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On Some Facets of the Partition Set of a Finite Set Read chapter Read first chapter

2016 | OriginalPaper | Chapter

9. Quality Measures in Clustering

Author : Israël César Lerman

Published in: Foundations and Methods in Combinatorial and Statistical Data Analysis and Clustering

Publisher: Springer London

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Abstract

The construction of a clustering criterion depends on the nature of the data and the mathematical structure retained for its representation. We saw in Chap. 2 two criteria associated with two different methods: the “Central partition” and the “Dynamic adaptative” methods, respectively. A formal definition of a criterion in Data Analysis may be expressed as follows: “The structure \(\sigma \) on the set E concerned to which the data representation belongs (e.g. similarity coefficient on E) is more general than that \(\tau \) of the structure sought (partition or ordered chain of partitions on E). The general strategy consists of injecting the family \(\Theta \) of the \(\tau \) structures on E into the family \(\Sigma \) of the \(\sigma \) structures on E. The objective is then to determine that or those of the elements of \(\Theta \) which are the most comparable with \(\sigma (E)\) (\(\sigma \) calculated on E). For this purpose, a criterion is built. Quite generally, a criterion is a ranking (preorder) relation (possibly partial) on the set \(\Theta \). For the latter, \(\tau (E)\) is preferred to \(\tau '(E)\) if and only if, in a given sense, \(\tau (E)\) is nearer \(\sigma (E)\) than \(\tau '(E)\)”.

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previous chapter The Notion of “Natural” Class, Tools for Its Interpretation. The Classifiability Concept
next chapter Building a Classification Tree
Footnotes
1
The rank notion used corresponds to that given in Sect. 3.​3.​3 of Chap. 3.
 
