Weakly dependent functional data

Siegfried Hörmann, P. Kokoszka

Research output: Contribution to journalArticle

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 σ-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.
LanguageEnglish
Pages1845-1884
Number of pages40
JournalThe annals of statistics
Volume38
Issue number3
StatusPublished - 2010

Fingerprint

Functional Data
Dependent Data
Time series
M-dependence
Functional Linear Model
Measures of Dependence
Moment
Change-point Detection
Autocovariance
Weak Dependence
Time Series Analysis
Cumulants
Principal Components
Long-run
Covariance matrix
Transactions
Continuous Time
Stochastic Processes
Consecutive
Scalar

Cite this

Hörmann, S., & Kokoszka, P. (2010). Weakly dependent functional data. The annals of statistics, 38(3), 1845-1884.

Weakly dependent functional data. / Hörmann, Siegfried; Kokoszka, P.

In: The annals of statistics, Vol. 38, No. 3, 2010, p. 1845-1884.

Research output: Contribution to journalArticle

Hörmann, S & Kokoszka, P 2010, 'Weakly dependent functional data' The annals of statistics, vol 38, no. 3, pp. 1845-1884.
Hörmann, Siegfried ; Kokoszka, P./ Weakly dependent functional data. In: The annals of statistics. 2010 ; Vol. 38, No. 3. pp. 1845-1884
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