Identification and spectral estimation of noisy multivariate autoregressive processes

Kernel structure of block Hankel and Toeplitz matrices and partial realization

This dissertation is concerned with the task of efficiently and accurately tracking the singular values and left singular vectors of a rapidly changing dominant column space of a matrix, in which a column of the matrix is replaced each time step. As part of this task, the dimension of this dominant subspace is determined automatically for each time step. Two methods for determining the singular values and left singular vectors of this dominant columnspace are presented. The first method, which is an exact method, will update all of the singular values and left singular vectors using the rank-two secular function. The derivation of this function, and its properties, are original contributions of this dissertation. This method requires a single O(n3) matrix product to rotate the singular vectors using direct multiplication, and all other computation is O(n2). The second method, the Improved Fast Adaptive Subspace Tracking (IFAST) method, will give accurate approximations of the r largest singular values and corresponding left singular vectors in O( nr2) time. An accuracy analysis of the approximation error of the second method, using the rank-two secular function from the first method, is presented. The block Hankel matrix structure is presented, which can give improved SNR for exponential signals in sensor arrays, at the expense of beam-width. The two rank determination methods of Shah and Tufts, where one is for time-series (Hankel) data and the other is for unstructured sensor array data, are combined into a single general method which works with unstructured, Hankel, and block Hankel matrices. An efficient way to calculate the thresholds used by this method is presented, allowing use in real-time applications.

Fast filtering of noisy autoregressive signals

Permutability of Toeplitz and Hankel matrices

The discrete-dipole approximation is used to study the problem of light scattering by homogeneous rectangular particles. The structure of the discrete-dipole approximation is investigated, and the matrix formed by this approximation is identified to be a symmetric, block-Toeplitz matrix. Special properties of block-Toeplitz arrays are explored, and an efficient algorithm to solve the dipole scattering problem is provided. Timings for conjugate gradient, Linpack, and block-Toeplitz solvers are given; the results indicate the advantages of the block-Toeplitz algorithm. A practical test of the algorithm was performed on a system of 1400 dipoles, which corresponds to direct inversion of an 8400 × 8400 real matrix. A short discussion of the limitations of the discrete-dipole approximation is provided, and some results for cubes and parallelepipeds are given. We briefly consider how the algorithm may be improved further.

Factorizations of Hermitian block Hankel matrices

CONTENTS Introduction Chapter I. The algebraic structure of the spaces of Toeplitz and Hankel matrices § 1. Basic definitions § 2. Basic notation § 3. Statement of the theorems § 4. Auxiliary lemmas § 5. Proof of the theorems Chapter II. Fast computations in some problems of linear algebra connected with Toeplitz and Hankel matrices § 1. Basic definitions and notation § 2. Statement of the theorems § 3. Auxiliary lemmas § 4. Proof of the theorems Chapter III. The fast prediction of stochastic processes § 1. The fast prediction of stationary stochastic processes § 2. The fast prediction of piecewise constant random functions Chapter IV. The structure of Toeplitz and Hankel integral operators § 1. Definitions § 2. Notation § 3. Statement of the theorems References

We study the solution of block system Tm,nx equals b by preconditioned conjugate gradient methods where Tm,n is an m X m block Toeplitz matrix with n X n Toeplitz blocks. This kind of systems occur in a variety of applications, such as the 2D digital signal processing and the discretization of 2D partial differential equations. We propose a new preconditioner for this kind of block systems. Our preconditioner is defined as the sum of block Toeplitz matrix with Toeplitz blocks and three sparse matrices with structure. Our numerical tests show that our preconditioner is superior to Level-1 and Level-2 circulant preconditioners.© (2000) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

This paper analyzes the recovery performance when a Toeplitz block matrix is used for sampling in compressed sensing. Different from current work that mainly discusses the restricted isometry property and the applications of such matrix, we provide an upper bound for the mean squared reconstruction error. By comparing with the random matrix in which its entries are drawn independently from certain probability distributions, the results show that the Toeplitz block matrix is efficient for compressed sampling. Simulation results validate that the sampling performance of Toeplitz block matrix can approach that of random matrix by choosing its parameters properly.

This paper mainly investigates long term equity return in the Taiwan stock market. We attempted to find the optimal model among and perform diagnostic analysis of the stochastic asset models, such as the independent lognormal model, the ARCH-type model, the Regime-Switching Lognormal model, and the Stochastic Log-Volatility model, all of which have been proposed by the American Academy of Actuaries. In addition, the most suitable long term equity model was determined. In the empirical study of the Taiwan stock market, the parameter estimations were carried out using maximum likelihood estimate and efficient moment of method. Following, we compared log-likelihood, Akaike Information Criterion and Schwartz Bayesian Criterion and drew Quantile-Quantile plots to test tails, to determine the appropriateness of models. Finally, Conditional Tail Expectation risk measures of stochastic equity models were estimated from simulation of the liability of a guaranteed minimum accumulation benefit contract. The results indicate that the generalized autoregressive conditional heteroskedastic and regime switching log normal models perform much better than other considered models for analyzing the Taiwan financial market.

