Multivariate Stationary Processes Research Articles

Mixed orthogonality graphs for continuous-time stationary processes

In this paper, we introduce different concepts of Granger causality and contemporaneous correlation for multivariate stationary continuous-time processes to model different dependencies between the component processes. Several equivalent characterisations are given for the different definitions, in particular by orthogonal projections. We then define two mixed graphs based on different definitions of Granger causality and contemporaneous correlation, the (mixed) orthogonality graph and the local (mixed) orthogonality graph. In these graphs, the components of the process are represented by vertices, directed edges between the vertices visualise Granger causal influences and undirected edges visualise contemporaneous correlation between the component processes. Further, we introduce various notions of Markov properties in analogy to Eichler (2012), which relate paths in the graphs to different dependence structures of subprocesses, and we derive sufficient criteria for the (local) orthogonality graph to satisfy them. Finally, as an example, for the popular multivariate continuous-time AR (MCAR) processes, we explicitly characterise the edges in the (local) orthogonality graph by the model parameters.

Extreme value analysis of high-dimensional Gaussian vector processes

The analysis of the extreme value pertaining to first-passage failure is a key consideration for engineering systems undergoing random vibration. Most of the literature focus on a univariate stochastic process, however for system reliability analysis, the extreme value of multiple dependent processes is required. Due to the problem complexity, analytical methods are scarce, and they are limited to low-dimensional problems (dimension refers to the number of component processes). This paper presents an analytical method for extreme analysis of multivariate stationary Gaussian processes. The method can efficiently solve high-dimensional problems, thus it has diverse applications such as analysing dynamic systems with many degrees-of-freedom, or evaluating space-time extremes of random fields by discretizing the spatial domain. A maximum process is defined, representing the maxima of the components at each time instant. The exact upcrossing rate of the maximum process is derived, and the Poisson approximation is invoked to estimate the extreme value distribution. The method is fast as it involves multiple one-dimensional integrals, which can be computed using Gaussian quadrature. Moreover, an accurate approximation is proposed that obviates the need for numerical integration. The proposed method is implemented on two examples, specifically a dynamic system and spreading ocean waves. Monte Carlo simulations, which are computationally demanding, are performed to verify the accuracy and correctness of the proposed method.

A frequency domain bootstrap for general multivariate stationary processes

Developing valid frequency domain bootstrap procedures for integrated periodogram statistics for multivariate time series is a challenging problem. This is mainly due to the fact that the distribution of such statistics depends on the fourth-order moment structure of the underlying multivariate process in nearly every scenario. Exceptions are some very special cases like nonparametric estimators of the spectral density matrix or Gaussian time series. In contrast to the univariate case, even additional structural assumptions – such as linearity of the multivariate process or a standardization of the statistic of interest – do not solve the problem. This paper proposes a new frequency domain bootstrap procedure for multivariate time series, the multivariate frequency domain hybrid bootstrap (MFHB), for integrated periodogram statistics as well as for functions thereof. Asymptotic validity of the MFHB procedure is established for these statistics and for a class of stationary multivariate processes satisfying rather weak dependence conditions ranging clearly beyond linear processes. The finite sample performance of the MFHB is investigated by means of simulations.

Efficient Method for Approximating the Joint Extreme Value Distribution of Multivariate Stationary Gaussian Processes

The joint extreme value distribution (JEVD) of multivariate random processes is important for evaluating the system reliability of a structure subjected to random vibrations. There are very limited analytical approaches for predicting the JEVD, and these approaches are only computationally viable for problems up to three dimensions. This paper presents an efficient approximate method, which is not limited to low-dimensional problems, for estimating the JEVD of multivariate stationary Gaussian processes. The proposed method modifies an existing method by using the Gauss-Legendre quadrature to evaluate the extreme value correlation coefficients under the bivariate Poisson assumption. Then, the JEVD is approximated using the Nataf transformation. The effectiveness of the proposed method is demonstrated via two numerical examples. The first example concerns the airgap problem of an offshore structure subjected to random waves, in which the extreme wave elevations are evaluated at six locations, and failure is defined as threshold exceedance for any location. The second example is a random vibration problem comprising a three degrees-of-freedom system. Previous studies focused on series system reliability; here three system reliabilities are examined, including series, parallel, and hybrid systems. The proposed method solves the problems efficiently, and it is found to provide an accurate prediction of the JEVD by comparison with Monte Carlo simulation.

