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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...
Citations
... Since X is stationary and ergodic, it readily follows that D x converges in distribution toward a complex normal distribution [27,4]. Since D s and D x are independent, the two above-mentioned steps thus only have to be performed for D s . ...
Accurate phase extraction from sinusoidal signals is a crucial task in various signal processing applications. While prior research predominantly addresses the case of asynchronous sampling with unknown signal frequency, this study focuses on the more specific situation where synchronous sampling is possible, and the signal’s frequency is known. In this framework, a comprehensive analysis of phase estimation accuracy in the presence of both additive and phase noises is presented. A closed-form expression for the asymptotic Probability Density Function (PDF) of the resulting phase estimator is validated by simulations depicting Root Mean Square Error (RMSE) trends in different noise scenarios. This estimator is asymptotically efficient, converging rapidly to its Cramèr-Rao Lower Bound (CRLB). Three distinct RMSE behaviours were identified based on SNR, sample count (N), and noise level: (i) saturation towards a random guess at low Signal to Noise Ratio (SNR), (ii) linear decrease with the square roots of N and SNR at moderate noise levels, and (iii) saturation at high SNR towards a noise floor dependent on the phase noise level. By quantifying the impact of sample count, additive noise, and phase noise on phase estimation accuracy, this work provides valuable insights for designing systems requiring precise phase extraction, such as phase-based fluorescence assays or system identification.
... The spectral analysis of functional time series has mainly been developed under Short Range Dependence (SRD). In this context, based on the weighted periodogram operator, a nonparametric framework is adopted in [20] . Particularly, the asymptotic normality of the functional discrete Fourier transform (fDFT), and the weighted periodogram operator of the curve data is proved under suitable summability conditions on the cumulant spectral density operators. ...
... The proposed test statistics operator based on the weighted periodogram operator is also formulated. Stationary SRD functional time series are characterized by the summability of the series of trace norms of the elements of the family of covariance operators {R τ , τ ∈ Z} (see, e.g., [20]). That is, X displays SRD if and only if ...
... Specifically, Lemmas 1-3 provide the asymptotic unbiasedness of the weighted periodogram operator in the space L 2 (M 2 d , dν ⊗ dν, C), and its consistency and asymptotic Gaussian distribution under H 0 . Their proofs can be obtained in the same way as in [20], where these results are established for the separable Hilbert space H = L 2 ([0, 1], C). The orthogonal expansion provided in Lemma 4 constitutes a key technical tool in the derivation of the proof of Theorem 1. ...
A statistical hypothesis test for long range dependence (LRD) in manifold-supported functional time series is formulated in the spectral domain. The proposed test statistic operator is based on the weighted periodogram operator. It is assumed that the elements of the spectral density operator family are invariant with respect to the group of isometries of the manifold. A Central Limit Theorem is derived to obtain the asymptotic Gaussian distribution of the proposed test statistics operator under the null hypothesis. The rate of convergence to zero, in the Hilbert--Schmidt operator norm, of the bias of the integrated empirical second and fourth order cumulant spectral density operators is established under the alternative hypothesis. The consistency of the test is derived, from the consistency, in the sense of the integrated mean square error, of the weighted periodogram operator under LRD. Our proposal to implement, in practice, the testing approach is based on the temporal-frequency-varying Karhunen-Lo\'eve expansion obtained here for invariant random Hilbert-Schmidt kernels on manifolds. A simulation study illustrates the main results regarding asymptotic normality and consistency, and the empirical size and power properties of the proposed testing approach.
... In this study, un-chart in Minitab Software was used. After analyzing and confirming the existence of time series components, using MATLAB software and using convolutional functions (conv) and moving average method [47,52,50]for the extraction and removal of the process component and of the Fourier series [16,79], was used to extract and remove the periodic component. ...
... Thus, there is a general interest in estimating these operators for which a reasonable (weak) dependence condition is required, see Dedecker et al. (2007) for a thorough review on weak dependence concepts. In fTSA, weak dependence in sense of cumulant mixing by Panaretos and Tavakoli (2013) is frequently used, but L p -m-approximability by Hörmann and Kokoszka (2010) even more often, as it usually follows from simple model assumptions. L p -m-approximability is used e.g. in functional linear regression, see Reimherr (2015) and Hörmann and Kidziński (2015), and parameter estimation of common fTS in separable Hilbert spaces , such as functional autoregressive (fAR), moving average (fMA), ARMA (fARMA), (generalized), autoregressive conditional heteroskedastic (f(G)ARCH) and invertible linear processes, see , Bosq (2000), Hörmann et al. (2013), Kühnert (2020), Kühnert et al. (2024), Kuenzer (2024), which requires the estimation of lagged covariance and cross-covariance operators of processes in Cartesian product ...
Estimating parameters of functional ARMA, GARCH and invertible processes requires estimating lagged covariance and cross-covariance operators of Cartesian product Hilbert space-valued processes. Asymptotic results have been derived in recent years, either less generally or under a strict condition. This article derives upper bounds of the estimation errors for such operators based on the mild condition L p-m-approximability for each lag, Cartesian power(s) and sample size, where the two processes can take values in different spaces in the context of lagged cross-covariance operators. Implications of our results on eigen elements and parameters in functional AR(MA) models are also discussed.
