Mathematical Tools for Data Mining

1. Front Matter

   Pages I-XII

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2. Set Theory

   1. Front Matter

      Pages 1-1

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   2. Sets, Relations, and Functions

      • Dan A. Simovici, Chabane Djeraba

      Pages 3-55

   3. Algebras

      • Dan A. Simovici, Chabane Djeraba

      Pages 57-77

   4. Graphs and Hypergraphs

      • Dan A. Simovici, Chabane Djeraba

      Pages 79-125

3. Partial Orders

   1. Front Matter

      Pages 127-127

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   2. Partially Ordered Sets

      • Dan A. Simovici, Chabane Djeraba

      Pages 129-172

   3. Lattices and Boolean Algebras

      • Dan A. Simovici, Chabane Djeraba

      Pages 173-224

   4. Topologies and Measures

      • Dan A. Simovici, Chabane Djeraba

      Pages 225-272

   5. Frequent Item Sets and Association Rules

      • Dan A. Simovici, Chabane Djeraba

      Pages 273-293

   6. Applications to Databases and Data Mining

      • Dan A. Simovici, Chabane Djeraba

      Pages 295-332

   7. Rough Sets

      • Dan A. Simovici, Chabane Djeraba

      Pages 333-348

4. Metric Spaces

   1. Front Matter

      Pages 349-349

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   2. Dissimilarities, Metrics, and Ultrametrics

      • Dan A. Simovici, Chabane Djeraba

      Pages 351-421

   3. Topologies and Measures on Metric Spaces

      • Dan A. Simovici, Chabane Djeraba

      Pages 423-458

   4. Dimensions of Metric Spaces

      • Dan A. Simovici, Chabane Djeraba

      Pages 459-493

   5. Clustering

      • Dan A. Simovici, Chabane Djeraba

      Pages 495-525

5. Combinatorics

   1. Front Matter

      Pages 527-527

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   2. Combinatorics

      • Dan A. Simovici, Chabane Djeraba

      Pages 529-549

   3. The Vapnik-Chervonenkis Dimension

      • Dan A. Simovici, Chabane Djeraba

      Pages 551-567

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.

• Boolean algebra
• Clustering
• combinatorics
• data mining
• database
• databases
• sets

From the reviews:

"The book is organized into four parts, with a total of 15 chapters. Each chapter … offers numerous exercises and references for further reading. … Overall, Simovici and Djeraba’s presentation of both the theoretical grounds and the practical aspects of the various data mining methodologies is good. … The book is intended for readers who have a data mining background … . It will help this audience to improve their knowledge of how different data mining strategies operate from a mathematical standpoint." (Aris Gkoulalas-Divanis, ACM Computing Reviews, February, 2009)

• University of Massachusetts, Boston, USA

  Dan A. Simovici

• University of Sciences and Technologies of Lille (USTL), France

  Chabane Djeraba

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