Stationary Multivariate Time Series Research Articles

On the asymptotic behaviour of the spectral function estimation of a multivariate stationary Gaussian time series

The abstracts (in two languages) can be found in the pdf file of the article.
 Original author name(s) and title in Russian and Lithuanian:
 Р. Бенткус. Об асимптотическом поведении оценки спектральной функции многомерной стационарной гауссовской последовательности
 R. Bentkus. Apie daugiamatės stacionarios Gauso sekos spektrinės funkcijos įvertinimo asimptotiką

Some Tests of Independence for Stationary Multivariate Time Series

Summary Let X(t)=(X1(t)X2(t)⋮XP(t)),t=⋯−1,0,1…, be a p-dimensional zero-mean stationary Gaussian time series possessing a spectral density matrix F(ω) = {fij(ω)}, i, j = 1, 2, …, p, satisfying some mild regularity conditions. Let F^(ωl), l=1, 2, … ,M, F^(ωl)={f^ij(ωl)} be suitably defined sample spectral density matrices for M values of ω, based on a record of length T ≫ M. We consider the following (null) hypotheses: H1: Xi(t), Xi(s) independent if i ≠ j, all s, t, H2: X1(t) independent of Xi(s), j = 2, …, p, all s, t, H3: X1(t), X1(t + τ) independent, all τ ≠ 0 (“white noise”). Approximate likelihood-ratio tests are found and the test statistics λi, i = 1, 2, 3, are found to be functions of the {F^(ωl), l=1, 2, … ,M}. The {F^(ωl)} converge in mean square to a family of independent complex Wishart matrices. As a consequence of this fact, log λ1 and log λ2 converge in first mean to random variables whose null densities can be given explicitly, being distributed as the logarithms of products of independent beta random variables with integer indices. Under the null hypothesis λ3 is distributed as λ˜3 where λ˜3 is Bartlett's statistic for homogeneity of variances. The asymptotic distributions of log λi, i = 1, 2, 3, under the alternative are discussed. Under the alternative, log λ3 tends in first mean to log λ˜3. A power series expansion for the characteristic function of log λ˜3 under the alternative is given.

Phase Free Estimation of Coherence

A phase free estimate of the coherence of a bivariate Gaussian process is presented. The technique is based on the usual independent, complex normal approximation to the distribution of the finite Fourier transform of a multivariate stationary time series, and the complex Wishart approximation to the distribution of spectrum estimates. If the spectral densities and coherence can be assumed to be constant over a wider frequency band than the phase can be assumed to be constant, the concept of inner and outer spectral windows would seem appropriate. Maximum likelihood estimates of the coherence are obtained using phase free marginal distributions at the inner window level. The results of simulations are presented showing the likelihood for various inner windows.

An Introduction to Polyspectra

The subject of this paper is the higher-order spectra or polyspectra of multivariate stationary time series. The intent is to derive (i) certain mathematical properties of polyspectra, (ii) estimates of polyspectra based on an observed stretch of time series, (iii) certain statistical properties of the proposed estimates and (iv) several applications of the results obtained. As might be expected, in lower order cases the polyspectrum reduces to spectra already considered. If one is considering a single time series, the first order polyspectrum is the usual power spectrum considered in [2], [14], [22], while the second order polyspectrum is the bispectrum considered in [12], [23], [28]. Also, if one is considering a pair of time series the first order polyspectrum is the cross-spectrum considered in [6], [10], [15]. For the case of a single time series the idea of a higher-order spectrum occurs in [3]. The idea has since been developed to a higher level of algebraic and analytic detail in [24]. Also in [24] the notion of considering a spectral representation for a cumulant rather than for a product moment occurs and is acknowledged to be due to Kolmogorov. Another related early paper is [18]. The present paper generalizes the definitions of these papers in the sense that $k$-dimensional time series are considered. Another contribution is a theorem indicating that for a broad class of processes one is wise to restrict consideration to cumulants rather than product moments. Finally it should be noted that the term polyspectrum is due to J. W. Tukey. I have perhaps used the term in a more restricted sense than he would wish in that I have reserved it for the Fourier transform of a cumulant (at the expense of other functions of moments).

Prediction of Multivariate Time Series

Abstract A method of linear prediction of stationary multivariate time series is discussed from the point of view of meteorological applications. Tests of significance are given and it is shown by examples that the method is practical even when the dimensionality of the series becomes quite large.