General Linear Structure Research Articles

Covariance Model with General Linear Structure and Divergent Parameters

For estimating the large covariance matrix with a limited sample size, we propose the covariance model with general linear structure (CMGL) by employing the general link function to connect the covariance of the continuous response vector to a linear combination of weight matrices. Without assuming the distribution of responses, and allowing the number of parameters associated with weight matrices to diverge, we obtain the quasi-maximum likelihood estimators (QMLE) of parameters and show their asymptotic properties. In addition, an extended Bayesian information criteria (EBIC) is proposed to select relevant weight matrices, and the consistency of EBIC is demonstrated. Under the identity link function, we introduce the ordinary least squares estimator (OLS) that has the closed form. Hence, its computational burden is reduced compared to QMLE, and the theoretical properties of OLS are also investigated. To assess the adequacy of the link function, we further propose the quasi-likelihood ratio test and obtain its limiting distribution. Simulation studies are presented to assess the performance of the proposed methods, and the usefulness of generalized covariance models is illustrated by an analysis of the U.S. stock market.

Structured condition numbers of structured Tikhonov regularization problem and their estimations

Both structured componentwise and structured normwise perturbation analysis of the Tikhonov regularization are presented. The structured matrices under consideration include: Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Structured normwise, mixed and componentwise condition numbers for the Tikhonov regularization are introduced and their explicit expressions are derived. For the general linear structure, based on the derived expressions, we prove structured condition numbers are smaller than their corresponding unstructured counterparts. By means of the power method and small sample statistical condition estimation (SCE), fast condition estimation algorithms are proposed. Our estimation methods can be integrated into Tikhonov regularization algorithms that use the generalized singular value decomposition (GSVD). For large scale linear structured Tikhonov regularization problems, we show how to incorporate the SCE into the preconditioned conjugate gradient (PCG) method to get the posterior error estimations. The structured condition numbers and perturbation bounds are tested on some numerical examples and compared with their unstructured counterparts. Our numerical examples demonstrate that the structured mixed condition numbers give sharper perturbation bounds than existing ones, and the proposed condition estimation algorithms are reliable. Also, an image restoration example is tested to show the effectiveness of the SCE for large scale linear structured Tikhonov regularization problems.

System identification of linear structures based on Hilbert–Huang spectral analysis. Part 2: Complex modes

AbstractA method, based on the Hilbert–Huang spectral analysis, has been proposed by the authors to identify linear structures in which normal modes exist (i.e., real eigenvalues and eigenvectors). Frequently, all the eigenvalues and eigenvectors of linear structures are complex. In this paper, the method is extended further to identify general linear structures with complex modes using the free vibration response data polluted by noise. Measured response signals are first decomposed into modal responses using the method of Empirical Mode Decomposition with intermittency criteria. Each modal response contains the contribution of a complex conjugate pair of modes with a unique frequency and a damping ratio. Then, each modal response is decomposed in the frequency–time domain to yield instantaneous phase angle and amplitude using the Hilbert transform. Based on a single measurement of the impulse response time history at one appropriate location, the complex eigenvalues of the linear structure can be identified using a simple analysis procedure. When the response time histories are measured at all locations, the proposed methodology is capable of identifying the complex mode shapes as well as the mass, damping and stiffness matrices of the structure. The effectiveness and accuracy of the method presented are illustrated through numerical simulations. It is demonstrated that dynamic characteristics of linear structures with complex modes can be identified effectively using the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.

