Article

Bootstrapping Frequency Domain Tests in Multivariate Time Series with an Application to Comparing Spectral Densities

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Abstract

We propose a general bootstrap procedure to approximate the null distribution of non-parametric frequency domain tests about the spectral density matrix of a multivariate time series. Under a set of easy-to-verify conditions, we establish asymptotic validity of the bootstrap procedure proposed. We apply a version of this procedure together with a new statistic to test the hypothesis that the spectral densities of not necessarily independent time series are equal. The test statistic proposed is based on an "L"2-distance between the non-parametrically estimated individual spectral densities and an overall, 'pooled' spectral density, the latter being obtained by using the whole set of "m" time series considered. The effects of the dependence between the time series on the power behaviour of the test are investigated. Some simulations are presented and a real life data example is discussed. Copyright (c) 2009 Royal Statistical Society.

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... Note that the assumptions (i) and (ii) on the weight function W and the bandwidths (b T ) T , respectively, in Theorem 1 are identical to the assumptions for multivariate time series used in Dette and Paparoditis [9]. Remark 1. ...
... , for which central limit theorems for generalized quadratic forms have to be invoked. Even in the finite dimensional case, central limit theorems for generalized quadratic forms are established under more structural assumptions on the underlying processes than those needed to deal with the sequence √ Tb(F X ,λ − F X ,λ ); see for instance Eichler [10] who uses summability conditions on the cumulants of all order or Dette and Paparoditis [9] who use linearity assumptions on the underlying vector processes. The technical challenges in dealing with the test statistic (3), also justify the additional structural assumptions imposed in this paper in order to establish the limiting distribution of U T , as compared to those used in van Delft and Dette [34]. ...
... Furthermore, the denumerator θ 0 can be estimated using the estimators of the spectral density operators involved in calculating the test statistic U T . A problem, however, occurs from the well-known fact that, even in the finite-dimensional case, the convergence of the distribution of such L 2 -norm based tests towards their limiting (Gaussian) distribution is very slow; see, e.g., Härdle and Mammen [15], Paparoditis [23] and Dette and Paparoditis [9]. In this case, bootstrap-based approaches may be very effective. ...
Article
The problem of comparing the entire second order structure of two functional processes is considered and a L2-type statistic for testing equality of the corresponding spectral density operators is investigated. The test statistic evaluates, over all frequencies, the Hilbert–Schmidt distance between the two estimated spectral density operators. Under certain assumptions, the limiting distribution under the null hypothesis is derived. A novel frequency domain bootstrap method is introduced, which leads to a more accurate approximation of the distribution of the test statistic under the null than the large sample Gaussian approximation derived. Under quite general conditions, asymptotic validity of the bootstrap procedure is established for estimating the distribution of the test statistic under the null. Furthermore, consistency of the bootstrap-based test under the alternative is proved. Numerical simulations show that, even for small samples, the bootstrap-based test has a very good size and power behavior. An application to a bivariate real-life functional time series illustrates the methodology proposed.
... Under the same stationarity assumptions, [JP15] and [DP09] consider permutation tests for the equality of spectral matrices of multivariate time series based on the integrated L 2 -distance between smoothed periodograms. As pointed out in [JP15], these tests are usually not consistent for non-smoothed periodograms, which is caused by the use of non-consistent spectral estimators. ...
... As pointed out in [JP15], these tests are usually not consistent for non-smoothed periodograms, which is caused by the use of non-consistent spectral estimators. As a consequence, the permutation tests in [JP15] and [DP09] require initial smoothing of the periodograms, thereby introducing an additional tuning parameter in the test procedure. In contrast, other than the minimum amount of smoothing required to ensure positive definiteness of the periodograms, the manifold signed-rank test does not require consistent estimators of the underlying spectra, as the manifold difference scores are asympotically independent of the underlying spectra under the null. ...
... In contrast, other than the minimum amount of smoothing required to ensure positive definiteness of the periodograms, the manifold signed-rank test does not require consistent estimators of the underlying spectra, as the manifold difference scores are asympotically independent of the underlying spectra under the null. The asymptotic test remains computationally efficient also for a large number of Fourier frequencies or higher-dimensional spectral matrices, whereas the exact tests in [JP15] and [DP09] become computationally more burdensome, as we need to calculate many randomized test statistics. The power of the asymptotic and exact tests is investigated under several different scenarios by means of simulations in Section 6.3. ...
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In multivariate time series analysis, the non-degenerate autocovariance or spectral density matrices of a second-order stationary time series are necessarily Hermitian and positive definite. In this paper, the concept of a statistical data depth is generalized to data observations in the non-Euclidean space of Hermitian positive definite matrices by exploiting the geometric properties of the space as a Riemannian manifold. First, the desired properties of a data depth function acting on the space of Hermitian positive definite matrices are presented. Second, two computationally efficient pointwise and integrated data depth functions that satisfy each of these requirements are introduced. Besides the characterization of central regions or the detection of outlying observations, the new data depths also provide a practical framework to perform rank-based hypothesis testing for samples of Hermitian positive definite matrices. The latter is illustrated by the analysis of collections of spectral matrices in several multivariate brain signal time series datasets.
