Analysis of Distribution Valued Dissimilarity Data

We deal with methods for analyzing complex structured data, especially, distribution valued data. Nowadays, there are many requests to analyze various types of data including spatial data, time series data, functional data and symbolic data. The idea of symbolic data analysis proposed by Diday covers a large range of data structures. We focus on distribution valued dissimilarity data and multidimensional scaling (MDS) for these kinds of data. MDS is a powerful tool for analyzing dissimilarity data. The purpose of MDS is to construct a configuration of the objects from dissimilarities between objects. In conventional MDS, the input dissimilarity data are assumed (non-negative) real values. Dissimilarities between objects are sometime given probabilistically; dissimilarity data may be represented as distributions. We assume that the distributions between objects i and j are non-central chi-square distributions \({\chi }^{2}(p,{\delta }_{ij}/{\gamma }_{ij})\) multiplied by a scalar (say \({\gamma }_{ij}\)), i.e. \({s}_{ij}\,\sim \,{\gamma }_{ij}{\chi }^{2}(p,{\delta }_{ij}/{\gamma }_{ij})\). We propose a method of MDS under this assumption; the purpose of the method is to construct a configuration; \({x}_{i}\,\sim \,N({\mu }_{i},{\alpha }_{i}^{2}{I}_{p}),i = 1,2,\cdots \,,n\).