Consistently recovering the signal from noisy functional data

TL;DR

This paper introduces a PCA-based method to consistently recover signals from noisy, discretely observed functional data without requiring smoothness assumptions, even with autocorrelated noise.

Contribution

It proposes a novel framework for signal recovery in functional data using factor model estimation, applicable under mild conditions including autocorrelated noise.

Findings

Consistent signal recovery achieved without smoothness assumptions

Method handles autocorrelated noise effectively

Theoretical guarantees under mild conditions

Abstract

In practice most functional data cannot be recorded on a continuum, but rather at discrete time points. It is also quite common that these measurements come with an additive error, which one would like eliminate for the statistical analysis. When the measurements for each functional datum are taken on the same grid, the underlying signal-plus-noise model can be viewed as a factor model. The signals refer to the common components of the factor model, the noise is related to the idiosyncratic components. We formulate a framework which allows to consistently recover the signal by a PCA based factor model estimation scheme. Our theoretical results hold under rather mild conditions, in particular we don't require specific smoothness assumptions for the underlying curves and allow for a certain degree of autocorrelation in the noise.