... Several authors have proposed tests based on 1 the integrated periodogram [see Anderson (1993) or Chen and Romano (1999) among others]. Because on one hand, test statistics based on the integrated periodogram are usually not distribution free and, on the other hand, the type of hypotheses that can be tested by the integrated periodogram is limited, alternative methods have been proposed which are based on estimates of the spectral density [see Taniguchi and Kondo (1993), Taniguchi et al. (1996), Paparoditis (2000), Dette and Spreckelsen (2003), Eichler (2008) or Dette and Paparoditis (2009), among others]. These methods usually yield a normal distribution as the asymptotic law of the corresponding test statistics, but require the specification of a smoothing parameter in order to get consistent estimates of the spectral density matrix. ...
... In Section 2 we introduce the necessary notation, the basic assumptions and explain the main principle of our approach in the case of testing the null hypothesis of a white noise process. Section 3 is devoted to the problem of comparing spectral densities of a multivariate time series [see Eichler (2008) or Dette and Paparoditis (2009)]. In all cases we show that the proposed test statistic is asymptotically normally distributed, and a simple goodness-of-fit test for the null hypothesis is proposed, which uses the quantiles of the standard normal distribution. ...
... Swanepoel and van Wyk (1986) considered two independent stationary autoregressive processes. Diggle and Fisher (1991) proposed graphical devices to compare periodograms, and a more recent reference is Dette and Paparoditis (2009), who proposed a bootstrap test for the problem of testing for equal spectral densities of m [not necessarily uncorrelated] time series {X j,t } t∈Z (j = 1, . . . , m), that is ...
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In this paper new tests for nonparametric hypotheses in stationary processes are proposed. Our approach is based on an estimate of the L^2-distance between the spectral density matrix and its best approximation under the null hypothesis. We explain the main idea in the problem of testing for a constant spectral density matrix and in the problem of comparing the spectral densities of several correlated stationary time series. The method is based on direct estimation of integrals of the spectral density matrix and does not require the specification of smoothing parameters. We show that the limit distribution of the proposed test statistic is normal and investigate the finite sample properties of the resulting tests by means of a small simulation study.
... Remark 3.3 A detailed discussion about the use of Theorem 3.1 can be found in Dette and Paparoditis (2009) and we only briefly mention the potential applications here. ...
... Carmona and Wang (1996), Coates and Diggle (1986), Swanepoel and van Wyk (1986) or Diggle and Fisher (1991) among others. Recently Dette and Paparoditis (2009) considered the case d = 2 and proposed to base a test for the hypothesis H 0 : f 11 = . . . = f dd on the statistic S T (Ψ) with the functional (1.2). ...
... Note that in the case d = 2 this result does not coincide with the corresponding statement in Dette and Paparoditis (2009) and that there is minor error in this reference. ...
Article
Full-text available
In a recent paper Eichler (2008) considered a class of non- and semiparametric hypotheses in multivariate stationary processes, which are characterized by a functional of the spectral density matrix. The corresponding statistics are obtained using kernel estimates for the spectral distribution and are asymptotically normal distributed under the null hypothesis and local alternatives. In this paper we derive the asymptotic properties of these test statistics under fixed alternatives. In particular we show also weak convergence but with a different rate compared to the null hypothesis.
... We now apply the above methodology to testing for equality of spectral densities i.e. Eichler [2008] and Dette and Paparoditis [2009] propose testing for equality of the spectral densities using an L 2 -distance, q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q ...
... This may explain why the proportion of rejection levels under the null using S β T are a little larger than those with S 0.25 T . Comparing our results to those reported in Dette and Paparoditis [2009], we see their frequency bootstrap procedure performs a little better. There are two possible explanations for this (i) they use a different test statistic, based on ratios (ii) the power transform may make the test a little conservative in rejecting the null. ...
Preprint
Inference for statistics of a stationary time series often involve nuisance parameters and sampling distributions that are difficult to estimate. In this paper, we propose the method of orthogonal samples, which can be used to address some of these issues. For a broad class of statistics, an orthogonal sample is constructed through a slight modification of the original statistic, such that it shares similar distributional properties as the centralised statistic of interest. We use the orthogonal sample to estimate nuisance parameters of weighted average periodogram estimators and L2L_{2}-type spectral statistics. Further, the orthogonal sample is utilized to estimate the finite sampling distribution of various test statistics under the null hypothesis. The proposed method is simple and computationally fast to implement. The viability of the method is illustrated with various simulations.
... Fan and Zhang [6] consider generalised likelihood ratio tests for the same hypothesis. There also is a rich literature on non-parametric comparison of the (multivariate) spectra of two time series, here some recent references include [7,8,9,10,11], but this list is by no means complete. ...
... here condition (A.2) from Lemma A.2 in [21] is satisfied with g(x) = x(| log x| + 1) d by (8). With Ψ(x) := x 6 the Orlicz norm ||X|| Ψ coincides with the L 6 norm ||X|| 6 = (E|X| 6 ) 1/6 so that we have, for any κ ∈ (0, 1) and sufficiently small ||a − b|| 1 , ...