Causal Vector Autoregression Enhanced with Covariance and Order Selection

A causal vector autoregressive (CVAR) model is introduced for weakly stationary multivariate processes, combining a recursive directed graphical model for the contemporaneous components and a vector autoregressive model longitudinally. Block Cholesky decomposition with varying block sizes is used to solve the model equations and estimate the path coefficients along a directed acyclic graph (DAG). If the DAG is decomposable, i.e., the zeros form a reducible zero pattern (RZP) in its adjacency matrix, then covariance selection is applied that assigns zeros to the corresponding path coefficients. Real-life applications are also considered, where for the optimal order p≥1 of the fitted CVAR(p) model, order selection is performed with various information criteria.

On consistency and sparsity for high-dimensional functional time series with application to autoregressions

Modelling a large collection of functional time series arises in a broad spectral of real applications. Under such a scenario, not only the number of functional variables can be diverging with, or even larger than the number of temporally dependent functional observations, but each function itself is an infinite-dimensional object, posing a challenging task. In this paper, we propose a three-step procedure to estimate high-dimensional functional time series models. To provide theoretical guarantees for the three-step procedure, we focus on multivariate stationary processes and propose a novel functional stability measure based on their spectral properties. Such stability measure facilitates the development of some useful concentration bounds on sample (auto)covariance functions, which serve as a fundamental tool for further convergence analysis in high-dimensional settings. As functional principal component analysis (FPCA) is one of the key dimension reduction techniques in the first step, we also investigate the non-asymptotic properties of the relevant estimated terms under a FPCA framework. To illustrate with an important application, we consider vector functional autoregressive models and develop a regularization approach to estimate autoregressive coefficient functions under the sparsity constraint. Using our derived non-asymptotic results, we investigate convergence properties of the regularized estimate under high-dimensional scaling. Finally, the finite-sample performance of the proposed method is examined through both simulations and a public financial dataset.

An adaptive multi-taper spectral estimation for stationary processes

The multi-taper spectral estimation method is effective in reducing both bias and variance of the spectrum of stationary stochastic processes. The number of tapers (NoT) in the method controls the bias and variance trade-off of the spectrum. In general, the NoT is empirically determined and constant at all frequencies, limiting the adjustment of the local bias and variance. This paper develops an adaptive multi-taper approach with the varying NoT at each frequency point for estimating the power spectral density (PSD) and coherence function of multivariate stationary processes. An iterative procedure with a stopping criterion is proposed to optimize the NoT at each frequency without manual tunning. The sine taper with a simple analytical expression and satisfactory leakage protection is employed. The proposed adaptive multi-taper approach is applied to three numerical examples, the structural response of a building model, a wind speed process, and an autoregressive process, which exhibit different spectral characteristics. The results of all examples show that the proposed approach outperforms Welch’s and traditional multi-taper methods in estimating the PSD and coherence with a smaller bias and variance.

Simulating multivariate stationary non-Gaussian process based on wavenumber–frequency spectrum and unified Hermite polynomial model

It is of importance to model the input excitations considering the spatial variation and non-Gaussianity for the reliability evaluation of long span structures. The existing multivariate stationary non-Gaussian simulation methods are mainly based on the target marginal distribution and cross-spectrum matrix. Due to the Cholesky decomposition of cross-spectrum matrix, the computational cost of these methods increases rapidly with the increase of spatial points. Therefore, this paper develops a novel simulation method based on the wavenumber–frequency spectrum and unified Hermite polynomial model (UHPM). Firstly, UHPM is extended to the homogenous and stationary non-Gaussian field. Then, a complete transformation model from homogenous and stationary non-Gaussian auto-correlation function (ACF) into underlying Gaussian ACF with its applicable range is proposed. Furthermore, two types of incompatibility between the first four marginal moments and wavenumber–frequency spectrum are discussed, and the corresponding remedies are provided. To facilitate the calculation, the 2D-Fast Fourier Transform (FFT) technique is embedded in the Wiener–Khintchine transformation and spectral representation method (SRM). Finally, a unified simulation framework for multivariate stationary non-Gaussian process based on its relationship to the homogenous and stationary non-Gaussian field is presented. Two numerical examples, involving the simulations of non-Gaussian wind velocities and ground motion accelerations, are investigated to verify the capabilities of the proposed method.

