Generalized Resampling Scheme With Application to Spectral Density Matrix in Almost Periodically Correlated Class of Time Series

2015 ◽  
Vol 37 (3) ◽  
pp. 369-404
Author(s):  
Łukasz Lenart
Keyword(s):  
Time Series ◽  
Density Matrix ◽  
Spectral Density ◽  
Spectral Density Matrix

Spectral Analysis for Bivariate Time Series with Long Memory

1996 ◽  
Vol 12 (5) ◽  
pp. 773-792 ◽  
Author(s):  
J. Hidalgo
Keyword(s):  
Time Series ◽  
Density Matrix ◽  
Spectral Density ◽  
Long Memory ◽  
Limit Theorems ◽  
Stationary Process ◽  
Absolute Summability ◽  
Spectral Density Matrix ◽  
Mixing Coefficients ◽  
Long Memory Processes

This paper provides limit theorems for spectral density matrix estimators and functionals of it for a bivariate covariance stationary process whose spectral density matrix has singularities not only at the origin but possibly at some other frequencies and, thus, applies to time series exhibiting long memory. In particular, we show that the consistency and asymptotic normality of the spectral density matrix estimator at a frequency, say λ, which hold for weakly dependent time series, continue to hold for long memory processes when λ lies outside any arbitrary neighborhood of the singularities. Specifically, we show that for the standard properties of spectral density matrix estimators to hold, only local smoothness of the spectral density matrix is required in a neighborhood of the frequency in which we are interested. Therefore, we are able to relax, among other conditions, the absolute summability of the autocovariance function and of the fourth-order cumulants or summability conditions on mixing coefficients, assumed in much of the literature, which imply that the spectral density matrix is globally smooth and bounded.

Asymptotic Expansions of the Distributions of Statistics Related to the Spectral Density Matrix in Multivariate Time Series and Their Applications

1990 ◽  
Vol 6 (1) ◽  
pp. 75-96 ◽  
Author(s):  
Masanobu Taniguchi ◽  
Koichi Maekawa
Keyword(s):  
Time Series ◽  
Density Matrix ◽  
Spectral Density ◽  
Asymptotic Expansions ◽  
Multivariate Time Series ◽  
Principal Component ◽  
Autoregressive Process ◽  
Reduced Form ◽  
Gaussian Stationary Process ◽  
Spectral Density Matrix

Let {X(t)} be a multivariate Gaussian stationary process with the spectral density matrix f0(ω), where θ is an unknown parameter vector. Using a quasi-maximum likelihood estimator θ̂ of θ, we estimate the spectral density matrix f0(ω) by fθ̂(ω). Then we derive asymptotic expansions of the distributions of functions of fθ̂(ω). Also asymptotic expansions for the distributions of functions of the eigenvalues of fθ̂(ω) are given. These results can be applied to many fundamental statistics in multivariate time series analysis. As an example, we take the reduced form of the cobweb model which is expressed as a two-dimensional vector autoregressive process of order 1 (AR(1) process) and show the asymptotic distribution of θ̂, the estimated coherency, and contribution ratio in the principal component analysis based on θ̂ in the model, up to the second-order terms. Although our general formulas seem very involved, we can show that they are tractable by using REDUCE 3.

scholarly journals Testing collinearity of vector time series

2019 ◽  
Vol 22 (2) ◽  
pp. 97-116
Author(s):  
Tucker S McElroy ◽  
Agnieszka Jach
Keyword(s):  
Time Series ◽  
Density Matrix ◽  
Spectral Density ◽  
Time Series Data ◽  
Series Data ◽  
Test Statistics ◽  
Rank Reduction ◽  
Spectral Density Matrix ◽  
Schur Complements ◽  
Vector Time Series

