Showing papers on "Singular value decomposition published in 1993"

The use of the L-curve in the regularization of discrete ill-posed problems

Abstract: Regularization algorithms are often used to produce reasonable solutions to ill-posed problems. The L-curve is a plot—for all valid regularization parameters—of the size of the regularized solution versus the size of the corresponding residual. Two main results are established. First a unifying characterization of various regularization methods is given and it is shown that the measurement of “size” is dependent on the particular regularization method chosen. For example, the 2-norm is appropriate for Tikhonov regularization, but a 1-norm in the coordinate system of the singular value decomposition (SVD) is relevant to truncated SVD regularization. Second, a new method is proposed for choosing the regularization parameter based on the L-curve, and it is shown how this method can be implemented efficiently. The method is compared to generalized cross validation and this new method is shown to be more robust in the presence of correlated errors.

On the early history of the singular value decomposition

Abstract: This paper surveys the contributions of five mathematicians—Eugenio Beltrami (1835–1899), Camille Jordan (1838–1921), James Joseph Sylvester (1814–1897), Erhard Schmidt (1876–1959), and Hermann Weyl (1885–1955)—who were responsible for establishing the existence of the singular value decomposition and developing its theory.

Voltage stability indices for stressed power systems

Abstract: The use of static voltage stability indices based on a singular value decomposition of the power flow Jacobian matrix and matrices derived from the Jacobian matrix is discussed. It is shown that such indices, together with the singular vectors, contain substantial and important information about the proximity to voltage instability and also about critical buses and disturbances from a voltage instability point of view. This is done by a theoretical analysis of the linear power flow equations and an analysis from model power systems as well as realistic power systems (1033 nodes). It is argued that indices based on these matrices are useful for the system analyst in planning and operations planning. >

Identification of observer/Kalman filter Markov parameters - Theory and experiments

Abstract: An algorithm to compute Markov parameters of an observer or Kalman filter from experimental input and output data is discussed. The Markov parameters can then be used for identification of a state space representation, with associated Kalman gain or observer gain, for the purpose of controller design. The algorithm is a non-recursive matrix version of two recursive algorithms developed in previous works for different purposes. The relationship between these other algorithms is developed. The new matrix formulation here gives insight into the existence and uniqueness of solutions of certain equations and gives bounds on the proper choice of observer order. It is shown that if one uses data containing noise, and seeks the fastest possible deterministic observer, the deadbeat observer, one instead obtains the Kalman filter, which is the fastest possible observer in the stochastic environment. Results are demonstrated in numerical studies and in experiments on an ten-bay truss structure.

Subspace-based signal analysis using singular value decomposition

Abstract: A unified approach is presented to the related problems of recovering signal parameters from noisy observations and identifying linear system model parameters from observed input/output signals, both using singular value decomposition (SVD) techniques. Both known and new SVD-based identification methods are classified in a subspace-oriented scheme. The SVD of a matrix constructed from the observed signal data provides the key step in a robust discrimination between desired signals and disturbing signals in terms of signal and noise subspaces. The methods that are presented are distinguished by the way in which the subspaces are determined and how the signal or system model parameters are extracted from these subspaces. Typical examples, such as the direction-of-arrival problem and system identification from input/output measurements, are elaborated upon, and some extensions to time-varying systems are given. >

Performance analysis for DOA estimation algorithms: unification, simplification, and observations

Abstract: Subspace based direction-of-arrival (DOA) estimation has motivated many performance studies, but limitations such as the assumption of an infinite amount of data and analysis of individual algorithms generally exist in these performance studies. The authors have previously proposed a unified performance analysis based on a finite amount of data and achieved a tractable expression for the mean-squared DOA estimation error for the multiple signal classification (MUSIC). Min-Norm, estimation of signal parameters using rotational invariance techniques (ESPRIT), and state-space realization algorithms. However, this expression uses the singular values and vectors of a data matrix, which are obtained by the highly nonlinear transformation of the singular value decomposition (SVD). Thus the effects of the original data parameters such as numbers of sensors and snapshots, source coherence and separations were not explicitly analyzed. The authors unify and simplify this previous result and derive a unified expression based on the original data parameters. They analytically observe the effects of these parameters on the estimation error. >

