Multivariate Stationary Stochastic Processes Research Articles

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.

Simulation of multivariate stationary stochastic processes using dimension-reduction representation methods

In view of the Fourier-Stieltjes integral formula of multivariate stationary stochastic processes, a unified formulation accommodating spectral representation method (SRM) and proper orthogonal decomposition (POD) is deduced. By introducing random functions as constraints correlating the orthogonal random variables involved in the unified formulation, the dimension-reduction spectral representation method (DR-SRM) and the dimension-reduction proper orthogonal decomposition (DR-POD) are addressed. The proposed schemes are capable of representing the multivariate stationary stochastic process with a few elementary random variables, bypassing the challenges of high-dimensional random variables inherent in the conventional Monte Carlo methods. In order to accelerate the numerical simulation, the technique of Fast Fourier Transform (FFT) is integrated with the proposed schemes. For illustrative purposes, the simulation of horizontal wind velocity field along the deck of a large-span bridge is proceeded using the proposed methods containing 2 and 3 elementary random variables. Numerical simulation reveals the usefulness of the dimension-reduction representation methods.

Different estimators of the spectral matrix: an empirical comparison testing a new shrinkage estimator

ABSTRACTIn this paper we propose a new non parametric estimator of the spectral matrix of a multivariate stationary stochastic process, with the main goal to locally improve the deficiencies of the smoothed periodogram in terms of mean square error of the estimates. Our estimator is based on a convex linear combination of the frequency averaged periodogram and an estimate of the true mean spectral matrix across frequencies. In a wide simulation study we show that our estimator turns out to be able to markedly improve the frequency averaged periodogram especially at central frequencies.

Simulation of Multivariate Stationary Gaussian Stochastic Processes: Hybrid Spectral Representation and Proper Orthogonal Decomposition Approach

By observing that the optimal basis for the proper orthogonal decomposition (POD) can be obtained from the cross power spectral density (XPSD) matrix of a multivariate stationary Gaussian stochastic process, the computational efficiency, in both time and memory consumption, of simulations of this process is improved by using a hybrid spectral representation and POD approach with negligible loss of accuracy. This hybrid approach actually simulates another multivariate process with many fewer variables in an optimal subspace obtained by the POD. This approach is straightforward, effective, and does not place any conditions on the XPSD matrices. Furthermore, the error induced by the reduction of variables is predictable and controllable prior to the simulation procedure. The spectral representation method (SRM) is discussed in a heuristic way. In this paper, a specific POD theorem is formally stated, proved, and related to XPSD matrices. A numerical example is given to demonstrate the effectiveness of this hybrid approach. This approach may also have potential applications for simulations of nonstationary non-Gaussian processes.

Completely regular multivariate stationary processes and the Muckenhoupt condition

In this paper we shall give a necessary and sufficient condition for a multivariate stationary stochastic process to be completely regular. Let us recall that a multivariate stationary stochastic process with discrete time 1 is a sequence of d-tuples x(n) = (x1(n), x2(n), . . . , xd(n)), n ∈ Z of scalar random variables such that E|xj(n)| <∞ and the correlation matrix Q(n, k) Q(n, k) = {Q(n, k)i,j}1≤i,j≤d := { Exi(n)xj(k) }

A unique representation theorem for the conditional expectation of stationary processes and application to dynamic estimation problems

In this paper, multivariate strict sense stationary stochastic processes are considered. It is shown that there exists a universal function by means of which the conditional expectation of any stationary process with respect to its past can be represented. This requires no ergodicity assumptions. The important implications of this result in the evaluation of the achievable performance in certain dynamic estimation problems with incomplete statistical information are also discussed.

Spectral characterization of the Wold–Zasuhin decomposition and prediction-error operator

Over thirty years ago Wiener and Masani pointed out in the introduction of their celebrated paper [31] that for a general multivariate stationary stochastic process no relation had been given for the prediction-error matrix in terms of the spectrum of the process. In particular it was unknown how to characterize the rank of the process in spectral terms (cf. Masani[12], p. 369, question 1). Despite explicit progress in this connection with certain regular processes, such as the series representations in [11, 19, 22, 32], or the iterative approach of [28, 29], and despite progress in the structure theory of degenerate processes ([8, 10, 14, 15, 26]), a general relation or series expression has remained elusive.

Canonical Correlation in Multivariate Time Series Analysis With an Application to One-Year-Ahead and Multiyear-Ahead Macroeconomic Forecasting

A simple one-period-ahead and multiperiod-ahead prediction procedure for multivariate time series is suggested, based on the canonical correlation technique. The prediction procedure is direct in the sense that no lag orders and parameters have to be estimated first, as in the usual ARMAX or VAR parameterizations of multivariate stationary stochastic processes. A best (in the mean squared error sense) predictor can be obtained directly using singular-value decompositions of covariance matrices. The procedure is used to forecast one-year-ahead and multiyear-ahead national growth rates of 14 countries for the years 1974–1984.

