Weakly dependent functional data

By Siegfried Hormann, Piotr Kokoszka. 2010

394 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Economics and Econometrics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/409074cce105ef82155ad4a2fa7ee35986402aa0

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Citations in the corpus

Structural breaks in time series (2013, background): “The time series character of functional data was for the first time systematically treated in Hörmann and Kokoszka (2010), who also highlighted how structural break procedures are affected by serial functional correlation.”
Convolutional autoregressive models for functional time series (2016, neutral): “All the theoretical and empirical results in the literature were developed based on the models and methods in Bosq (2000), including Hörmann and Kokoszka (2010), Horváth, Huskova, and Kokoszka (2010), Aue, Noriho, and Hörmann (2012), Horváth and Kokoszka (2012), Horáth, Kokoszka, and Reeder (2012), ”
Testing the structural stability of temporally dependent functional observations and application to climate projections (2011, methodology): “As pointed out by Hörmann and Kokoszka [13], the BGHK test is not applicable to functional time series because it does not take the temporal dependence of ηi’s into account.”
Functional data analysis in the Banach space of continuous functions (2017, background): “[27] Siegfried Hörmann and Piotr Kokoszka.”
Weak invariance principles for sums of dependent random functions (2013, methodology): “Hörmann and Kokoszka [19] call the processes satisfying (1.1)–(1.4) L2 m-decomposable processes.”
Testing for stationarity of functional time series in the frequency domain (2017, background): “Hörmann & Kokoszka (2010) introduced a general weak dependence concept for stationary functional time series, while van Delft & Eichler (2016) provided a framework for locally stationary functional time series.”
Asymptotic normality of the principal components of functional time series (2013, methodology): “Hörmann and Kokoszka [17] and Hörmann et al. [16] discuss examples which include bilinear and ARCH-type functional processes.”
A Plug‐in Bandwidth Selection Procedure for Long‐Run Covariance Estimation with Stationary Functional Time Series (2016, methodology): “We refer the reader to Ferraty and Vieu (2006) and Hörmann and Kokoszka (2010) for an overview of the fields of functional data analysis and functional time-series analysis, and to Horváth et al. (2014), Tavakoli and Panaretos (2016), Aue et al. (2014), Aston and Kirch (2012a), Aston and Kirch…”
Long-Range Dependent Curve Time Series (2020, background): “…and Ziegelmann 2010); the other extends certain linear and nonlinear sequences and martingale or m-dependent approximation techniques (Bosq 2000; Hörmann and Kokoszka 2010; Horváth and Kokoszka 2012; Rice and Shang 2017), where it is usually assumed that Xt = g(εt , εt−1, . . .), where g : S∞ →…”
Tests for Error Correlation in the Functional Linear Model (2010, methodology): “Several examples of functional sequences satisfying (A2), (A4), and (A5) can be found in Hörmann and Kokoszka (2010) and Aue et al. (2010).”
Estimation in functional linear quantile regression (2012, background): “Note that [25] discussed weakly dependent functional data.”
Prediction of functional ARMA processes with an application to traffic data (2016, background): “In H¨ormann and Kokoszka [13] it is shown that, under L 4 − m approximability (a weak dependence concept for functional processes), empirical estimators of mean and covariance of the functional process are √ n -consistent.”
Estimation of the mean of functional time series and a two‐sample problem (2011, methodology): “The dependence condition that we use is satisfied by many examples of functional time series, including the linear and auto-regressive conditional heteroscedasticity type of processes; see Hörmann and Kokoszka (2010) and Aue et al. (2012).”
Cramér–Karhunen–Loève representation and harmonic principal component analysis of functional time series (2013, background): “See Kokoszka [26] for a review of recent developments in dependent functional data.”
Dynamic functional principal components (2012, methodology): “Turning to the proof of proposition 4, we first establish the following lemma, which is an extension to lag h autocovariance operators of a consistency result from Hörmann and Kokoszka (2010) on the empirical covariance operator.”
Uniform change point tests in high dimension (2015, result): “[6, 26, 30]), the literature in the latter case is rather sparse compared to the univariate theory.”