## Mathematical Classification and ClusteringI am very happy to have this opportunity to present the work of Boris Mirkin, a distinguished Russian scholar in the areas of data analysis and decision making methodologies. The monograph is devoted entirely to clustering, a discipline dispersed through many theoretical and application areas, from mathematical statistics and combina torial optimization to biology, sociology and organizational structures. It compiles an immense amount of research done to date, including many original Russian de velopments never presented to the international community before (for instance, cluster-by-cluster versions of the K-Means method in Chapter 4 or uniform par titioning in Chapter 5). The author's approach, approximation clustering, allows him both to systematize a great part of the discipline and to develop many in novative methods in the framework of optimization problems. The optimization methods considered are proved to be meaningful in the contexts of data analysis and clustering. The material presented in this book is quite interesting and stimulating in paradigms, clustering and optimization. On the other hand, it has a substantial application appeal. The book will be useful both to specialists and students in the fields of data analysis and clustering as well as in biology, psychology, economics, marketing research, artificial intelligence, and other scientific disciplines. Panos Pardalos, Series Editor. |

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### Contents

Classes and Clusters | 1 |

a Review | 2 |

12 Forms and Purposes of Classification | 18 |

13 Table Data and Its Types | 25 |

14 ColumnConditional Data and Clustering | 33 |

15 Clustering Problems for Comparable Data | 41 |

16 Clustering Problems for Aggregable Data | 53 |

Geometry of Data Sets | 59 |

Partition Square Data Table | 229 |

51 Partition Structures | 230 |

52 Admissibility in Agglomerative Clustering | 246 |

53 Uniform Partitioning | 254 |

54 Additive Clustering | 263 |

55 Structured Partition and Block Model | 268 |

56 Aggregation of Mobility Tables | 278 |

Partition Rectangular Data Table | 285 |

21 ColumnConditional Data | 60 |

22 Transformation of Comparable Data | 78 |

23 LowRank Approximation of Data | 91 |

Clustering Algorithms a Review | 109 |

31 A Typology of Clustering Algorithms | 110 |

32 A Survey of Clustering Techniques | 128 |

33 Interpretation Aids | 158 |

Single Cluster Clustering | 169 |

41 Subset as a Cluster Structure | 170 |

Heuristics and Criteria | 178 |

43 Moving Center | 194 |

ColumnConditional Data | 198 |

ComparableAggregable Data | 206 |

46 Multi Cluster Approximation | 217 |

61 Bilinear Clustering for Mixed Data | 286 |

62 KMeans and Bilinear Clustering | 298 |

63 ContributionBased Analysis of Partitions | 308 |

64 Partitioning in Aggregable Tables | 320 |

Hierarchy as a Clustering Structure | 329 |

71 Representing Hierarchy | 330 |

72 Monotone Equivariant Methods | 348 |

73 Ultrametrics and Tree Metrics | 354 |

74 Split Decomposition Theory | 363 |

75 Pyramids and Robinson Matrices | 375 |

76 A Linear Theory for Binary Hierarchies | 384 |

399 | |

423 | |

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### Common terms and phrases

AL(S analysis approximation average similarity biclustering bilinear model binary Boolean box clustering centroid classes classification cluster structure clustering algorithm clustering methods clusters found coefficient components concept conceptual clustering contingency contingency table contribution correlation corresponding criteria d-splits data matrix data scatter data set data table decomposition defined Discussion dissimilarity elements entities entity-to-variable entries equal equation equivalence relation equivalent Euclidean distance example formula fuzzy cluster implies indicator functions intensional interpretation involved iteration K-Means kind least-squares criterion Let us consider linear local search maximizing maximum means measure minimizing minimum spanning tree Mirkin moving center node clusters Obviously optimal pair parameters presented principal cluster principal component analysis problem procedure Proof quantitative represented residual satisfies scalar product Section SEFIT seriation similarity matrix single cluster spanning tree split standard point Statement step subsets threshold graph tree metric ultrametric values variable space variance vectors weights zero

### Popular passages

Page 400 - H.-J. Bandelt and A. Dress. Reconstructing the shape of a tree from observed dissimilarity data.

Page 411 - Arabie, P. (1994). The analysis of proximity matrices through sums of matrices having (anti-)Robinson forms.

Page 410 - In: IJ Cox, P. Hansen, and B. Julesz (Eds.) Partitioning Data Sets. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, American Mathematical Society, 105-116.

Page 401 - Baulieu, FB (1989) A classification of presence/absence based dissimilarity coefficients. Journal of Classification, 6, 233-46.

Page 405 - WS (1982). GENNCLUS: New models for general nonhierarchical clustering analysis. Psychometrika, 47, 446-449.