Canonical Correlation in Multivariate Time Series Analysis with an Application to One-Year-Ahead and Multiyear-Ahead Macroeconomic Forecasting

A simple one-period-ahead and multiperiod-ahead prediction procedure for multivariate time series is suggested, based on the canonical correlation technique. The prediction procedure is direct in the sense that no lag orders and parameters have to be estimated first, as in the usual ARMAX or VAR parameterizations of multivariate stationary stochastic processes. A best (in the mean squared error sense) predictor can be obtained directly using singular-value decompositions of covariance matrices. The procedure is used to forecast one-year-ahead and multiyear-ahead national growth rates of 14 countries for the years 1974–1984.

Autoregressive representations of multivariate stationary stochastic processes

Consider a q-variate weakly stationary stochastic process {Xn} with the spectral density W. The problem of autoregressive representation of {Xn} or equivalently the autoregressive representation of the linear least squares predictor of Xn, based on the infinite past is studied. It is shown that for every W in a large class of densities, the corresponding process has a mean convergent autoregressive representation. This class includes as special subclasses, the densities studied by Masani (1960) and Pourahmadi (1985). As a consequence it is shown that the condition W-1∈Lqxq1 or minimality of {Xn} is dispensable for this problem. When W is not in this class or when W has zeros of order 2 or more, it is shown that {Xn} has a mean Abel summable or mean compounded Cesaro summable autoregressive representation.

On the Weiner-Masani Algorithm for Finding the Generating Function of Multivariate Stochastic Processes

It is shown that the algorithms for determining the generating function and prediction error matrix of multivariate stationary stochastic processes developed by Weiner and Masani (1957, 1958) and later by Masani (1960) will work in some more general setting.

On Determining the Predictor of Nonfull-Rank Multivariate Stationary Random Processes

Algorithms for determining the generating function and the predictor for some nonfull-rank multivariate stationary stochastic processes are obtained. In fact, it is shown that the well-known algorithms given by Wiener and Masani [Acts Math., 99 (1958), pp. 93–137] for the full-rank case are valid in certain nonfull-rank cases exactly in the same form.

On the angle between past and future for multivariate stationary stochastic processes

A characterization for the positivityof the angle between past and future of multivariate stationary stochastic processes is established. In order to prove the results a lemma is proved which is of independent interest, and which is very useful in other areas of prediction theory as well.

On the Mean Convergence of the Best Linear Interpolator of Multivariate Stationary Stochastic Processes

It is shown that a necessary and sufficient condition, for the existence of a mean-convergent series for the linear interpolator of a $q$-variate stationary stochastic process $\{X_n\}$ with density matrix $W$, is that the Fourier series of the isomorph of the linear interpolator should converge in the norm of $L^2(W)$, and this happens if the past and future of the process are at positive angle. This provides a positive answer to a question of H. Salehi (1979) concerning the square summability of the inverse of $W$ and improves upon the work of Rozanov (1960) and Salehi (1979).

The determination of optimum structures for the state space representation of multivariate stochastic processes

When identifying a model for a multivariate stationary stochastic process, an important problem is that of determining the structure of the state-variable model. Several overlapping parameterizations can usually be fitted to a given process, and the question arises as to which structure leads to the most accurate parameter estimates. The accuracy of parameter estimates is often measured by the determinant of the Fisher information matrix. We show that all admissible structures will give asymptotically the same value to this criterion. For finite data some structures may still be better than others, and two heuristic structure estimation methods are analyzed. Some simulation results are also presented.

On the Problem of Structure Selection for the Identification of Stationary Stochastic Processes

Several "overlapping" but uniquely identifiable parametrizations can usually be fitted to the same multivariate stationary stochastic process. We show that these parametrizations are defined by a finite set of intrinsic invariants, and that they all give the same value to the determinant of the Fisher information matrix.

On the construction of Wold decomposition for multivariate stationary processes

A method to construct the Wold decomposition for multivariate stationary stochastic processes x k , k ∈ Z, is presented. The method is based on orthogonal decompositions for x k , k ∈ Z, obtained by forming orthogonal projections of x k , k ∈ Z, onto its component processes x k j , k ∈ Z, j = 1, …, q. The method does not give a complete solution to the Wold decomposition problem.

