Detection of points of impact in classification of Gaussian functional data
We consider the problem of binary classification of Gaussian functional data, assuming that all relevant information is contained in the values of the functions at a finite number of points (called points of impact). We propose a new variable selection method to identify these points, enabling the replacement of functional observations with finite-dimensional vectors while minimizing information loss.
Many existing methods rely on the analysis of dependence measures between the response variable and the function values at each point. Recent findings in regression suggest a direct link between impact points and the non-differentiability of the covariance function. We show that, under mild conditions, the points where distance covariance is non-differentiable coincide with the points of impact. Thus, identifying impact points reduces to detecting non-differentiability in distance covariance. We propose a detection method and evaluate its performance through simulations across various scenarios.
Keywords: Functional data analysis functional classification points of impact variable selection distance covariance.