Fourier analysis of stationary time series in function space

By Victor M. Panaretos, Shahin Tavakoli. 2013

155 citations recorded by the source corpus.

We develop the basic building blocks of a frequency domain framework for drawing statistical inferences on the second-order structure of a stationary sequence of functional data. The key element in such a context is the spectral density operator, which generalises the notion of a spectral density matrix to the functional setting, and characterises the second-order dynamics of the process. Our main tool is the functional Discrete Fourier Transform (fDFT). We derive an asymptotic Gaussian representation of the fDFT, thus allowing the transformation of the original collection of dependent random functions into a collection of approximately independent complex-valued Gaussian random functions. Our results are then employed in order to construct estimators of the spectral density operator based on smoothed versions of the periodogram kernel, the functional generalisation of the periodogram matrix. The consistency and asymptotic law of these estimators are studied in detail. As immediate consequences, we obtain central limit theorems for the mean and the long-run covariance operator of a stationary functional time series. Our results do not depend on structural modelling assumptions, but only functional versions of classical cumulant mixing conditions, and are shown to be stable under discrete observation of the individual curves.

Subfields: Economics and Econometrics, Finance, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/91632d9768a50a1eae8d3b076c393c457409c2db

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Citations in the corpus

Testing for stationarity of functional time series in the frequency domain (2017, methodology): “(4.2) A proof of these facts can be found in Panaretos & Tavakoli (2013).”
Functional Data Analysis (2016, background): “These include functional designs and domain selection problems and also dependent functional data such as functional time series, with many recent interesting developments, e.g. Panaretos & Tavakoli (2013).”
Dynamic functional principal components (2012, methodology): “The methodology relies on a frequency domain analysis for functional data: a topic which is still in its infancy (see, for instance, Panaretos and Tavakoli (2013a)).”
Testing stationarity of functional time series (2014, result): “We note however the recent work of Panaretos and Tavakoli (2013a), Panaretos and Tavakoli (2013b) and Hörmann et al. (2013) who advance the spectral analysis of stationary functional time series.”
A Plug‐in Bandwidth Selection Procedure for Long‐Run Covariance Estimation with Stationary Functional Time Series (2016, background): “Similar conditions to these are studied in the seminal work of Panaretos and Tavakoli (2013) on the topic Fourier analysis of functional time series, and in Zhang (2016).”
Peter Hall, functional data analysis and random objects (2016, background): “…a complete theory of functional auto-regressive models was already presented in Bosq (2000), this area recently has met with increasing interest, with models both in the spectral as well as time domains and also spatio-temporal extensions (Horvath and Kokoszka, 2012; Panaretos and Tavakoli, 2013).”
Locally Stationary Functional Time Series (2016, methodology): “1. It will become clear in the next section that a representation under these extra conditions excludes time-varying FAR(p) models, which is why we will not impose them. Similar to proposition 2.1 of Panaretos and Tavakoli (2013b), it is easily shown that the Fourier transform of the autocovariance ”
Kaleidoscope: An Efficient, Learnable Representation For All Structured Linear Maps (2020, methodology): “For example, the DFT is a crucial step in the standard ﬁlter bank speech preprocessing pipeline (Jurafsky & Martin, 2014), and is commonly used when dealing with time series data in general (Panaretos & Tavakoli, 2013).”
White Noise Testing and Model Diagnostic Checking for Functional Time Series (2016, methodology): “Our test is constructed based on the periodogram function recently X. Zhang / Journal of Econometrics 194 (2016) 76–95 77 introduced by Panaretos and Tavakoli (2013a) and it has nontrivial power against Pitman’s local alternatives that are within a √ T - neighborhood of the null hypothesis, where T…”
Cramér–Karhunen–Loève representation and harmonic principal component analysis of functional time series (2013, methodology): “Basic Definitions and Assumptions Our basic assumptions concerning the smoothness of the curves {Xt(τ)} and the strength of dependence between the elements of the sequence {Xt} follow those in Panaretos and Tavakoli [33].”