Adaptive Hausdorff distances and dynamic clustering of symbolic interval data

Abstract

This paper presents a partitional dynamic clustering method for interval data based on adaptive Hausdorff distances. Dynamic clustering algorithms are iterative two-step relocation algorithms involving the construction of the clusters at each iteration and the identification of a suitable representation or prototype (means, axes, probability laws, groups of elements, etc.) for each cluster by locally optimizing an adequacy criterion that measures the fitting between the clusters and their corresponding representatives. In this paper, each pattern is represented by a vector of intervals. Adaptive Hausdorff distances are the measures used to compare two interval vectors. Adaptive distances at each iteration change for each cluster according to its intra-class structure. The advantage of these adaptive distances is that the clustering algorithm is able to recognize clusters of different shapes and sizes. To evaluate this method, experiments with real and synthetic interval data sets were performed. The evaluation is based on an external cluster validity index (corrected Rand index) in a framework of a Monte Carlo experiment with 100 replications. These experiments showed the usefulness of the proposed method.

An Improved Automatic Algorithm for Global Eddy Tracking Using Satellite Altimeter Data

“…The HD has been widely used in image matching and shape comparison tasks and can be applied to measure the similarity between two patterns of points in images denoting edges [38][39][40]. Given two finite point sets = {a1, a2, a3... am} and = {b1, b2, b3... bn}, the HD is defined as follows: …”

Clustering of Symbolic Data based on Affinity Coefficient: Application to a Real Data Set

“…Table 2 shows part of the corresponding data matrix (De Carvalho et al, 2006b); the complete data matrix is available in the SODAS (Symbolic Official Data Analysis System) Software. According to De Carvalho et al (2006b), several studies carried out in French Guyana indicated abnormal levels of mercury contamination in some Amerindian populations. This contamination was connected to their consumption of contaminated freshwater fish.…”

“…In order to study this phenomenon, the data set mentioned above was collected by researchers from The symbolic objects (species) were grouped into four a priori clusters according to diet. The a priori classification is as follows (De Carvalho et al, 2006b): In order to apply the affinity coefficient a WW (k,k), with equal weights ( j =1/p), a transformed data matrix was computed, according to that described in Bacelar- Nicolau et al (2009Nicolau et al ( , 2010. Each interval variable (generalized column) gave a sub-table with a suitable number of columns corresponding to a set of elementary intervals.…”

“…Some dissimilarity coefficients for the particular case of interval data can be found in the literature (see e.g. Chavent and Lechevallier, 2002;Chavent et al, 2003;Souza and De Carvalho, 2004;De Carvalho et al, 2006a, 2006b). Bacelar-Nicolau et al (2009) use a similarity coefficient, namely, a generalized affinity coefficient (Matusita, 1951;Bacelar-Nicolau, 1980, 1988, 2000, for clustering data units described by interval variables, within the scope of hierarchical clustering of complex and heterogeneous data.…”

Self-organizing map for symbolic data

“…For considering the cluster validity problem on symbolic data, we will set up an XB-like validity index for symbolic data under the FSCM along with the XB validity index proposed by Xie and Beni [47] in Section 5. We finally mention that, because symbolic interval data (or only called interval data) are most appeared in real applications, there are various methods for clustering interval data in the literature, such as Chavent et al [6], de Carvalho et al [11,12,[14][15][16][17], de Souza et al [18], Guru et al [25,26] and Irpino and Verde [31], etc. We next propose the S-SOM for symbolic data.…”