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Finding parametric models that accurately describe the dependence structure of observed data is a central task in the analysis of time series. Classical frequency domain methods provide a popular set of tools for fitting and diagnostics of time series models, but their applicability is seriously impacted by the limitations of covariances as a measure of dependence. Motivated by recent developments of frequency domain methods that are based on copulas instead of covariances, we propose a novel graphical tool that allows to access the quality of time series models for describing dependencies that go beyond linearity. We provide a thorough theoretical justification of our approach and show in simulations that it can successfully distinguish between subtle differences of time series dynamics, including non-linear dynamics which result from GARCH and EGARCH models. We also demonstrate the utility of the proposed tools through an application to modeling returns of the S&P 500 stock market index.
... where the d 1 × d 1 matrix f Z,12 (ω) is the cross-spectral matrix of the processes X t and Y t and f Z, 11 (ω) and f Z, 22 (ω) are the spectral matrices of X t and Y t respectively. We consider testing ...
... The L 2 norm above in (12) on the spectral matrices is similar to the statistics considered in Eichler [9] and Dette and Paparoditis [11] wherein the problem of testing equality of spectral matrices is discussed. Suppose that Assumptions 1,2 are satisfied, an application of Theorem 3.5 of Eichler [9] yields, under H 0 , ...
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In some applications, it is important to compare the stochastic properties of two multivariate time series that have unequal dimensions. A new method is proposed to compare the spread of spectral information in two multivariate stationary processes with different dimensions. To measure discrepancies, a frequency specific spectral ratio (FS-ratio) statistic is proposed and its asymptotic properties are derived. The FS-ratio is blind to the dimension of the stationary process and captures the proportion of spectral power in various frequency bands. Here we develop a technique to automatically identify frequency bands that carry significant spectral power. We apply our method to track changes in the complexity of a 32-channel local field potential (LFP) signal from a rat following an experimentally induced stroke. At every epoch (a distinct time segment from the duration of the experiment), the nonstationary LFP signal is decomposed into stationary and nonstationary latent sources and the complexity is analyzed through these latent stationary sources and their dimensions that can change across epochs. The analysis indicates that spectral information in the Beta frequency band (12–30 Hertz) demonstrated the greatest change in structure and complexity due to the stroke.
... That is, a test which evaluates the differences between the entire, infinite dimensional, structure of the two spectral density operators compared. Notice that in the finite-dimensional case, i.e., for (univariate or multivariate) real-valued time series, such tests have been developed among others by Eichler (2008) and Dette and Paparoditis (2009). ...
... A problem in implementing the above test occurs from the well-known fact that, even in the finite-dimensional case, the convergence of such L 2 -norm based test statistics towards their limiting distribution is very slow; see, e.g., Härdle and Mammen (1993), Paparoditis (2000) and Dette and Paparoditis (2009 ...
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The problem of testing equality of the entire second order structure of two independent functional processes is considered. A fully functional L2L^2-type test is developed which evaluates, over all frequencies, the Hilbert-Schmidt distance between the estimated spectral density operators of the two processes. Under the assumption of a linear functional process, the asymptotic behavior of the test statistic is investigated and its limiting distribution under the null hypothesis is derived. Furthermore, a novel frequency domain bootstrap method is developed which leads to a more accurate approximation of the distribution of the test statistic under the null than the large sample Gaussian approximation derived. Asymptotic validity of the bootstrap procedure is established under very general conditions and consistency of the bootstrap-based test under the alternative is proved. Numerical simulations show that, even for small samples, the bootstrap-based test has a very good size and power behavior. An application to a bivariate real-life functional time series is also presented.
... It is, therefore, important to develop a fully functional test for the testing problem at hand. Notice that in the finite-dimensional case, i.e., for (univariate or multivariate) real-valued time series, such tests have been developed among others by Eichler (2008) and Dette and Paparoditis (2009). ...
... A problem in implementing the above test occurs from the well-known fact that, even in the finite-dimensional case, the convergence of such L 2 -norm based test statistics towards their limiting distribution is very slow; see, e.g., Härdle and Mammen (1993), Paparoditis (2000) and Dette and Paparoditis (2009). In this case, bootstrap-based approaches may be very effective. ...
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The problem of testing equality of the entire second order structure of two independent functional linear processes is considered. A fully functional L2L^2-type test is developed which evaluates, over all frequencies, the Hilbert-Schmidt distance between the estimated spectral density operators of the two processes. The asymptotic behavior of the test statistic is investigated and its limiting distribution under the null hypothesis is derived. Furthermore, a novel frequency domain bootstrap method is developed which approximates more accurately the distribution of the test statistic under the null than the large sample Gaussian approximation obtained. Asymptotic validity of the bootstrap procedure is established and consistency of the bootstrap-based test under the alternative is proved. Numerical simulations show that, even for small samples, the bootstrap-based test has very good size and power behavior. An application to meteorological functional time series is also presented.
... We now apply the above methodology to testing for equality of spectral densities i.e. Eichler [2008] and Dette and Paparoditis [2009] propose testing for equality of the spectral densities using an L 2 -distance, q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q ...
... This may explain why the proportion of rejection levels under the null using S β T are a little larger than those with S 0.25 T . Comparing our results to those reported in Dette and Paparoditis [2009], we see their frequency bootstrap procedure performs a little better. There are two possible explanations for this (i) they use a different test statistic, based on ratios (ii) the power transform may make the test a little conservative in rejecting the null. ...
