Weakly dependent functional data

TL;DR

This paper introduces a moment-based dependence measure for functional time series, analyzing its effects on statistical procedures like PCA, covariance estimation, change point detection, and linear modeling, especially under weak dependence.

Contribution

It proposes a new dependence notion for functional data and studies its impact on key statistical methods, bridging the gap between independence and dependence in functional time series.

Findings

Dependence affects PCA and covariance estimation in functional data.

Weak dependence can preserve robustness of certain statistical procedures.

The new measure applies to both linear and nonlinear functional time series.

Abstract

Functional data often arise from measurements on fine time grids and are obtained by separating an almost continuous time record into natural consecutive intervals, for example, days. The functions thus obtained form a functional time series, and the central issue in the analysis of such data consists in taking into account the temporal dependence of these functional observations. Examples include daily curves of financial transaction data and daily patterns of geophysical and environmental data. For scalar and vector valued stochastic processes, a large number of dependence notions have been proposed, mostly involving mixing type distances between $\sigma$-algebras. In time series analysis, measures of dependence based on moments have proven most useful (autocovariances and cumulants). We introduce a moment-based notion of dependence for functional time series which involves $m$-dependence. We show that it is applicable to linear as well as nonlinear functional time series. Then we investigate the impact of dependence thus quantified on several important statistical procedures for functional data. We study the estimation of the functional principal components, the long-run covariance matrix, change point detection and the functional linear model. We explain when temporal dependence affects the results obtained for i.i.d. functional observations and when these results are robust to weak dependence.