Asymptotic independence ex machina: Extreme value theory for the diagonal SRE model

We consider multivariate stationary processes satisfying a stochastic recurrence equation of the form , where are i.i.d. random vectors and are i.i.d. diagonal matrices and (Mt) are i.i.d. random variables. We obtain a full characterization of the vector scaling regular variation properties of , proving that some coordinates Xt, i and Xt, j are asymptotically independent even though all coordinates rely on the same random input (Mt). We prove the asynchrony of extreme clusters among marginals with different tail indices. Our results are applied to some multivariate autoregressive conditional heteroskedastic (BEKK‐ARCH and CCC‐GARCH) processes and to log‐returns. Angular measure inference shows evidences of asymptotic independence among marginals of diagonal SRE with different tail indices.

A high-efficiency simulation method of wind field and its application on transmission line

Generally, the fluctuating wind is simplified as several independent one-dimensional multivariate stationary Gaussian processes in simulating a natural wind field. The correlation in the lateral, longitudinal and vertical directions should all be considered in the simulation of longitudinal wind field for the large-span spatial structures. In fact, this type of structure has lots of simulation points. The calculation amount of wind field simulation by the harmonic superposition method depends on the scale of cross-spectral density matrix, which is directly related to the number of simulated points, leading to a low efficiency when generating the time-varying wind speed. This paper innovatively proposes a high-efficiency simulation method for the longitudinal wind field based on Taylor's hypothesis. Subsequently, the effectiveness of the proposed wind field method was verified by the numerical simulation. Finally, the dynamic responses of a transmission tower-line system under the wind loadings generated with the new method and traditional method are calculated and compared. The percentages difference of the mean and maximum axial force at the main tower members are less than 0.02% and 1%, respectively, indicating the effectiveness of the proposed time delay method. The results also show that the proposed simulation method of wind field can not only ensure the simulation accuracy, but also significantly improve the efficiency of wind speed generation, which is suitable for the wind load simulation of large-span spatial structures.

Simulation of ergodic multivariate stochastic processes: An enhanced spectral representation method

The spectral representation method (SRM) has been extensively used in the simulation of multivariate stationary Gaussian stochastic processes. In this study, an enhanced SRM for the simulation of ergodic multivariate stochastic processes is developed. Through shifting a frequency increment, the enhanced SRM offers a faster convergence rate of probabilistic characteristics in comparison with conventional SRM. The computational efficiency is enhanced as the Cholesky decomposition is only required at single-indexed frequencies. Additionally, a two-dimensional fast Fourier transform (FFT) algorithm is proposed to expedite simultaneously estimation of the double summation of cosine functions in both frequency and space dimensions. Accordingly, compared to the traditional FFT-based SRM, in which FFT is only used for the summation in the frequency dimension, the simulation efficiency is further enhanced significantly. The numerical example concerning the simulation of wind velocity field demonstrates that the proposed method offers a faster convergence rate and a higher level of efficiency.

A Multi-Taper S-Transform Method for Spectral Estimation of Stationary Processes

The S-transform (ST) is a method of time-frequency analysis of time series. We develop here the Multi-taper S-transform (MTST) for spectral estimation of multi-variate stationary processes through replacing the scalable Gaussian window of the ST with scalable, adjustable orthogonal time-frequency Hermite functions. The MTST is shown to reduce bias and variance of power spectral density (PSD) estimates and coherence over the entire frequency domain and compares favourably with the estimates obtained with the Welch and Multi-taper methods. The MTST method has been successfully applied to data from two anemometers in Hong Kong, during Typhoon Mangkhut, for the estimation of PSD and coherence.

Modeling Spectral Properties in Stationary Processes of Varying Dimensions with Applications to Brain Local Field Potential Signals

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.

Modeling and Estimation of Multivariate Discrete and Continuous Time Stationary Processes

In this paper, we give an autoregressive model of order 1 type of characterization covering all multivariate strictly stationary processes indexed by the set of integers. Consequently, under square integrability, we derive continuous time algebraic Riccati equations for the parameter matrix of the characterization. This provides us with a natural way to define the corresponding estimator. In addition, we show that the estimator inherits consistency from autocovariances of the stationary process. Furthermore, the limiting distribution is given by a linear function of the limiting distribution of the autocovariances. We also present the corresponding existing results of the continuous time setting paralleling them to the discrete case treated in this paper.

