SOME OPTIMAL DEPTH FUNCTIONS IN CLASSIFICATION AND TESTING OF HOMOGENEITY FOR FUNCTIONAL DATA

Abstract

Notions of data depth have been discussed in literature with its extension to classification. In this study, maximum depth classification was considered for functional data with the aim of identifying optimal depth function under certain data structures. The data structures include the presence of location outlier, scale outlier, symmetric and asymmetric outlier. The performance of depth functions in maximum depth classification was investigated in terms of probability of misclassification using simulation and analysis of real data sets. The probability of misclassification was estimated in terms of mean proportion of misclassification. The optimal depth functions in maximum depth classification were identified. Also, depth-oriented Wilcoxon type tests of homogeneity was proposed for functional data. The performances of depth functions in the test of homogeneity were presented in terms of power of the test using simulation and analysis of real data sets. For the two sample problem, each observation in a population or sample was first converted to the distribution value of its depth function. Then, the Wilcoxon rank sum test was applied. For the three or more samples, each observation in a population or sample was first converted to the distribution value of its depth function. Then, the Kruskal-Wallis test was applied. An optimal depth functions in the test of homogeneity were identified as the one with the highest power. Modified band depth was found to be optimal in maximum depth classification under different contamination structures. For test of homogeneity based on Wilcoxon rank sum test, modified band depth and spatial depth were found to be the optimal depth functions. For Kruskal-wallis test, all the depth functions performed well, except modified band depth.