Actes - Société Francophone de Classification

SFC 2009
Moreover, the mixture h m is such that minimizes JS divergence within hth cluster.
The final partition’s quality can be evaluated using the index obtained by ratio between
divergence, analogously to the way proposed by Chavent et al. [CHA 03]
5. Computation procedure
According to (4), the JS divergence among objects belonging to h th cluster has the following expression:
d
( h)
i
= H ( m ) " ( ) H ( f )
JS
h
#
! !
i
C
h
$
h
$
i
( h)
and after simple passages, we obtained the following expression for d :
d
' p
$ ( ( )
$
j
= % ! H f ) + H ( ci
) "
i
# ! C $ % & j=
" i h 1
#
( h)
i
JS H ( mh
) " ! ( h)
JS
B
d JS and total JS
Then, we can easily compute JS dissimilarities among objects in a cluster, computing copula entropy, marginal
entropies and mixture entropy separately. To obtain these quantities, numerical integration procedure, based on
adaptive methods, can be used. Subsequently, the W
JS
d quantity can be computed.
The proposed clustering algorithm allows us to find simultaneously the best partition of symbolic objects,
according to the chosen criterion, and a suitable model to describing dependence inside observations.
6. Bibliography
[BOC 00] BOCK H.H., DIDAY E., Analysis of Symbolic Data, Expanatory methods for extracting statistical
informations from Complex data, Studies in Classification, Data Analysis and Knowledge Organization,
Springer Verlag, 2000.
[CHA 03] CHAVENT M., DE CARVALHO F.A.T., LECHEVALLIER Y., VERDE R., Trois nouvelles méthodes de
classification automatique des données symbolique de type intervalle, Revue de Statistique Appliquée, vol. 4,
2003, p. 5-29.
[PAP 91] PAPOULIS A., Probability, Random Variables and Stochastic Process, McGraw-Hill,1991.
[SHA 71] SHANNON C.E., WEAVER W., La teoria matematica delle comunicazioni, Etas Kompass, 1971.
[SKL 59] SKLAR A., Fonctions de répartition à n dimension et leurs marges. Publications de l’Institut de
Statistique de l’Université de Paris, vol. 8, 1959, p. 229-231
[VER 08] VERDE E., IRPINO A., Comparing Histogram Data Using Mahalanobis-Wasserstein Distance,
Proceeding in Compstat 2008: Proceedings in Computational Statistics, Heidelberg, Physica-Verlag
Springer, 2008.
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