Summary We investigate the collinearity of vector time series in the frequency domain, by examining the rank of the spectral density matrix at a given frequency of interest. Rank reduction corresponds to collinearity at the given frequency. When the time series is nonstationary and has been differenced to stationarity, collinearity corresponds to co-integration at a particular frequency. We examine rank through the Schur complements of the spectral density matrix, testing for rank reduction via assessing the positivity of these Schur complements, which are obtained from a nonparametric estimator of the spectral density. New asymptotic results for the test statistics are derived under the fixed bandwidth ratio paradigm; they diverge under the alternative, but under the null hypothesis of collinearity the test statistics converge to a non-standard limiting distribution. Subsampling is used to obtain the limiting null quantiles. A simulation study and an empirical illustration for 6-variate time series data are provided.

scholarly journals Modal Participation Estimated from the Response Correlation Matrix

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Rune Brincker ◽  
Sandro D. R. Amador ◽  
Martin Juul ◽  
Manuel Lopez-Aenelle
Keyword(s):  
Density Matrix ◽  
Spectral Density ◽  
Correlation Matrix ◽  
Analytical Form ◽  
Mode Shapes ◽  
Transformation Matrices ◽  
Spectral Density Matrix ◽  
Spectral Density Functions ◽  
Input Load ◽  
The Time Domain

In this paper, we are considering the case of estimating the modal participation vectors from the operating response of a structure. Normally, this is done using a fitting technique either in the time domain using the correlation function matrix or in the frequency domain using the spectral density matrix. In this paper, a more simple approach is proposed based on estimating the modal participation from the correlation matrix of the operating responses. For the case of general damping, it is shown how the response correlation matrix is formed by the mode shape matrix and two transformation matrices T1 and T1 that contain information about the modal parameters, the generalized modal masses, and the input load spectral density matrix Gx. For the case of real mode shapes, it is shown how the response correlation matrix can be given a simple analytical form where the corresponding real modal participation vectors can be obtained in a simple way. Finally, it is shown how the real version of the modal participation vectors can be used to synthesize empirical spectral density functions.

scholarly journals Some Elements of Operational Modal Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Rune Brincker
Keyword(s):  
Density Matrix ◽  
Spectral Density ◽  
Modal Analysis ◽  
Mode Shape ◽  
Fourier Integral ◽  
Operational Modal Analysis ◽  
Difficult Case ◽  
Domain Identification ◽  
Spectral Density Matrix ◽  
Random Responses

This paper gives an overview of the main components of operational modal analysis (OMA) and can serve as a tutorial for research oriented OMA applications. The paper gives a short introduction to the modeling of random responses and to the transforms often used in OMA such as the Fourier series, the Fourier integral, the Laplace transform, and the Z-transform. Then the paper introduces the spectral density matrix of the random responses and presents the theoretical solutions for correlation function and spectral density matrix under white noise loading. Some important guidelines for testing are mentioned and the most common techniques for signal processing of the operating signals are presented. The algorithms of some of the commonly used time domain and frequency domain identification techniques are presented and finally some issues are discussed such as mode shape scaling, and mode shape expansion. The different techniques are illustrated on the difficult case of identifying the three first closely spaced modes of the Heritage Court Tower building.

Complex Coherence Constraint for the Definition of the Spectral Density Matrix

2018 ◽  
Vol 61 (1) ◽  
pp. 7-19
Author(s):  
Zhihua Liu ◽  
Chenguang Cai ◽  
Yan Xia ◽  
Ming Yang
Keyword(s):  
Density Matrix ◽  
Spectral Density ◽  
Random Vibration ◽  
Multiple Input Multiple Output ◽  
New Approach ◽  
Spectral Density Matrix ◽  
Multiple Input ◽  
Input Multiple Output ◽  
Recursive Formulas ◽  
Definition Of

Abstract The cross spectral density (CSD) for a multiple-input/multiple-output (MIMO) random vibration is typically defined by the complex coherence consisting of the modulus and the phase. The purpose of this paper is to present a constraint for the complex coherence to allow the CSD to be defined more easily. The study of the complex coherence constraint is based on Cholesky decomposition of the spectral density matrix (SDM). The complex coherence must be bounded in the interior or on the boundary of a constraint circle to ensure a physically realizable random vibration. This paper proposes a new approach to define the complex coherences of the SDM by using recursive formulas based on the constraint circle.

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