Exploring Multivariate Modality by Unsupervised Mixture of Cubic B-Splines in 1-D Using Model Selection Criteria

Abstract

This paper proposes and considers a new distribution-free technique based on the work of Atilgan and Bozdogan (1990, 1992) to explore multivariate modality of a given multivariate data set using an unsupervised mixture of cubic B-Spline in density estimation in one dimensional spece. In the multivariate case, this is achieved by utilizing the Mahalanobis (1936) distance of each point from the multivariate mean (centroid), Mahalanobis distance data depth (MDD), jackknife Mahalanobis distance data depth (JMDD), principal components (PC) transformation, and common PC transformation. Analysis is carried out under the assumption that we do not know a priori of the classification or the grouping of the data. These dimension reduction techniques result in a better description of the shape of the underlying distribution which has many attractive properties. The EM algorithm is developed for the mixture of cubic B-splines in an interactive symbolic computational environment to obtain the maximum likelihood estimators of the parameters and to evaluate model selection criteria such as AIC (Akaike, 1973), CAIC (Bozdogan, 1987), and ICOMP (Bozdogan 1988,1990, 1993, 1994) in objectively detecting the modality of the data and in density estimation to determine the most parsimonious fit. A real numerical example is shown with multivariate data set with a known number of mixture clusters and configurations to illustrate the versatility and efficiency of the proposed approach in detecting the multimodality and the density concentration points (clusters) of multivariate data with a high degree of accuracy in univariate space.