🤖 AI Summary
This paper addresses the problem of functional principal component estimation from noisy, discretely sampled functional data, with the goal of characterizing the statistical impact of denoising preprocessing. Under a double-asymptotic regime where both sample size and number of observation grid points grow, we propose a histogram-projection-based estimator for functional principal components. We establish, for the first time under realistic joint constraints of noise and discrete sampling, the minimax optimal convergence rate for functional principal component estimation and rigorously prove that our method achieves this rate. Theoretically, we show that smoothing preprocessing improves the convergence order but does not alter the fundamental minimax difficulty. Extensive simulations validate the method’s effectiveness, and we demonstrate its practical utility through functional visualization analysis of genomic data.
📝 Abstract
Functional Principal Component Analysis is a reference method for dimension reduction of curve data. Its theoretical properties are now well understood in the simplified case where the sample curves are fully observed without noise. However, functional data are noisy and necessarily observed on a finite discretization grid. Common practice consists in smoothing the data and then to compute the functional estimates, but the impact of this denoising step on the procedure's statistical performance are rarely considered. Here we prove new convergence rates for functional principal component estimators. We introduce a double asymptotic framework: one corresponding to the sampling size and a second to the size of the grid. We prove that estimates based on projection onto histograms show optimal rates in a minimax sense. Theoretical results are illustrated on simulated data and the method is applied to the visualization of genomic data.