Article
Inference for statistics of a stationary time series often involve nuisance parameters and sampling distributions that are difficult to estimate. In this paper, we propose the method of orthogonal samples, which can be used to address some of these issues. For a broad class of statistics, an orthogonal sample is constructed through a slight modification of the original statistic, such that it shares similar distributional properties as the centralised statistic of interest. We use the orthogonal sample to estimate nuisance parameters of weighted average periodogram estimators and L2L_{2}-type spectral statistics. Further, the orthogonal sample is utilized to estimate the finite sampling distribution of various test statistics under the null hypothesis. The proposed method is simple and computationally fast to implement. The viability of the method is illustrated with various simulations.
... Most existing methods in the literature, regardless of in frequency or time domain, were designed to compare only two time series. These methods include Coates and Diggle (1986), Diggle and Fisher (1991), Maharaj (2002), Alonso and Maharaj (2006), Lund, Bassily, and Vidakovic (2009), Dette and Paparoditis (2009), Decowski and Li (2015), Salcedo, Porto, and Morettin (2012), Jin and Wang (2016), Grant and Quinn (2017), Zhang and Tu (2018), Li and Lu (2018), Cirkovic and Fisher (2021), Jin (2021) and many others. ...
... To include other hypotheses, Eichler (2008) considered the testing problem described by the integrated squared (Euclidean) norm of the function of spectral density matrices. In the same spirit of Eichler (2008), tests for the equality of spectra are proposed using the bootstrap method (Dette and Paparoditis, 2009) and the randomization method (Jentsch and Pauly, 2015). On the other hand, supremum-type statistics are considered by Woodroofe and Van Ness (1967), Hannan (1970, Thorem 12, Chapter V.5), Rudzkis (1993), and Wu and Zaffaroni (2018). ...
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Coherence is a widely used measure to assess linear relationships between time series. However, it fails to capture nonlinear dependencies. To overcome this limitation, this paper introduces the notion of residual spectral density as a higher-order extension of the squared coherence. The method is based on an orthogonal decomposition of time series regression models. We propose a test for testing the existence of the residual spectrum and derive its fundamental properties. A numerical study illustrates finite sample performance of the proposed method. An application of the method shows that the residual spectrum can effectively detect brain connectivity.
... ,Dette and Paparoditis (2009),Dette and Hildebrandt (2012),Jentsch and Pauly (2015),Dette et al. (2011a) andDette et al. (2011b), to name a few. Among them, to test two specific null hypotheses,Dette et al. (2011a) proposed a method that requires an appropriate summation of the periodogram. ...
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In this paper, we propose a new frequency domain test for pairwise time reversibility at any specific couple of quantiles of two-dimensional marginal distribution. The proposed test is applicable to a very broad class of time series, regardless of the existence of moments and Markovian properties. By varying the couple of quantiles, the test can detect any violation of pairwise time reversibility. Our approach is based on an estimator of the L2-distance between the imaginary part of copula spectral density kernel and its value under the null hypothesis. We show that the limiting distribution of the proposed test statistic is normal and investigate the finite sample performance by means of a simulation study. We illustrate the use of the proposed test by applying it to stock price data.
... Many references have considered the comparison, classification and clustering of two or several time series. For example, De Souza and Thomson (1982), Coates and Diggle (1986), Potscher and Reschenhofer (1988), Diggle and Fisher (1991), Dargahi-Noubary (1992), Diggle and al Wasel (1997), Kakizawa et al. (1998), Timmer et al. (1999), Maharaj [(1999);(2000);(2002);(2005)], Caiado et al. (2006), Eichler (2008), Fokianos and Savvides (2008), Caiado et al. (2009), Dette and Paparoditis (2009), Dette et al. (2010), Hildebrandt (2011), Jentsch (2012), Jentsch and Pauly (2012), Salcedo et al. (2012), Jentsch and Pauly (2014), Triacca (2016), studied these subjects for stationary time series. ...
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Full-text available
In time series analysis, comparing spectral densities of several processes with almost periodic spectra is an interested problem. The aim of this paper is to give an approach to test the equality among spectral densities of several independent almost periodically correlated (cyclostationary) processes. This approach is based on the limiting distribution for the periodogram and the discrete Fourier transform. The simulation results indicate that the approach well acts.
... Many references have considered the comparison, classification and clustering of two or several processes. For example, De Souza and Thomson (1982), Coates and Diggle (1986), Potscher and Reschenhofer (1988), Diggle and Fisher (1991), Dargahi-Noubary (1992), Diggle and al Wasel (1997), Kakizawa et al. (1998), Timmer et al. (1999), Maharaj [(1999);(2000);(2002);(2005)], Caiado et al. (2006), Eichler (2008), Fokianos and Savvides (2008), Caiado et al. (2009), Dette and Paparoditis (2009), Dette et al. (2010), Hildebrandt (2011), Jentsch (2012), Jentsch and Pauly (2012), Salcedo et al. (2012), Jentsch and Pauly (2014), Triacca (2016), studied these subjects for stationary time series. ...
Preprint
Full-text available
In time series analysis, comparing spectral densities of several processes with almost periodic spectra is an interested problem. The aim of this paper is to give an approach to test the equality among spectral densities of several independent almost periodically correlated (cyclostationary) processes. This approach is based on the limiting distribution for the periodogram and the discrete Fourier transform. The simulation results indicate that the approach well acts.