Literature
1.
Benzécri, J.-P.: L’Analyse des Données, 1 La Taxinomie. Dunod (1973)
2.
Benzécri, J.P.: Analyse factorielle des proximités. Publications de l’Institut de Statistique de l’Université de Paris, (13–14):13:235–282, 14:219–246 (1964–1965)
3.
Bertin, J.: La Graphique et le traitement graphique de l’Information. Flammarion (1977)
4.
Bertin, J.: Graphic and graphic information process, translated by William J. Berg and Paul Scott. de Gruyter (1981)
5.
Daniels, H.E.: The relation between measures of correlation in the universe of sample permutations. Biometrika 33, 129–135 (1944)MathSciNetCrossRefMATH
6.
de la Vega, W.F.: Techniques de classification automatique utilisant un indice de ressemblance. Revue Francaise de Sociologie, (8–4):506–520 (1967)
7.
de la Vega, W.F.: Quelques propriétés des hiérarchies de classification. In Gardin, J.-C. (ed.) Archéologie et Calculateurs, pp. 329–343. Centre National de la Recherche Scientifique (1970)
8.
Ganter, B., Wille, R.: Formal Concept Analysis, Mathematical Foundations. Springer, New York (1999)CrossRefMATH
9.
Ghazzali, N.: Comparaison et réduction d ’ arbres de classification, en relation avec des problèmes de quantification en imagerie numérique. Ph.D. thesis, Université de Rennes 1, mai 1992
10.
Govaert, G.: Classification croisée, Doctorat d’Etat. Ph.D. thesis, University of Paris 6 (1983)
11.
Govaert, G.: La classification croisée. La Revue de Modulad 4:9–36 (1989)
12.
Hartigan, J.A.: Direct clustering of a data matrix. J. Am. Stat. Assoc. 67(337), 123–129 (1972)CrossRef
13.
Hartigan, J.A.: Clustering Algorithms. Wiley, New York (1975)MATH
14.
Huang, Z.: Extensions to the k-means algorithm for clustering large data sets with categorical values. Data Min. Knowl. Discov. 2, 283–304 (1998)CrossRef
15.
Hubert, L.J.: Inference procedures for the evaluation and comparison of proximity matrices. In: Felsenstein, J. (ed.) Numerical Taxonomy, pp. 209–228. Springer, New York (1983)CrossRef
16.
Kendall, D.G.: Seriation from abundance matrices. In: Kendall, D.G., Hodson, F.R., Tautu, P. (eds.) Mathematics in the Archaeological and Historical Sciences, pp. 215–252. Edinburgh University Press, Edinburgh (1971)
17.
Lefebvre, B., Losfeld, J.: Formalisation constructive de la notion de classe polythétique pour un tableau de données binaires. In Diday, E., et al. (eds.) Analyse des Données et Informatique. IRIA (1979)
18.
Leredde, H.: La méthode des pôles d’attraction; La méthode des pôles d’agrégation: deux nouvelles familles de classification automatique et sériation, Volume I: méthodes et exemples réels, Volume II: Programmes. Ph.D. thesis, University of Paris 6 (1979)
19.
Lerman, I.-C.: Les bases de la classification automatique. Gauthier-Villars (1970)
20.
Lerman, I.C.: On two criteria of classification. In: Cole, A.J. (ed.) Numerical Taxonomy, pp. 114–128. Academic Press, New York (1969)
21.
Lerman, I.C.: Analyse du phénomène de la sériation. Revue Mathématique et Sciences Humaines 38:39–57 (1972)
22.
23.
Lerman, I.C.: Group methodology in production management. Appl. Stoch. Models Data Anal. 2, 153–165 (1986)CrossRef
24.
Lerman, I.C.: Comparing classification tree structures: a special case of comparing \(q\)-ary relations. RAIRO Oper. Res. 33, 339–365 (1999)MathSciNetCrossRefMATH
25.
Lerman, I.C., Ghazzali, N.: What do we retain from a classification tree? In: Diday, E., Lechevallier, Y. (eds.) Symbolic-Numeric Data Analysis and Learning, pp. 27–42. Nova Science, New York (1991)
26.
Lerman, I.C., Leredde, H.: La méthode des pôles d’attraction. In Diday, E., et al. (eds.) Analyse des Données et Informatique, pp. 37–50. IRIA (1977)
27.
Lerman, I.C., Pinto da Costa, J., Silva, H.: Validation of very large data sets clustering by means of a nonparametric linear criterion. In: Bock, H.-H., Jajuga, K., Sokolowski, A. (eds.) Classification, Clustering and Data Analysis, pp. 147–157. Springer, New York (2002)CrossRef
28.
Lerman, I.C., Rouxel, F.: Comparing classification tree structures: a special case of comparing \(q\)-ary relations ii. RAIRO Oper. Res. 34, 251–281 (2000)MathSciNetCrossRefMATH
29.
30.
Madeira, S.C., Oliveira, A.L.: Biclustering algorithms for biological data analysis. IEEE Trans. Comput. Biol. Bioinf. 1, 24–45 (2004)CrossRef
31.
Mannila, H.: Finding total and partial orders from data for seriation. Lect. Notes Comput. Sci. 5255, 16–25 (2008)CrossRef
32.
Milligan, G.W., Cooper, M.C.: An examination of procedures for determining the number of clusters in a data set. Psychometrika 2:159–179 (1985)
33.
Mollière, J.-L.: What’s the real number of clusters. In: Gaul, W., Schader, M. (eds.) Classification as a Tool of Research, pp. 311–320. North-Holland, Amsterdam (1986)
34.
35.
Nicolau, M.H.: Analyse d’un algorithme de classification. Ph.D. thesis, University of Paris 6 (1972)
36.
Pieraut-le, G., van Meter, K.: Étude génétique de la construction d’une propriété relationnelle: La propriété de passage. CNRS (1976)
37.
Shepard, R.N.: The analysis of proximities: multidimensional scaling with unknown distance function. Psychometrika 27, 219–246 (1962)MathSciNetCrossRefMATH
38.
Tanay, A., Sharan, R., Shamir, R.: Biclustering algorithms: a survey. In: Aluru, S. (ed.) Handbook of Computational Molecular Biology, pp. 1–20. Chapman, Boca Raton (2004)
39.
Warren, S.S.: Cubic clustering criterion. Report A - 108, SAS Institute Inc. (1983)
40.
Wilderjans, T.F., Depril, D., Mechelen, I.V.: Additive biclustering: a comparison of one new and two existing ALS algorithms. J. Classif. 30, 56–74 (2013)MathSciNetCrossRef
41.
Zahn, C.T.: Approximating symmetric relations by equivalence relations. S.I.A.M. J. Appl. Math. 12, 840–847 (1964)MathSciNetMATH
Metadata
Title
Quality Measures in Clustering
Author
Israël César Lerman
Copyright Year
2016
Publisher
Springer London
Book
Foundations and Methods in Combinatorial and Statistical Data Analysis and Clustering

Print ISBN: 978-1-4471-6791-4

Electronic ISBN: 978-1-4471-6793-8

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-1-4471-6793-8_9

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