System identification of linear structures based on Hilbert–Huang spectral analysis. Part 1: normal modes

AbstractBased on the Hilbert–Huang spectral analysis, a method is proposed to identify multi‐degree‐of‐freedom (MDOF) linear systems using measured free vibration time histories. For MDOF systems, the normal modes have been assumed to exist. In this method, the measured response data, which are polluted by noises, are first decomposed into modal responses using the empirical mode decomposition (EMD) approach with intermittency criteria. Then, the Hilbert transform is applied to each modal response to obtain the instantaneous amplitude and phase angle time histories. A linear least‐square fit procedure is proposed to identify the natural frequency and damping ratio from the instantaneous amplitude and phase angle for each modal response. Based on a single measurement of the free vibration time history at one appropriate location, natural frequencies and damping ratios can be identified. When the responses at all degrees of freedom are measured, the mode shapes and the physical mass, damping and stiffness matrices of the structure can be determined. The applications of the proposed method are illustrated using three linear systems with different dynamic characteristics. Numerical simulation results demonstrate that the proposed system identification method yields quite accurate results, and it offers a new and effective tool for the system identification of linear structures in which normal modes exist. Copyright © 2003 John Wiley & Sons, Ltd.

Macroeconomic modeling with asymmetric vector autoregressions

VARs typically employ the same number of lags for each variable. Consequently, they often estimate many insignificant coefficients. The “asymmetric VAR” (AVAR), defined as a VAR for which each variable may have a unique number of lags, is shown to be the reduced form for a general linear structure. A particular economic structure is developed to compare and contrast parameter estimates from AVAR and VAR specifications. Qualitatively similar results are obtained from each model. Parameters in the AVARs frequently have smaller standard errors than the VAR, suggesting AVARs may obtain more efficient estimates. Important questions about macroeconomic structure are addressed with these models.

Steady State Response Prediction of General Linear Structures Coupled by Local Nonlinear Elements

This short paper is concerned with a direct application of the techniques developed by the authors in two previous publications. It is considered as a “booster” to the existing alternating frequency/time domain method which already has many important advantages over most other methods.

Blind Identification of Linear Structures

The blind identification of possibly singular linear structures is considered in this paper. It is shown, in a constructive proof, that general linear structures is identifiable by its output statistics if and only if all sources are non-Gaussian. A set of sufficient conditions for the identifiability of linear structures using only the fourth-order cumulants are also given that generalize existing identifiability conditions

Asymptotic modal analysis and statistical energy analysis of dynamical systems

The well known text by Lyon on Statistical Energy Analysis (SEA) [R. H. Lyon, Statistical Energy Analysis of Dynamical Systems: Theory and Applications (MIT Press, Cambridge, MA, 1975)] is the standard reference on the subject. As originally conceived SEA was based upon several plausible hypotheses. Various authors have considered the underlying basis for SEA and thereby advanced our understanding of it. In the present paper, the relationship between classical modal analysis (CMA) and SEA is studied. It is shown how the results of SEA may be obtained as an asymptotic limit of CMA. The present work only considers the response of a single general linear structure under a random or sinusoidal load. However future plans will consider (a) the structural acoustic response of an enclosure bounded by a flexible wall and excited by external forces and (b) two (or more) coupled structural systems. Here the emphasis is on (1) a new derivation of the SEA results from CMA for a single structure under a random load, (2) the generalization of the usual SEA result to show that, asymptotically, all points on the structure have the same response. In the literature this is sometimes invoked as an assumption. Here the result is derived as part of the asymptotic limit of CMA. (3) A numeriCal example which displays the manner in which the asymptotic limit is approached for random loading and (4) an extension of the usual SEA result for sinusoidal loading. In view of the above, the present results might be called more appropriately Asymptotic Modal Analysis (AMA) rather than SEA.

Effects of the estimation of covariance matrix parameters in the generalized multivariate linear model

We Consider the generalized multivariate linear model and assume the covariance matrix of the p x 1 vector of responses on a given individual can be represented in the general linear structure form described by Anderson (1973). The effects of the use of estimates of the parameters of the covariance matrix on the generalized least squares estimator of the regression coefficients and on the prediction of a portion of a future vector, when only the first portion of the vector has been observed, are investigated. Approximations are derived for the covariance matrix of the generalized least squares estimator and for the mean square error matrix of the usual predictor, for the practical case where estimated parameters are used.