On the Bilateral Prediction Error Matrix of a Multivariate Stationary Stochastic Process

Previous article Next article On the Bilateral Prediction Error Matrix of a Multivariate Stationary Stochastic ProcessA. G. Miamee and H. SalehiA. G. Miamee and H. Salehihttps://doi.org/10.1137/0510023PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA formula for the bilateral prediction error matrix of a multivariate stationary stochastic process in terms of the spectral density and its generalized inverse is given.[1] A. N. Kolmogoroff, Stationary sequences in Hilbert's space, Bolletin Moskovskogo Gosudarstvenogo Universiteta. Matematika, 2 (1941), 40pp, (In Russian.) MR0009098 Google Scholar[2] A. Makagon and , A. Weron, Wold-Cramér concordance theorems for interpolation of q-variate stationary processes over locally compact Abelian groups, J. Multivariate Anal., 6 (1976), 123–137 10.1016/0047-259X(76)90024-5 MR0426126 0332.60026 CrossrefISIGoogle Scholar[3] P. Masani, U. Grenander, Cramér's theorem on monotone matrix-valued functions and the Wold-decomposition, Probability and statistics: The Harald Cramér volume (edited by Ulf Grenander), Almquist & Wiksell, Stockholm, 1959, 175–189 MR0124929 0095.12501 Google Scholar[4] P. Masani, The prediction theory of multivariate stochastic processes. III. Unbounded spectral densities, Acta Math., 104 (1960), 141–162 MR0121952 0096.11505 CrossrefISIGoogle Scholar[5] James B. Robertson and , Milton Rosenberg, The decomposition of matrix-valued measures, Michigan Math. J., 15 (1968), 353–368 10.1307/mmj/1029000039 MR0239044 0167.14602 CrossrefISIGoogle Scholar[6] Habib Salehi, Application of the Hellinger integrals to q-variate stationary stochastic processes, Ark. Mat., 7 (1968), 305–311 (1968) MR0236991 0164.19001 CrossrefISIGoogle Scholar[7] Habib Salehi and , John K. Scheidt, Interpolation of q-variate weakly stationary stochastic processes over a locally compact abelian group, J. Multivariate Anal., 2 (1972), 307–331 10.1016/0047-259X(72)90019-X MR0326833 0243.60027 CrossrefGoogle Scholar[8] N. Wiener and , P. Masani, The prediction theory of multivariate stochastic processes. I. The regularity condition, Acta Math., 98 (1957), 111–150 MR0097856 0080.13002 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Canonical correlation and reduction of multiple time seriesAnnals of the Institute of Statistical Mathematics, Vol. 46, No. 4 | 1 Jan 1994 Cross Ref On the predictor of non-full-rank bivariate stochastic processesJournal of Multivariate Analysis, Vol. 29, No. 1 | 1 Apr 1989 Cross Ref On Determining the Predictor of Nonfull-Rank Multivariate Stationary Random ProcessesA. G. MiameeSIAM Journal on Mathematical Analysis, Vol. 18, No. 4 | 17 July 2006AbstractPDF (818 KB) Volume 10, Issue 2| 1979SIAM Journal on Mathematical Analysis217-445 History Submitted:27 May 1977Published online:17 February 2012 InformationCopyright © 1979 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0510023Article page range:pp. 247-252ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

Operators as spectral integrals of operator-valued functions from the study of multivariate stationary stochastic processes

P. Masani and the author have previously answered the question, “When is an operator on a Hilbert space H the integral of a complex-valued function with respect to a given spectral (projection-valued) measure?” In this paper answers are given to the question, “When is a linear operator from H q to H p the integral of a spectral measure?”; here the values of the integrand are linear operators from the square-summable q-tuples of complex numbers to the square-summable p-tuples of complex numbers, and our spectral measure for H q is the “inflation” of a spectral measure for H . In the course of this paper, we make available tools for handling the spectral analysis of q-variate weakly stationary processes, 1 ≤ q ≤ ∞, which should enable researchers to deal in the future with the case q = ∞. We show as one application of our theory that if U = ∫( in0, 2 π] e − iθ E( dθ) is a unitary operator on H and if T is a bounded linear operator from H q to H q (1 ≤ q ≤ ∞) which is a prediction operator for each stationary process ( U nx ) −∞ ∞ ⊆ H q (for each x = (x i) i j ∈ H q , U n x = ( U n x i ) i=1 q ), then T is a spectral integral, ∫( 0,2 π )] Φ( θ) E( dθ), and the Banach norm of T, | T| B = ess sup | Φ( θ)| B .

Best Linear Unbiased Estimation for Multivariate Stationary Processes

The general linear hypothesis is formulated for a multivariate stationary stochastic process. The best (minimum variance) linear unbiased estimates are derived for the regression functions and it is shown that many signal estimation problems are special cases of the general linear model. Several examples are presented illustrating the technique for particular multivariate processes.