Many neuroscience experiments record sequential trajectories where each trajectory consists of oscillations and fluctuations around zero. Such trajectories can be viewed as zero‐mean functional data. When there are structural breaks in higher‐order moments, it is not always easy to spot these by mere visual inspection. Motivated by this challenging problem in brain signal analysis, we propose a detection and testing procedure to find the change point in functional covariance. The detection procedure is based on the cumulative sum statistics (CUSUM). The fully functional testing procedure relies on a null distribution which depends on infinitely many unknown parameters, though in practice only a finite number of these parameters can be included for the hypothesis test of the existence of change point. This paper provides some theoretical insights on the influence of the number of parameters. Meanwhile, the asymptotic properties of the estimated change point are developed. The effectiveness of the proposed method is numerically validated in simulation studies and an application to investigate changes in rat brain signals following an experimentally‐induced stroke.
A new classification method for functional data is proposed in this article. 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 LFPs 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 second‐moment structure as the discriminating feature and uses the Hilbert–Schmidt norm to measure the discrepancy between the second‐moment structure 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 adaptively determines the basis functions (discriminative feature functions) that account for the major discrepancy. In addition, the selected discriminative feature functions provide insights into 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.
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