... clustering of two or numerous time series processes were investigated in different time-domain and frequency-domain methods by several scientists [see e.g. De Souza and Thomson [1], Coates and Diggle [2], Potscher and Reschenhofer [3], Diggle and Fisher [4], Dargahi-Noubary [5], Diggle and al Wasel [6], Kakizawa et al. [7], Timmer et al. [8], Maharaj [9,10], Caiado et al. [11], Eichler [12], Fokianos and Savvides [13], Caiado et al. [14], Dette and Paparoditis [15], Dette et al. [16], Dette and Hildebrandt [17], Jentsch [18], Jentsch and Pauly [19], Salcedo et al. [20], Jentsch and Pauly [21], Mahmoudi et al. [22], Triacca [23]. Nearly all these approaches can be utilized to the stationary processes or the non-stationary processes (which are transformable to stationary processes by using differencing). ...
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Our primary objective in this article is to compare the spectral densities of numerous cyclostationary processes. By using the limiting distributions of the periodogram and the discrete Fourier transform, a novel approach is introduced to compare the spectral densities of independent cyclostationary processes. Also, the ability of the introduced approach is studied by employing simulated and real datasets.
... Most of the existing methods have been developed to compare different time series through their spectral densities or the autocovariance functions. To compare two stationary time series which are independent of each other, the methods include Coates and Diggle (1986), Diggle and Fisher (1991), Fokianos and Savvides (2008), Lund et al. (2009), Dette and Paparoditis (2009), Decowski and Li (2015), Jin and Wang (2016), Grant and Quinn (2017) and others. For comparing stationary time series that are dependent on each other, the methods include Alonso and Maharaj (2006), Jin (2011), Jentsch andPauly (2015) and Jin et al. (2019). ...
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Statistical comparison of time series is useful for the detection of mechanical damage and many other real-world applications. New methods have been proposed to check whether two semi-stationary time series have the same normalized dynamics. The proposed methods differ from traditional methods in that they are based on the Laplace periodogram, which is a robust tool to analyze the serial dependence of time series. Via the method of estimating equations, a generalized score statistic and an order selected statistic are developed for the comparison. Their asymptotic distributions under the null are obtained. The proposed methods are applicable to compare two semi-stationary time series which may be dependent on each other. They also can be used to compare two time series whose traditional spectral densities or autocovariance structures may not exist. A Monte Carlo simulation study illustrates the validity of the asymptotic results and the finite sample performance. The proposed methods have been applied to an analysis of non-stationary vibration signals for mechanical damage detection.
... Really, they may be interested in making comparisons between the probabilistic behaviors of several time series models fitted on several independent datasets. There are many studies on comparison, classification, discrimination, and clustering of two or several stationary time series models [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48], cyclostationary (periodically correlated) time series models [49,50], almost cyclostationary time series models [51,52], and non-stationary time series models [53][54][55][56][57]. ...
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This work is devoted to apply the parametric and nonparametric techniques to construct test of hypothesis about the equality of the probabilistic behaviors of several time series models with fractional Brownian motion errors fitted on several independent datasets. The accuracy and power of the introduced method are studied using the simulated and real datasets. The results indicate that the introduced approach is more powerful than other alternative approaches, in non-stationary cases. Ó 2020 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
... Furthermore, a PSD comparison based on statistical hypothesis using different statistics has been applied, namely, the Kolmogorov-Smirnov type statistic, the χ 2 statistic, the Kullback-Leibler type statistic and the L 2 distance [62,63]. In addition, Georgiou [64] applied a pseudo Riemannian metric to calculate distances of PSDs from different random processes for a prediction problem. ...
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Composite structures undergo a gradual damage evolution from initial inter-fibre cracks to extended damage up to failure. However, most composites could remain in service despite the existence of damage. Prerequisite for a service extension is a reliable and component-specific damage identification. Therefore, a vibration-based damage identification method is presented that takes into consideration the gradual damage behaviour and the resulting changes of the structural dynamic behaviour of composite rotors. These changes are transformed into a sequence of distinct states and used as an input database for three diagnostic models, based on the Kullback-Leibler divergence, the two-sample Kolmogorov-Smirnov test and a statistical hidden Markov model. To identify the present damage state based on the damage-dependent modal properties, a sequence-based diagnostic system has been developed, which estimates the similarity between the present unclassified sequence and obtained sequences of damage-dependent vibration responses. The diagnostic performance evaluation delivers promising results for the further development of the proposed diagnostic method.
... Li and Lu [6] developed a wavelet method by comparing the wavelet coefficients of two periodograms. Additional related works include [7][8][9][10][11][12][13]. ...
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It is an important problem to compare two time series in many applications. In this paper, a computational bootstrap procedure is proposed to test if two dependent stationary time series have the same autocovariance structures. The blocks of blocks bootstrap on bivariate time series is employed to estimate the covariance matrix which is necessary in order to construct the proposed test statistic. Without much additional effort, the bootstrap critical values can also be computed as a byproduct from the same bootstrap procedure. The asymptotic distribution of the test statistic under the null hypothesis is obtained. A simulation study is conducted to examine the finite sample performance of the test. The simulation results show that the proposed procedure with the bootstrap critical values performs well empirically and is especially useful when time series are short and non-normal. The proposed test is applied to an analysis of a real data set to understand the relationship between the input and output signals of a chemical process.
