On a definition of a functional depth through finite projections: a FPCA approach
Résumé
In this paper, we present a new functional depth called Principal Component functional Depth (PCD) for square-integrable processes X over a compact set. This depth involves a generic multivariate depth function which is evaluated at the projection of the function on the basis formed by the first J vectors of the Karhunen-Loève decomposition of X. We first investigate whether our Principal Component Decomposition (PCD) satisfies the desirable properties of statistical functional depths, following the axiomatization of [21]. In a second step, we present a consistent estimator for our PCD and establish its uniform consistency with a convergence rate. Finally, we complement our study with simulations and various real-world applications in supervised classification, demonstrating that our maximum PCD classifier equals or outperforms conventional competitors.
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