Variation Pattern Classification of Functional Data

A new classification method for functional data is proposed in this paper. This work is motivated by the need to identify features that discriminate between neurological conditions on which local field potentials (LFPs) were recorded. Regardless of the condition, these local field potentials have zero mean and thus the first moments of these random processes do not have discriminating power. We propose the variation pattern classification (VPC) method {which employs the (auto-)covariance operators as the discriminating features} and uses the Hilbert-Schmidt norm to measure the discrepancy between the (auto-)covariance operators of different groups. The proposed VPC method is demonstrated to be sensitive to the discrepancy, {potentially leading to a higher rate of classification}. One important innovation lies in the dimension reduction where the VPC method data-adaptively determines the basis functions (discriminative feature functions) that account for the major discrepancy. In addition, the selected discriminative feature functions provide insights on the discrepancy between different groups because they reveal the features of variation pattern that differentiate groups. Consistency properties are established and, furthermore, simulation studies and the analysis of rat brain LFP trajectories empirically demonstrate the advantages and effectiveness of the proposed method.