... For instance, in parametric approaches, clusters are usually built based upon similarity of their estimated parameters (see e.g., Kalpakis et al., 2001;Corduas and Piccolo, 2008) some of which take a Bayesian approach (Bauwens and Rombouts, 2007;Frühwirth-Schnatter and Kaufman, 2008;Juárez and Steel, 2010). Nonparametric methods are often based upon comparing similarity of the estimated power spectra, which is a research topic on its own (see e.g., Coates and Diggle, 1986;Eichler, 2008;Dette and Paparoditis, 2009;Dette and Hildebrandt, 2012;Jentsch and Pauly, 2015). This approach is taken in, i.a., Kakizawa et al. (1998); Savvides et al. (2008); Fokianos and Promponas (2011); Holan and Ravishanker (2018). ...
Preprint
Due to the surge of data storage techniques, the need for the development of appropriate techniques to identify patterns and to extract knowledge from the resulting enormous data sets, which can be viewed as collections of dependent functional data, is of increasing interest in many scientific areas. We develop a similarity measure for spectral density operators of a collection of functional time series, which is based on the aggregation of Hilbert-Schmidt differences of the individual time-varying spectral density operators. Under fairly general conditions, the asymptotic properties of the corresponding estimator are derived and asymptotic normality is established. The introduced statistic lends itself naturally to quantify (dis)-similarity between functional time series, which we subsequently exploit in order to build a spectral clustering algorithm. Our algorithm is the first of its kind in the analysis of non-stationary (functional) time series and enables to discover particular patterns by grouping together `similar' series into clusters, thereby reducing the complexity of the analysis considerably. The algorithm is simple to implement and computationally feasible. As a further application we provide a simple test for the hypothesis that the second order properties of two non-stationary functional time series coincide.
... The third kind of methods for detecting changepoints is based on hypothesis tests. Dette and Paparoditis [12] and Dette and Hildebrandt [11] proposed an approach to test the equality of spectrum between two successive segments. This idea iis interesting but it does not take into account the nature and the structure of the dependence between these successive segments. ...
Preprint
This paper intends to develop tools for characterizing non-linear spectral dependence between spontaneous brain signals. We use parametric copula models (both bivariate and vine models) applied on the magnitude of Fourier coefficients rather than using coherence. The motivation behind this work is an experiment on rats that studied the impact of stroke on the connectivity structure (dependence) between local field potentials recorded at various channels. We address the following major questions. First, we ask whether one can detect any changepoint in the regime of a brain channel for a given frequency band based on a difference between the cumulative distribution functions modeled for each epoch (small window of time). Our proposed approach is an iterative algorithm which compares each successive bivariate copulas on all the epochs range, using a bivariate Kolmogorov-Smirnov statistic. Second, we ask whether stroke can alter the dependence structure of brain signals; and examine whether changes in dependence are present only in some channels or generalized across channels. These questions are addressed by comparing Vine-copulas models fitted for each epoch. We provide the necessary framework and show the effectiveness of our methods through the results for the local field potential data analysis of a rat.
... Fan and Zhang (2004) consider generalised likelihood ratio tests for the same hypothesis. There also is a rich literature on non-parametric comparison of the (multivariate) spectra of two time series, here some recent references include Diggle and Fisher (1991); Dette and Paparoditis (2009); McElroy and Holan (2009) ;Jentsch and Pauly (2015); Chau et al. (2017), but this list is by no means complete. ...
Article
Finding parametric models that accurately describe the dependence structure of observed data is a central task in the analysis of time series. Classical frequency domain methods provide a popular set of tools for fitting and diagnostics of time series models, but their applicability is seriously impacted by the limitations of covariances as a measure of dependence. Motivated by recent developments of frequency domain methods that are based on copulas instead of covariances, we propose a novel graphical tool that allows to access the quality of time series models for describing dependencies that go beyond linearity. We provide a thorough theoretical justification of our approach and show in simulations that it can successfully distinguish between subtle differences of time series dynamics, including non-linear dynamics which result from GARCH and EGARCH models.
... It ensures that the GPFb and Fmaxb tests asymptotically keep the nominal type I error level under the null hypothesis and, as we will see, their powers tend to one for local alternatives. Other testing procedures having such a property are proposed inChung and Romano (2016),Dette and Paparoditis (2009),Konietschke et al. (2015),Pauly et al. (2015),Smaga (2015Smaga ( , 2017a andZhang et al. (2013). ...
Article
In real data analysis, it is often interesting to consider a general linear hypothesis testing (GLHT) problem for functional data, which includes the one-way ANOVA, post hoc or contrast analysis as special cases. Existing tests for this GLHT problem include a L²-norm-based test and an F-type test but their theoretical properties have not been investigated. In addition, for functional one-way ANOVA, simulation studies in the literature indicate that they are less powerful than the globalizing pointwise F (GPF) test and the Fmax-test. The GPF and Fmax-test enjoy several other good properties. They are scale-invariant in the sense that their test statistics do not change if we multiply each of functional curves with a non-zero function of the observed locations. In this paper, the GPF and Fmax-test are adapted to the above GLHT problem. Their theoretical properties, for example root-n consistency as well as those of the L²-norm-based and F-type tests are established. Intensive simulation studies are carried out to compare the finite-sample behavior of the tests under consideration in scenarios reflecting various practical characteristics of functional data. Simulation results indicate that the GPF test has higher power than other tests when the functional data are less correlated, and the Fmax-test has higher power than other tests when the functional data are moderately or highly correlated. These results are also confirmed by application of the GPF and Fmax tests to the corneal surface data coming from medical industry. This application suggests the new methods may help to make more clear and sure decisions in practice. For a convenient application of the considered testing procedures, their implementation is developed in the R programming language.
