The history of the development of statistical hypothesis testing in time series analysis is reviewed briefly and it is pointed out that the hypothesis testing procedure is not adequately defined as the procedure for statistical model identification. The classical maximum likelihood estimation procedure is reviewed and a new estimate minimum information theoretical criterion (AIC) estimate (MAICE) which is designed for the purpose of statistical identification is introduced. When there are several competing models the MAICE is defined by the model and the maximum likelihood estimates of the parameters which give the minimum of AIC defined by AIC = (-2)log-(maximum likelihood) + 2(number of independently adjusted parameters within the model). MAICE provides a versatile procedure for statistical model identification which is free from the ambiguities inherent in the application of conventional hypothesis testing procedure. The practical utility of MAICE in time series analysis is demonstrated with some numerical examples.

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A new look at the statistical model identification

Reference-based stochastic subspace identification for output-only modal analysis

Given a mean square continuous stochastic vector process y with stationary increments and a rational spectral density $\Phi $ such that $\Phi (\infty )$ is finite and nonsingular, consider the problem of finding all minimal (wide sense) Markov representations (stochastic realizations) of y. All such realizations are characterized and classified with respect to deterministic as well as probabilistic properties. It is shown that only certain realizations (internal stochastic realizations) can be determined from the given output process y. All others (external stochastic realizations) require that the probability space be extended with an exogeneous random component. A complete characterization of the sets of internal and external stochastic realizations is provided. It is shown that the state process of any internal stochastic realization can be expressed in terms of two steady-state Kalman–Busy filters, one evolving forward in time over the infinite past and one backward over the infinite future. An algori...

On the Stochastic Realization Problem

Reference based stochastic subspace identification in civil engineering

The techniques and analysis presented in this thesis provide new methods to solve optimization problems posed on Riemannian manifolds. These methods are applied to the subspace tracking problem found in adaptive signal processing and adaptive control. A new point of view is offered for the constrained optimization problem. Some classical optimization techniques on Euclidean space are generalized to Riemannian manifolds. Several algorithms are presented and their convergence properties are analyzed employing the Riemannian structure of the manifold. Specifically, two new algorithms, which can be thought of as Newton's method and the conjugate gradient method on Riemannian manifolds, are presented and shown to possess quadratic and superlinear convergence, respectively. These methods are applied to several eigenvalue and singular value problems, which are posed as constrained optimization problems. New efficient algorithms for the eigenvalue problem are obtained by exploiting the special homogeneous space structure of the constraint manifold. It is shown that Newton's method applied to the Rayleigh quotient on a sphere converges cubically, and that the Rayleigh quotient iteration is an efficient approximation of Newton's method. The Riemannian version of the conjugate gradient method applied to this function gives a new algorithm for finding the eigenvectors corresponding to the extreme eigenvalues of a symmetric matrix. The Riemannian version of the conjugate gradient method applied to a generalized Rayleigh quotient yields a superlinearly convergent algorithm for computing the k eigenvectors corresponding to the extreme eigenvalues of an n-by-n matrix. This algorithm requires $O(nk\sp2)$ operations and O(k) matrix-vector multiplications per step. Several gradient flows are analyzed that solve eigenvalue and singular value problems. The new optimization algorithms are applied to the subspace tracking problem of adaptive signal processing. A new algorithm for subspace tracking is given, which is based upon the conjugate gradient method applied to the generalized Rayleigh quotient. The results of several numerical experiments demonstrating the convergence properties of the new algorithms are given.

Geometric optimization methods for adaptive filtering

This paper exploits the concept of a predictor space in the minimal realization problem for systems generating an analytic impulse response matrix. The predictor space constructed, by stochastic input and output processes forms the state space for the stochastic system representation, where a system is represented by the basis of the predictor space and the innovation process of input. The minimal realization problem is then solved for a given analytic impulse response matrix by defining a stochastic system driven by white noise whose input-output covariance equals the given impulse response matrix. It is shown that the coefficient matrices of the stochastic system representation constitute a solution to the minimal realization problem for the deterministic system with given impulse response matrix. The paper provides a unifying overview to many aspects of the realization problem and its algorithms.

Stochastic theory of minimal realization

An optimal control problem for a class of distributed parameter systems with bilinear type controls applied to the coefficient matries is formulated The necessary conditions for optimal bilinear controls from the variational technique are given The conjugate gradient minimization algorithm is presented with a numerical example

James Clary

Papers

Stochastic theory of minimal realization

Optimal control of bilinear distributed parameter systems