  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.