... Many methods have been developed to compare the second-order dynamics of time series. These methods in the frequency domain include Coates and Diggle (1986), Diggle and Fisher (1991), Maharaj (2002), Fokianos and Savvides (2008), Dette andPaparoditis (2009), Jin (2011), Preuss and Hildebrandt (2013), Lu and Li (2013), Jentsch and Pauly (2015), and Decowski and Li (2015). The methods in the time domain include Alonso and Maharaj (2006), Lund et al. (2009), and Jin and Wang (2016). ...
Article
In this paper, a new frequency-domain test is proposed to check the equality of spectral densities of two or more stationary time series. The proposed test is able to deal with multiple independent time series of different lengths naturally, based on some regression models of log periodograms. The asymptotic null distribution of the proposed test is obtained. The consistency is shown under any fixed alternative and a sequence of local alternatives. A simulation study is conducted to examine the finite sample performance of the test. By jointly modeling all log periodograms, the test is empirically robust when multiple time series are mutually dependent to some extent. It also works well for non-Gaussian time series. The proposed test is applied to compare several vibrational signals for damage detection of a mechanical system.
... Unlike popular methods (e.g. Dette and Paparoditis, 2009;Maharaj, 2002;Jentsch and Pauly, 2015), the proposed test statistics are independent of any estimator of spectra, avoiding the need to select smoothing parameters. The advantage of the proposed statistics is twofold. ...
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Based on periodogram-ratios of two univariate time series at different frequency points, two tests are proposed for comparing their spectra. One is an Anderson-Darling-like statistic for testing the equality of two time-invariant spectra. The other is the maximum of Anderson-Darling-like statistics for testing the equality of two spectra no matter that they are time-invariant and time-varying. Both of two tests are applicable for independent or dependent time series. Several simulation examples show that the proposed statistics outperform those that are also based on periodogram-ratios but constructed by the Pearson-like statistics.
... We then quantified for each locus its deviation from the mean spectrum S. Variances are typically compared by variance ratios, rather than taking differences. Dette & Paparoditis (2009) proposed testing hypotheses about consistency among multiple spectra by dividing each spectrum by the mean spectrum, subtracting the expected value of 1 from each ratio and summing the squared deviations, so that larger sums of squares would indicate stronger evidence against the null hypothesis. However, this approach was not successful with the simulated data, as the selected loci tended to have lower sums of squares than neutral loci ( Fig. 1). ...
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The spatial signature of microevolutionary processes structuring genetic variation may play an important role in the detection of loci under selection. However, the spatial location of samples has not yet been used to quantify this. Here, we present a new two-step method of spatial outlier detection at the individual and deme levels using the power spectrum of Moran eigenvector maps (MEM). The MEM power spectrum quantifies how the variation in a variable, such as the frequency of an allele at a SNP locus, is distributed across a range of spatial scales defined by MEM spatial eigenvectors. The first step (Moran spectral outlier detection: MSOD) uses genetic and spatial information to identify outlier loci by their unusual power spectrum. The second step uses Moran spectral randomization (MSR) to test the association between outlier loci and environmental predictors, accounting for spatial autocorrelation. Using simulated data from two published papers, we tested this two-step method in different scenarios of landscape configuration, selection strength, dispersal capacity and sampling design. Under scenarios that included spatial structure, MSOD alone was sufficient to detect outlier loci at the individual and deme levels without the need for incorporating environmental predictors. Follow-up with MSR generally reduced (already low) false-positive rates, though in some cases led to a reduction in power. The results were surprisingly robust to differences in sample size and sampling design. Our method represents a new tool for detecting potential loci under selection with individual-based and population-based sampling by leveraging spatial information that has hitherto been neglected.
... Such property of a resampling procedure is desirable. For instance, the bootstrap tests of Dette and Paparoditis (2009) and Konietschke et al. (2015) or the permutation tests of Pauly et al. (2015) and Smaga (2015) also have this property. The reason for this is that the property ensures that when the null hypothesis is true, the asymptotic distribution of Q n (W n ) and asymptotic conditional distribution of Q * n (W * n ) are the same. ...
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The nonparametric and parametric bootstrap methods for multivariate hypothesis testing are developed. They are used to approximate the null distribution of the test statistics proposed by Duchesne and Francq (2015 Duchesne, P., Francq, C. (2015). Multivariate hypothesis testing using generalized and {2}-inverses - with applications. Statistics 49: 475–496.[Taylor & Francis Online], [Web of Science ®]), resulting in bootstrap testing procedures. In the problem of testing for the mean vector of a multivariate distribution, the asymptotic validity of the bootstrap methods is proved. The finite sample performance of the new solutions is demonstrated by means of Monte Carlo simulation studies. They indicate that for small sample size, the bootstrap tests provide a better finite sample properties than the asymptotic tests considered by Duchesne and Francq (2015 Duchesne, P., Francq, C. (2015). Multivariate hypothesis testing using generalized and {2}-inverses - with applications. Statistics 49: 475–496.[Taylor & Francis Online], [Web of Science ®]).