ОПТИМАЛЬНА ОБРОБКА СИГНАЛІВ ЗОБРАЖЕНЬ НА ФОНІ ШУМОВИХ ПЕРЕШКОД В ІНФОРМАЦІЙНІЙ СИСТЕМІ З АДАПТИВНОЮ АНТЕННОЮ РЕШІТКОЮ

The method to restore image signals against the arbitrary intensity noise background in the information radio engineering system with adaptive antenna array has been developed. In order to use methods developed for processing one-dimensional signals for image recovery in the information system with adaptive antenna array, the transition from the two-dimensional array to vector representation is carried out. Mathematical model of narrowband signal formed at input of antenna array elements in space-time sense is obtained. Correlation matrices of image carrier signals, interference and noise are considered and features of adaptive processing of image signals coming from several sources are observed. An expression was found for the likelihood function if the incoming vector process is a multivariate stationary Gaussian process with a non-zero mean. According to the maximum likelihood criterion, the expression for the system of optimal independent parametric weight vectors necessary for image signals restoring against the arbitrary intensity noise background coming from several independent sources in the information system with adaptive antenna array is obtained. In accordance to this system, a signal processing algorithm is built in the adaptive processor of N-dimensional adaptive antenna array. A simulation model of image signal restores coming from one source in the information system with adaptive antenna array against the arbitrary intensity noise background coming from several independent sources is built. It is shown that the use of weight coefficients calculated on the basis of the correlation matrix of observations, due to its properties, does not allow dividing the set of correlated image signals. The direction of further development of the obtained results in the class of invariant methods is determined.

JH-singularity and JH-regularity of multivariate stationary processes over LCA groups

JH-singularity and JH-regularity of multivariate stationary processes over LCA groups

Multivariate scale-free temporal dynamics: From spectral (Fourier) to fractal (wavelet) analysis

Abstract The Fourier transform (or spectral analysis) has become a universal tool for data analysis in many different real-world applications, notably for the characterization of temporal/spatial dynamics in data. The wavelet transform (or multiscale analysis) can be regarded as tailoring spectral estimation to classes of signals or functions defined by scale-free dynamics. The present contribution first formally reviews these connections in the context of multivariate stationary processes, and second details the ability of the wavelet transform to extend multivariate scale-free temporal dynamics analysis beyond second-order statistics (Fourier spectrum and autocovariance function) to multivariate self-similarity and multivariate multifractality. Illustrations and qualitative discussions of the relevance of scale-free dynamics for macroscopic brain activity description using MEG data are proposed.

Baxter’s inequality for finite predictor coefficients of multivariate long-memory stationary processes

For a multivariate stationary process, we develop explicit representations for the finite predictor coefficient matrices, the finite prediction error covariance matrices and the partial autocorrelation function (PACF) in terms of the Fourier coefficients of its phase function in the spectral domain. The derivation is based on a novel alternating projection technique and the use of the forward and backward innovations corresponding to predictions based on the infinite past and future, respectively. We show that such representations are ideal for studying the rates of convergence of the finite predictor coefficients, prediction error covariances, and the PACF as well as for proving a multivariate version of Baxter’s inequality for a multivariate FARIMA process with a common fractional differencing order for all components of the process.

Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues

A univariate Hawkes process is a simple point process that is self-exciting and has clustering effect. The intensity of this point process is given by the sum of a baseline intensity and another term that depends on the entire past history of the point process. Hawkes process has wide applications in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we prove a functional central limit theorem for stationary Hawkes processes in the asymptotic regime where the baseline intensity is large. The limit is a non-Markovian Gaussian process with dependent increments. We use the resulting approximation to study an infinite-server queue with high-volume Hawkes traffic. We show that the queue length process can be approximated by a Gaussian process, for which we compute explicitly the covariance function and the steady-state distribution. We also extend our results to multivariate stationary Hawkes processes and establish limit theorems for infinite-server queues with multivariate Hawkes traffic.

Measure-invariance of copula functions as tool for testing no-arbitrage assumption

Copulas, which are invariant under margins’ transforms induced by some change of measure, are investigated. It is emphasized that this particular kind of transforms induced by some change of measure, largely used in pricing techniques, preserves the invariance of the aggregation operator and a sufficient condition to assure it is proved. The discussion is extended to the time-preserving of measure-invariance; a characterization of its stability in time for multivariate stationary processes, based on the dynamic copula representation (see Cherubini et al., 2011), is provided. Finally a measure invariance-based statistical test for the absence of arbitrage opportunity assumption and its preservation in time is proposed and an empirical experiment based on quotes of S&P 500 futures and options traded on the Chicago Mercantile Exchange (CME) is discussed.