Call connection technique

Abstract: In an environment in which simultaneous voice/data (SVD) modems are provisioned in the public switched network for separating and separately routing voice and data calls from users having SVD modems, at least one user who is already employing a first connection through an SVD modem provisioned in the public switched network, for interacting with a remote destination over one of the two channels provided by an SVD modem, may have a second connection, over the second of the two channels provided by an SVD modem, automatically established for him The second connection may be established in response to a request by one of the users In a particular embodiment of the invention, the data channels supplied from each SVD modem of each of said users are routed to a common data application, eg, a game, and a direct connection is established between the voice channels of the SVD modems in response to a command from the data application in response to the request of one of the users Such a feature is called "voice assist" or, in the gaming context, "talk and play" A similar "data assist" feature can to automatically establish a data connection between users who are interacting with a common application over their voice channels A context-sensitive voice assist feature is also provided to automatically establish a voice connection between a user of an application and an appropriate person, given the context in which the user requests the voice connection

Limited-data computed tomography algorithms for the physical sciences

Abstract: Five limited-data computed tomography algorithms are compared. The algorithms used are adapted versions of the algebraic reconstruction technique, the multiplicative algebraic reconstruction technique, the Gerchberg–Papoulis algorithm, a spectral extrapolation algorithm descended from that of Harris [J. Opt. Soc. Am. 54, 931–936 (1964)], and an algorithm based on the singular value decomposition technique. These algorithms were used to reconstruct phantom data with realistic levels of noise from a number of different imaging geometries. The phantoms, the imaging geometries, and the noise were chosen to simulate the conditions encountered in typical computed tomography applications in the physical sciences, and the implementations of the algorithms were optimized for these applications. The multiplicative algebraic reconstruction technique algorithm gave the best results overall; the algebraic reconstruction technique gave the best results for very smooth objects or very noisy (20-dB signal-to-noise ratio) data. My implementations of both of these algorithms incorporate a priori knowledge of the sign of the object, its extent, and its smoothness. The smoothness of the reconstruction is enforced through the use of an appropriate object model (by use of cubic B-spline basis functions and a number of object coefficients appropriate to the object being reconstructed). The average reconstruction error was 1.7% of the maximum phantom value with the multiplicative algebraic reconstruction technique of a phantom with moderate-to-steep gradients by use of data from five viewing angles with a 30-dB signal-to-noise ratio.

The singular value decomposition and long and short spaces of noisy matrices

Abstract: Using geometrical, algebraic, and statistical arguments, it is clarified why and when the singular value decomposition is successful in so-called subspace methods. First the concepts of long and short spaces are introduced, and a fundamental asymmetry in the consistency properties of the estimates is discussed. The model, which is associated with the short space, can be estimated consistently, but the estimates of the original data, which follow from the long space, are always inconsistent. An expression is found for the asymptotic bias in terms of canonical angles, which can be estimated from the data. This allows all equivalent reconstructions of the original signals to be described as a matrix ball, the center of which is the minimum variance estimate. Remarkably, the canonical angles also appear in the optimal weighting that is used in weighted subspace fitting approaches. The results are illustrated with a numerical simulation. A number of examples are discussed. >

Updating a rank-revealing ULV decomposition

Abstract: A ULV decomposition of a matrix A of order n is a decomposition of the form $A = ULV^H $, where U and V are orthogonal matrices and L is a lower triangular matrix. When A is approximately of rank k, the decomposition is rank revealing if the last $n - k$ rows of L are small. This paper presents algorithms for updating a rank-revealing ULV decomposition. The algorithms run in $O( n^2 )$ time, and can be implemented on a linear array of processors to run in $O( n )$ time.

13 Computation using the QR decomposition

Abstract: Publisher Summary The QR decomposition is one of the most basic tools for statistical computation, yet it is also one of the most versatile and powerful. The chapter discusses the problem of perfect or nearly perfect linear dependencies, the complex QR decomposition, the QR decomposition in regression, the essential properties of the QR decomposition, and the use of Householder transformations to compute the QR decomposition. A classical alternative to the Householder QR algorithm is the Gram–Schmidt method. Least-squares regression estimates can also be found using either the Cholesky factorization or the singular value decomposition. The QR decomposition approach has been observed to offer excellent numerical properties at reasonable computational cost, while providing factors, Q and R , which are quite generally useful. Although the QR decomposition is a long established technique, it is not in stasis. The QR decomposition is a key to stable and efficient solutions to many least-squares problems. A diverse collection of examples in statistics is given in the chapter and it is certain that there are many more problems for which orthogonalization algorithms can be exploited. In addition, the QR decomposition provides ready insight into the statistical properties of regression estimates.

