Consistently recovering the signal from noisy functional data

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 do not require specific smoothness assumptions for the underlying curves and allow for a certain degree of autocorrelation in the noise.