Geometric Empirical Bayesian Model for Classification of Functional Data Under Diverse Sampling Regimes
Functional data analysis (FDA) is focused on various statistical tasks, including inference, for observations that vary over a continuum, which are not effectively addressed by multivariate methods. A feature of these functional observations is the presence of two distinct forms of variability: amplitude that describes differences in magnitudes of features, e.g., extrema, and phase that describes differences in timings of amplitude features. One area of focus in FDA is the classification of new observations based on previously observed training data that has been split into predefined classes. Existing methods fail to directly account for both phase and amplitude variability, and work under the restrictive assumption that functional observations are measured on a common, fine grid over the input domain. In this work, we address these issues directly by formulating a Bayesian hierarchical model for irregular, fragmented or sparsely sampled functional observations, where training data from different classes are available. Our approach builds on a recently developed inferential framework for incomplete functional observations and the elastic FDA framework for characterizing amplitude and phase variability. The approach operates by inferring individual parameters that separately track amplitude and phase, which can be combined to infer complete functions underlying each observation, and a class parameter, which can be used to discern the class membership of an observation based on the training data. We validate the proposed framework using simulation studies and real data applications, and showcase the advantages of this perspective when both amplitude and phase variability are present in the data.