Minimax estimation of Functional Principal Components from noisy discretized functional data

TL;DR

This paper investigates the statistical properties of functional principal component analysis (FPCA) when applied to noisy, discretized functional data, providing new convergence rates and optimal estimation methods.

Contribution

It introduces a double asymptotic framework and demonstrates that histogram-based projections achieve minimax optimal rates for FPCA with noisy, discretized data.

Findings

Proves new convergence rates for FPCA estimators.

Shows histogram-based projections are minimax optimal.

Validates results through simulations and genomic data analysis.

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.