10.7 Metrics on Partitions406