... This problem has found considerable interest in the literature [see for example Jenkins (1961) or De Souza and Thomson (1982) for some early results], but in the nonparametric situation nearly all proposed procedures are only reasoned by simulation studys or heuristic proofs, see Coates and Diggle (1986), Pötscher and Reschenhofer (1988), Diggle and Fisher (1991) and Maharaj (2002) among many others. Most recently Eichler (2008), Dette and Paparoditis (2009), Dette et al. (2010) and Dette and Hildebrandt (2011) provided mathematical details for the above testing problem using different L 2 -type statistics, but nevertheless in all mentioned articles it is always required that the different time series have the same length, which is typically not the case in practice. Caiado and Pena (2009) considered different metrics for the comparison of time series with unequal sample sizes in a simulation study and Jentsch and Pauly (2011) provided a theoretical result, which however does not yield a consistent test as it was also pointed out by the authors. ...
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This paper deals with the comparison of several stationary processes with unequal sample sizes. We provide a detailed theoretical framework on the testing problem for equality of spectral densities in the bivariate case, but also present the generalization to the m dimensional case and to other statistical applications like testing for zero correlation or clustering of time series data with different length. We prove asymptotic normality of an appropriately standardized version of the test statistic both under the null and the alternative and investigate the finite sample properties of our method in a comprehensive simulation study. Furthermore we apply our approach to cluster financial time series data with different sample length.
... Many previous methods have been proposed to test the equality of the autocovariance (autocorrelation) structures, or equivalently, the corresponding (normalized) spectral densities of two different time series. Related works on this problem include Coates and Diggle (1986), Diggle and Fisher (1991), Maharaj (2002), Fokianos and Savvides (2008), Dette and Paparoditis (2009), Lund et al. (2009), Lu and Li (2013), Jin (2011), Carsten and Pauly (2012) and Decowski and Li (2015). The last three recent works aforementioned can be applied to compare two different time series of unequal lengths. ...
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An order selection test is proposed to check the equality of two independent stationary time series in their correlation structures. The asymptotic distribution of the order selection test statistic under the null hypothesis is obtained. Furthermore, it is shown that the proposed test is consistent not only under any fixed alternative hypothesis but also under a sequence of local alternative hypotheses. A simulation study is conducted to examine the finite sample performance of the test in comparison to some existing methods. The proposed test is also applied to an analysis of a biomedical data set.
... Due to the equivalence between the spectral density information and the autocovariance structure of a short memory stationary time series, different test statistics have been proposed in the frequency domain [1][2][3][4][5][6] as well as the time domain [7] for the comparison. Coates and Diggle [1] proposed some test statistics on the log ratio of two periodograms to test the equality in the spectral densities of two independent stationary time series. ...
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In this paper, we propose some tests in the time domain to check the equality of autocorrelation structure of multiple () independent stationary time series. To develop the tests, multiple time series are divided into two groups in different ways. Given that k time series are in one group and time series are in the other group, test statistics , are the maximum Mahalanobis difference between 2 groups of time series over all possible ways of such grouping. The asymptotic distributions under the null are derived. An extensive simulation is conducted to check the finite sample properties of the test statistics. Suggestion is given on the selection of test statistics under different situations based on the simulation. The paper provides some options to compare multiple time series when the length of time series is relatively short and M is relatively large. An application of the tests on vibrational data analysis is discussed.
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In this paper, we propose tests for the existence of random effects and interactions for two-way models with dependent errors. We prove that the proposed tests are asymptotically distribution-free which have asymptotically size τ{{\tau }} and are consistent. We elucidate the nontrivial power under the local alternative when a sample size tends to infinity and the number of groups is fixed. A simulation study is performed to investigate the finite-sample performance of the proposed tests. In the real data analysis, we apply our tests to the daily log-returns of 24 stock prices from six countries and four sectors. We find that there is no strong evidence to support the existence of substantial differences in the log-return across countries, nor to the existence of interactions between countries and sectors. However, there exists random effect differences in the daily log-return series across different sectors.
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Based on periodogram-ratios of two univariate time series at different frequency points, two tests are proposed for comparing their spectra. One is an Anderson-Darling-like statistic for testing the equality of two time-invariant spectra. The other is the maximum of Anderson-Darling-like statistics for testing the equality of two time-varying spectra. Both of two tests are applicable for independent or dependent time series. Several simulation examples show that the proposed statistics outperform those that are also based on periodogram-ratios but constructed by the Pearson-like statistics.
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1 Stationary Time Series.- 2 Hilbert Spaces.- 3 Stationary ARMA Processes.- 4 The Spectral Representation of a Stationary Process.- 5 Prediction of Stationary Processes.- 6* Asymptotic Theory.- 7 Estimation of the Mean and the Autocovariance Function.- 8 Estimation for ARMA Models.- 9 Model Building and Forecasting with ARIMA Processes.- 10 Inference for the Spectrum of a Stationary Process.- 11 Multivariate Time Series.- 12 State-Space Models and the Kalman Recursions.- 13 Further Topics.- Appendix: Data Sets.
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