SVDPACKC (Version 1.0) User''s Guide

Abstract: SVDPACKC comprises four numerical (iterative) methods for computing the singular value decomposition (SVD) of large sparse matrices using ANSI C. This software package implements Lanczos and subspace iteration-based methods for determining several of the largest singular triplets (singular values and corresponding left- and right-singular vectors) for large sparse matrices. The package has been ported to a variety of machines ranging from supercomputers to workstations: CRAY Y-MP, IBM RS/6000-550, DEC 5000-100, HP 9000-750, SPARCstation 2, and Macintosh II/fx. This document {\it (i)} explains each algorithm in some detail, {\it (ii)} explains the input parameters for each program, {\it (iii)} explains how to compile/execute each program, and {\it (iv)} illustrates the performance of each method when we compute lower rank approximations to sparse {\it term-document} matrices from information retrieval applications. A user-friendly software interface to the package for UNIX-based systems and the Macintosh II/fx is also described.

Two-dimensional Prony modeling and parameter estimation

Abstract: A new method for estimating the two-dimensional (2D) exponential modes and amplitude coefficients in a Prony model is presented. This method involves two parts, each utilizing a 1D singular value decomposition-based technique, and is capable of locating frequencies anywhere in the 2D frequency plane. Simulations are shown which demonstrate the performance of the algorithm. >

A method and system for near real-time analysis and display of electrocardiographic signals

Abstract: The multidimensional ECG processing and display system (60) of the present invention may be used with an electrocadiographic (ECG) monitoring system. Input ECG data (61) from multiple, sequential time intervals is collected and formatted into a two-dimensional matrix using the processing function (62). The two-dimensional matrix is decomposed using singular value decomposition (SVD) to obtain its corresponding singular values and singular vectors, a compressed form of the matrix. The singular vectors are analyzed and filtered to identify and enhance signal components of interest using the subspace processing function (63) and the signal processing function (64). Selected singular vectors are transformed into their frequency domain representations by the Fast Fourier Transform (FFT), or related techniques.

Practical transfer function estimation and its application to wide frequency range representation of transformers

Abstract: A widely applicable, general methodology for estimation of transfer function parameters from frequency response data is presented. The procedure is based on the solution of a linear least squares problem by the singular value decomposition (SVD). The condition of the problem is discussed and approaches referred to as shifting and scaling are introduced in order to reduce the condition number. To extend the application to practical cases with measurement errors and/or a large number of poles, a partitioned estimation method with Gauss-Seidel iterations is developed. An iterative improvement process with constraints on the poles is applied to increase the accuracy and to avoid the possibility of obtaining unstable poles. The application of the suggested method of estimation to the representation of transformers is presented with practical examples. Either transfer function or state equation representation can be obtained for transformers described by their terminal frequency responses. >

Actuator Placement Optimization by Genetic and Improved Simulated Annealing Algorithms.

Abstract: Concluding Remarks The following have been mathematically proven and shown by a numerical example: 1) SVD, as is well known, is a very strong numerical analysis tool for solving ill-conditioned linear equations, but it can be computationally costly when there is only small rank deficiency. 2) As was shown here, ED is a first-order approximation of SVD, and as was shown in Ref. 2, ED is computationally very efficient for large matrices with small rank deficiency, because only the first nonzero and all zero eigenvalues and eigenvectors are required. 3) As has been shown in this Note, SED is a second-order approximation of SVD and only requires the solution of two sets of linear equations. However, unless one obtains the first nonzero eigenvalue, the errors involved will be difficult to bound accurately.

A minimal realization approach to reduced-order modelling and modal analysis for power system response signals

Abstract: The authors report on a numerical scheme for compressing in parametric form small signal electromechanical responses of multimachine power systems, originating from transient stability programs (TSPs) or actual field testing. The result is achieved by using a multi-input multi-output (MIMO) minimal realization algorithm based on singular value decomposition (SVD), which can explicitly take into account the critical impact of the input interactions. The resulting parametric model is a reduced order representation of the underlying complex system, yet it is optimal (in the least-squares sense). Besides compact storage of damping information, the balanced state-space realization as such retains the principal components of the response signals, and could thus be useful for the tuning of static volt-ampere reactive (VAr) systems (SVSs) and power system stabilizers (PSSs). When it is transformed in the modal space, the model also provides insight into modal interaction mechanisms. Several examples are included for illustration purposes and other applications and improvements are also discussed. >

Regularized image reconstruction using SVD and a neural network method for matrix inversion

Abstract: Two methods of matrix inversion are compared for use in an image reconstruction algorithm. The first is based on energy minimization using a Hopfield neural network. This is compared with the inverse obtained using singular value decomposition (SVD). It is shown for a practical example that the neural network provides a more useful and robust matrix inverse. >

A singular value decomposition (SVD) based method for suppressing ocean clutter in high frequency radar

Abstract: The suppression of ocean clutter signals in high-frequency (HF) ground-wave radar is treated. The technique proposed is based on a new method of tracking time-varying frequencies of superimposed harmonics by singular value decomposition (SVD) and rank reduction of a Hankel matrix of the time-series data. A similar but reverse signal processing technique is developed in which the frequency components of the clutter signals are removed and another reduced rank Hankel matrix is constructed, and hence a new time series can be extracted. Computer-synthesized data were used to simulate the clutter signals, and it was found that the level of the clutter signals was substantially suppressed. >

Backward perturbation analysis of certain characteristic subspaces

Abstract: This paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemAx=?x, the generalized eigenvalue problem βAx=?Bx, and the singular value decomposition of a matrixA. This paper also gives residual bounds for certain eigenvalues, generalized eigenvalues and singular values.