This report documents the Multi-Model Analysis (MMA) computer code. MMA can be used to evaluate results from alternative models of a single system using the same set of observations for all models. As long as the observations, the observation weighting, and system being represented are the same, the models can differ in nearly any way imaginable. For example, they may include different processes, different simulation software, different temporal definitions (for example, steady-state and transient models could be considered), and so on. The multiple models need to be calibrated by nonlinear regression. Calibration of the individual models needs to be completed before application of MMA. MMA can be used to rank models and calculate posterior model probabilities. These can be used to (1) determine the relative importance of the characteristics embodied in the alternative models, (2) calculate model-averaged parameter estimates and predictions, and (3) quantify the uncertainty of parameter estimates and predictions in a way that integrates the variations represented by the alternative models. There is a lack of consensus on what model analysis methods are best, so MMA provides four default methods. Two are based on Kullback-Leibler information, and use the AIC (Akaike Information Criterion) or AICc (second-order-bias-corrected AIC) model discrimination criteria. The other two default methods are the BIC (Bayesian Information Criterion) and the KIC (Kashyap Information Criterion) model discrimination criteria. Use of the KIC criterion is equivalent to using the maximum-likelihood Bayesian model averaging (MLBMA) method. AIC, AICc, and BIC can be derived from Frequentist or Bayesian arguments. The default methods based on Kullback-Leibler information have a number of theoretical advantages, including that they tend to favor more complicated models as more data become available than do the other methods, which makes sense in many situations. Many applications of MMA will be well served by the default methods provided. To use the default methods, the only required input for MMA is a list of directories where the files for the alternate models are located. Evaluation and development of model-analysis methods are active areas of research. To facilitate exploration and innovation, MMA allows the user broad discretion to define alternatives to the default procedures. For example, MMA allows the user to (a) rank models based on model criteria defined using a wide range of provided and user-defined statistics in addition to the default AIC, AICc, BIC, and KIC criteria, (b) create their own criteria using model measures available from the code, and (c) define how each model criterion is used to calculate related posterior model probabilities. The default model criteria rate models are based on model fit to observations, the number of observations and estimated parameters, and, for KIC, the Fisher information matrix. In addition, MMA allows the analysis to include an evaluation of estimated parameter values. This is accomplished by allowing the user to define unreasonable estimated parameter values or relative estimated parameter values. An example of the latter is that it may be expected that one parameter value will be less than another, as might be the case if two parameters represented the hydraulic conductivity of distinct materials such as fine and coarse sand. Models with parameter values that violate the user-defined conditions are excluded from further consideration by MMA. Ground-water models are used as examples in this report, but MMA can be used to evaluate any set of models for which the required files have been produced. MMA needs to read files from a separate directory for each alternative model considered. The needed files are produced when using the Sensitivity-Analysis or Parameter-Estimation mode of UCODE_2005, or, possibly, the equivalent capability of another program. MMA is constructed using

Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix T_n(f), as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated generating function f. A restriction is that f has to be polynomial, monotone, and scalar-valued. In this paper we focus on the case where mathbf {f} is an stimes s matrix-valued trigonometric polynomial with sge 1, and T_n(mathbf {f}) is the block Toeplitz matrix generated by mathbf {f}, whose size is N(n,s)=sn. The case s=1 corresponds to that already treated in the literature. We numerically derive conditions which ensure the existence of an asymptotic expansion for the eigenvalues. Such conditions generalize those known for the scalar-valued setting. Furthermore, following a proposal in the scalar-valued case by the first author, Garoni, and the third author, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric block Toeplitz matrices with a high level of accuracy and a low computational cost. The resulting algorithm is an eigensolver that does not need to store the original matrix, does not need to perform matrix-vector products, and for this reason is called matrix-less. We use the asymptotic expansion for the efficient computation of the spectrum of special block Toeplitz structures and we provide exact formulae for the eigenvalues of the matrices coming from the mathbb {Q}_p Lagrangian Finite Element approximation of a second order elliptic differential problem. Numerical results are presented and critically discussed.

On algebras of Toeplitz plus Hankel matrices

Triangular factorization of general N × N Hermitian matrices requires O(N 3) operations. We present a procedure that factors certain structured matrices in O(N 2) operations. The family of structured matrices considered in this paper includes Toeplitz and Hankel matrices, as well as a variety of ‘Toeplitz-like’ and ‘Hankel-like’ matrices (e.g. inverses, sums and products of Toeplitz and Hankel matrices).

The paper concludes the issue of constructing a statistical model for a satellite image of an urban area. In a previous study (2006) we have introduced the Gaussian mixture as an appropriate model for an urban area when treated as a single object. Here the choice of the number of components of the mixture is addressed as a model selection problem. The Akaike Information Criterion (AIC) is used for choosing the best fitting Gaussian mixture. The EM algorithm is used for the estimation of the parameter and the optimal, parsimonious model is obtained by minimizing the AIC value, under some supplementary conditions on the weights of the different mixture components, so that the identified components are significant and well separated.