... which is continuous in ω and bounded; see Panaretos and Tavakoli [2013] and Hörmann, Kidzinski and Hallin (2013). We assume that the observations Y 1 , ..., Y n at hand are obtained as ...
Change-points in functional time series can be detected using the CUSUM-statistic, which is a non-linear functional of the partial sum process. Various methods have been proposed to obtain critical values for this statistic. In this paper we use the functional autoregressive sieve bootstrap to imitate the behavior of the partial sum process and we show that this procedure asymptotically correct estimates critical values under the null hypothesis. We also establish the consistency of the corresponding bootstrap based test under local alternatives. The finite sample performance of the procedure is studied via simulations under the null -hypothesis and under the alternative.
... Dimensionality reduction techniques for functional time series data have been in development. Panaretos and Tavakoli (2013) proposed a Fourier analysis for stationary functional time series. Hörmann et al. (2015) proposed a dynamic functional principal component analysis (DFPCA). ...
In this paper, we explore dimension reduction for time series of functional data within both stationary and non-stationary frameworks. We introduce a functional framework of generalized dynamic principal component analysis (GDPCA). The concept of GDPCA aims for better adaptation to possible nonstationary features of the series. We define the functional generalized dynamic principal component (GDPC) as static factor time series in a functional dynamic factor model and obtain the multivariate GDPC from a truncation of the functional dynamic factor model. GDFPCA uses a minimum squared error criterion to evaluate the reconstruction of the original functional time series. The computation of GDPC involves a two-step estimation of the coefficient vector of the loading curves in a basis expansion. We provide a proof of the consistency of the reconstruction of the original functional time series with GDPC converging in mean square to the original functional time series. Monte Carlo simulation studies indicate that the proposed GDFPCA is comparable to dynamic functional principal component analysis (DFPCA) when the data generating process is stationary, and outperforms DFPCA and FPCA when the data generating process is non-stationary. The results of applications to real data reaffirm the findings in simulation studies.
... Indeed, in this framework, an extended approach respect to the one given in [15] is adopted, since in that paper only the context of fractional integration of functional time series can be addressed. Note that, in our manifold scale varying spectral anal-ysis, Short Range Dependence (SRD) condition assumed in [19] is not required (see also [7], and [6] where SRD spherical functional time series are introduced and analyzed). ...
... Their spherical functional values are displayed at times t = 5, 7,9,11,13,15,17,19,21,23,25,27,29, 31, 33, 35, 37, 39 32 ...
... t =5,7,9,11,13,15,17,19,21,23,25, 27, 29, 31, 33, 35, 37, 39 ...
A functional nonlinear regression approach, incorporating time information in the covariates, is proposed for temporal strong correlated manifold map data sequence analysis. Specifically, the functional regression parameters are supported on a connected and compact two--point homogeneous space. The Generalized Least--Squares (GLS) parameter estimator is computed in the linearized model, having error term displaying manifold scale varying Long Range Dependence (LRD). The performance of the theoretical and plug--in nonlinear regression predictors is illustrated by simulations on sphere, in terms of the empirical mean of the computed spherical functional absolute errors. In the case where the second--order structure of the functional error term in the linearized model is unknown, its estimation is performed by minimum contrast in the functional spectral domain. The linear case is illustrated in the Supplementary Material, revealing the effect of the slow decay velocity in time of the trace norms of the covariance operator family of the regression LRD error term. The purely spatial statistical analysis of atmospheric pressure at high cloud bottom, and downward solar radiation flux in Alegria et al. (2021) is extended to the spatiotemporal context, illustrating the numerical results from a generated synthetic data set.
... This renders conventional common factor analysis uninformative or even inadequate when working with time series. For instance, as shown by Brillinger (1981) and Forni et al. (2000) for multivariate time series and by Panaretos and Tavakoli (2013) and Hörmann et al. (2015) for functional time series, common factors might inadequately capture serial dependencies due to overlooked autocovariance structures in the original dataset. ...
We propose a novel approximate factor model tailored for analyzing time-dependent curve data. Our model decomposes such data into two distinct components: a low-dimensional predictable factor component and an unpredictable error term. These components are identified through the autocovariance structure of the underlying functional time series. The model parameters are consistently estimated using the eigencomponents of a cumulative autocovariance operator and an information criterion is proposed to determine the appropriate number of factors. The methodology is applied to yield curve modeling and forecasting. Our results indicate that more than three factors are required to characterize the dynamics of the term structure of bond yields.
R Package: https://github.com/ottosven/dffm
... In this study, unchart in Minitab Software was used. After analyzing and confirming the existence of time series components, using MATLAB software and using convolutional functions (conv) and moving average method (Hyndman, 2011, Johnston et al., 1999, James, 1968 for the extraction and removal of the process component and of the Fourier series (Bloomfield, 2004, Panaretos andTavakoli, 2013), was used to extract and remove the periodic component. ...
... The approach of going to the Fourier domain analysis bypasses this issue. There are further related papers dealing with Fourier domain analysis: e.g., Aue and van Delft [17] and Panaretos and Tavakoli [18,19]. ...
