Consistently recovering the signal from noisy functional data

  • Fatima Jammoul (Speaker)

Activity: Talk or presentationTalk at conference or symposiumScience to science

Description

We consider noisy functional data $Y_t(s_i) = X_t(s_i) + u_{ti}$ that has been recorded at a discrete set of observation points. Naturally, the goal is to recover the underlying signal $X_t$. Commonly, this is done by non-parametric smoothing approaches, e.g. kernel smoothing or spline fitting. These methods act function by function and do not take the overall presented information into consideration. We argue that it is often more accurate to take the entire data set into account, which can help recover systematic properties of the underlying signal. Other approaches using functional principal components do just that, but require strong assumptions on the smoothness of the underlying signal. We show that under very mild assumptions, the signal may be viewed as the common components of a factor model. Using this discovery, we develop a PCA driven approach to recover the signal and show consistency. 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. We demonstrate the applicability of our approach with simulation experiments and real life data analysis. Our considerations show that even in settings that are advantageous for competing methods, the factor model approach provides competitive results. In particular we observe that for growing sample size, the factor model approach shows an improving fit, which is not the case for classic spline smoothers. The proposed method performs particularly well in cases of rough data and provides insight into the nature of underlying functional structure in real life data cases.
Period28 Sep 2021
Event title15th German Probability and Statistics Days: GPSD 2021
Event typeConference
LocationVirtuell, Germany
Degree of RecognitionInternational