Dynamic Functional Principal Components

Abstract

We address the problem of dimension reduction for time series of functional data .X t : t 2 Z/. Such functional time series frequently arise, for example, when a continuous time process is segmented into some smaller natural units, such as days. Then each X t represents one intraday curve. We argue that functional principal component analysis, though a key technique in the field and a benchmark for any competitor, does not provide an adequate dimension reduction in a time series setting. Functional principal component analysis indeed is a static procedure which ignores the essential information that is provided by the serial dependence structure of the functional data under study. Therefore, inspired by Brillinger's theory of dynamic principal components, we propose a dynamic version of functional principal component analysis which is based on a frequency domain approach. By means of a simulation study and an empirical illustration, we show the considerable improvement that the dynamic approach entails when compared with the usual static procedure.

Principal component analysis: a review and recent developments

“…Sometimes, as well as correlations between the p variables, there is a dependence structure between the n observations. A 'dynamic' version of functional PCA is proposed in [24], which is relevant when there are correlations between the observed curves, as well as the obvious correlation within the curves. It is based on an idea first suggested by Brillinger [25] for vector time series and uses frequency domain analysis.…”

On the asymptotic normality of kernel estimators of the long run covariance of functional time series

“…Nearly all stationary time series models based on independent innovations satisfy condition (2.1), including linear processes in function spaces, and the functional ARCH and GARCH processes, see [8,22]. Condition (2.4) specifies the level of dependence that is allowed within the sequence in terms of how well it can be approximated in the L 2 sense by finite dependent processes, and thus defines a version of L p -mapproximability for functional time series, see [23].…”

Discussion of “analysis of spatio-temporal mobile phone data: a case study in the metropolitan area of Milan” by P. Secchi, S. Vantini, and V. Vitelli

“…Two problems have received much attention: smoothing taking into account the spatial dependence and prediction of functional signals at unobserved spatial locations. Dimension reduction for dependent functional data has been studied in various scenarios: time series of functional data (Hörmann et al 2015), spatially correlated multilevel functional data (Staicu et al 2010), and spatial functional data (Hörmann and Kokoszka 2013;Liu et al 2014). These papers mainly rely on techniques from FPCA.…”