Functional Finite Mixture Modelling and Estimation

Abstract

Functional data analysis is a branch of statistics that studies models for information represented by functions. Meanwhile, finite mixture models serve as a conerstone in the field of cluster analysis, offering a flexible probabilisitic framework for the representation of heterogeneous data. These models posit that the observed data are drawn from a mixture of several different probability distributions from the same family, where each is conventionally thought to represent a distinct group within the overall population. However, their representation in terms of densities makes their application to function-valued random variables, the foundation of functional data analysis, difficult. Herein, we utilize density surrogates derived from the Karhunen-Loeve expansion to circumvent this discrepancy and develop functional finite mixture models for the clustering of functional data. Models developed for real-valued and vector-valued functions of a single variable. Estimation of all models is done using the expectation-maximization algorithm, and copious amounts of simulations and data examples are provided to demonstrate the properties and performance of the methodologies. Additionally, we present a new estimation approach to be used in tandem with the stochastic expectation-maximization algorithm. This estimation method offers increased precision in estimation with respect to the algorithm chain length when compared to averaging the chain. Asymptotic properties of the estimator are derived, and simulation studies are given to demonstrate its performance.