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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.
List of contents
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 × B of theorem 4.5.1.- 5.3 Other aspects of the case E = A × B.- 5.4 Prefilters.- 5.5 Ultrametrics and reflexive level foliations.- 5.6 On generalisations of indexed hierarchies.- 5.7 Benzécri structures.- 5.8 Subdominants.- Acknowledgements.- Appendix B: Proofs.- References.- 6 The residuationmodel 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.
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. Thus, classification is close to factorial analysis methods and to multi-dimensional scaling methods.
