A nonparametric bayesian approach for modeling and comparison of functional data

Abstract

The current advances of technology provides, among other things, several ways of collecting data, which enlarges the possibility of studying new phenomena. Researches focused on studying the functional relation between a variable and some quantity (usually time) produce the called functional data. The main feature of this kind of data is that they are registered using devices that can record values almost continuously over time. Suppose two groups of functional data and the interest is to evaluate the similarity of the groups over some range of time. This work proposes a method to compare the groups using predictive samples. The method submit data to a smoothing step using orthonormal functions series and the coefficients of the series are then used to model functional data, due to the bijective relation between the target functions and their respective coefficients. The goal is to estimate the multivariate density associated to the coefficients of each group. Under nonparametric Bayesian context, the densities were estimated using Dirichlet Process Mixture model. Comparison of the functional data groups were performed using a dissimilarity index based on some L2-distance and estimated using the predcitive samples of the fitted DPM model. The index has a great interpretative appeal and constitute an useful tool for data analysis. Furthermore, it is proposed a bayesian scheme to test the homogeneity of groups of functional data based on the distance between the distributions of the processes for each instant of time. A quick simulation study is presented, as well as preliminary analysis in real functional data set.