Design of identification experiments for robust control. A geometric approach for bivariate processes

Abstract: A new approach to the design of experiments is proposed to identify linear multiple-input-multiple-output (MIMO) models that will provide robust control. The experimental designs for identification are based on minimizing uncertainties in the structure of the multivariate model rather than simply the magnitude of identification error. The experimental designs are derived for steady-state robust stability of 2×2 systems using a geometric approach. This approach leads to a simple and unified design approach based on the singular value decomposition (SVD) of the process gain matrix. These robust or control-relevant identification designs for MIMO systems differ considerably from traditional designs developed for single-output systems. Typically in these new multivariate designs, the inputs are correlated, they are not binary sequences, and the magnitudes of the perturbations in low-gain directions are much larger than those in high-gain directions. The results are extended to identification under closed-loop conditions. Dual composition control of distillation processes is used to illustrate the physical interpretations and the effectiveness of the SVD-based design

Method and system to enhance medical signals for real-time analysis and high-resolution display

Abstract: A medical processing and display system that may be used with a medical monitoring device. This system enhances the medical data it receives. Once received, the medical data is reformatted into a two-dimensional matrix, X. A history database and other information are concatenated with the two-dimensional matrix. The concatenated matrix is decomposed using singular value decomposition ("SVD") to obtain corresponding left and right singular vectors, L and R, respectively, and singular values D. Selected singular vectors are transformed to their autocorrelation matrix form, which are concatenated, then decomposed using SVD to their corresponding singular vectors P, Pt, and singular values D. Certain of the singular vectors P are selected to filter out signal components of interest. The singular values D of the autocorrelation matrix are modified and used to adjust the weights of the associated singular vectors P. The weighted singular vectors are then combined and the resulting coefficients are used as a FIR filter to enhance the original singular vectors L and/or Rt to enhance singular values Le, Re. Enhanced medical data containing the features of interest Xe is generated from the enhanced original matrix singular values Le and Re, and modified singular values De, and the results may be displayed to a diagnotician in various formats.

Nonlinear noise reduction: A case study on experimental data.

Abstract: We apply a recently proposed nonlinear noise-reduction method to time sequences from two different experiments. We demonstrate that it is not difficult to choose the parameters of this algorithm, even though we use no other information about the underlying dynamics than the data themselves. The noise reduction is very robust with respect to changes in the choice of parameters. The reliability of the result is tested by an analysis of the corrections. We discuss the effect of noise reduction on estimates of dimensions, entropies, and Liapunov exponents. For comparison we process one of the sets, densely sampled Taylor-Couette flow data, with a global filter based on singular value decomposition.

Eigenvalues and eigenvectors of linearized elastic inversion

Abstract: Small contrasts in the parameters allow the linearization of elastic modeling and inversion. Although this assumption simplifies parameter estimation, it also impoverishes information: a linearized model, besides being an approximation, requires shrinking the data space to precritical angles. This work investigates the two basic questions regarding the type of information that can be retrieved and how to retrieve it. The method is the singular value decomposition of the transfer function between data and parameters. This tool allows a simple interpretation of the information contained in reflections, and it outlines a general estimation procedure. A medium characterized by a uniform background is considered; reflections are linearized according to the Born approximation. P-P reflections are analyzed. The simplicity of the model allows a theoretical study of the effects of velocity errors in the overburden. The analysis is performed for two different sets of elastic parameters. In both cases the conclusion...

Singular value decomposition-based MA order determination of non-Gaussian ARMA models

Abstract: Singular-value-decomposition (SVD)-based moving-average (MA) order determination of non-Gaussian processes using higher-order statistics is addressed. It is shown that the MA order determination of autoregressive moving-average (ARMA) models is equivalent to the rank determination of a certain error matrix, and a SVD approach is proposed. Its simplified form is applied to pure MA models. To improve the robustness of the order selection, a combination of the SVD and the product of diagonal entries (PODE) test is proposed. Some interesting applications of the two SVD approaches are presented, and simulations verify their performance. >