Title | Classification and Dissimilarity Analysis [electronic resource] / edited by Bernard Van Cutsem |
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Imprint | New York, NY : Springer New York, 1994 |

Connect to | http://dx.doi.org/10.1007/978-1-4612-2686-4 |

Descript | XIII, 238 p. 4 illus. online resource |

SUMMARY

Classifying objects according to their likeness seems to have been a step in the human process of acquiring knowledge, and it is certainly a basic part of many of the sciences. Historically, the scientific process has involved classification and organization particularly in sciences such as botany, geology, astronomy, and linguistics. In a modern context, we may view classification as deriving a hierarchical clustering of objects. Thus, classification is close to factorial analysis methods and to multi-dimensional scaling methods. It provides a mathematical underpinning to the analysis of dissimilarities between objects

CONTENT

1 Introduction -- 1.1 Classification in the history of Science -- 1.2 Dissimilarity analysis -- 1.3 Organisation of this publication -- 1.4 References -- 2 The partial order by inclusion of the principal classes of dissimilarity on a finite set, and some of their basic properties -- 2.1 Introduction -- 2.2 Preliminaries -- 2.3 The general structures of dissimilarity data analysis and their geometrical and topological nature -- 2.4 Inclusions -- 2.5 The convex hulls -- 2.6 When are the inclusions strict? -- 2.7 The inclusions shown are exhaustive -- 2.8 Discussion -- Acknowledgements -- References -- 3 Similarity functions -- 3.1 Introduction -- 3.2 Definitions. Examples -- 3.3 The WM (DP) forms -- 3.4 The WM(D) form -- Appendix: Some indices of dissimilarity for categorical variables -- References -- 4 An order-theoretic unification and generalisation of certain fundamental bijections in mathematical classification. I -- 4.1 Introduction and overview -- 4.2 A few notes on ordered sets -- 4.3 Predissimilarities -- 4.4 Bijections -- 4.5 The unifying and generalising result -- 4.6 Further properties of an ordered set -- 4.7 Stratifications and generalised stratifications -- 4.8 Residual maps -- 4.9 On the associated residuated maps -- 4.10 Some applications to mathematical classification -- Acknowledgements -- Appendix A: Proofs -- References -- 5 An order-theoretic unification and generalisation of certain fundamental bijections in mathematical classification. II -- 5.1 Introduction and overview -- 5.2 The case E = A ร{151} B of theorem 4.5.1 -- 5.3 Other aspects of the case E = A ร{151} B -- 5.4 Prefilters -- 5.5 Ultrametrics and reflexive level foliations -- 5.6 On generalisations of indexed hierarchies -- 5.7 Benzรฉcri structures -- 5.8 Subdominants -- Acknowledgements -- Appendix B: Proofs -- References -- 6 The residuation model for the ordinal construction of dissimilarities and other valued objects -- 6.1 Introduction -- 6.2 Residuated mappings and closure operators -- 6.3 Lattices of objects and lattices of values -- 6.4 Valued objects -- 6.5 Lattices of valued objects -- 6.6 Notes and conclusions -- Acknowledgements -- References -- 7 On exchangeability-based equivalence relations induced by strongly Robinson and, in particular, by quadripolar Robinson dissimilarity matrices -- 7.1 Overview -- 7.2 Preliminaries -- 7.3 Quadripolar Robinson matrices of order four -- Equivalence relations induced by strongly Robinson matrices -- 7.5 Reduced forms -- 7.6 Limiting r-forms of strongly Robinson matrices -- 7.4 Limiting r-forms of quadripolar Robinson matrices -- References -- 8 Dimensionality problems in L1-norm representations -- 8.1 Introduction -- 8.2 Preliminaries and notations -- 8.3 Dimensionality for semi-distances of Lp-type -- 8.4 Dimensionality for semi-distances of L1-type -- 8.5 Numerical characterizations of semi-distances of L1-type -- 8.6 Appendices -- References -- Unified reference list

Mathematics
Probabilities
Mathematics
Probability Theory and Stochastic Processes