Showing papers on "Singular value decomposition published in 1989"

Orthogonal least squares methods and their application to non-linear system identification

Abstract: Identification algorithms based on the well-known linear least squares methods of gaussian elimination, Cholesky decomposition, classical Gram-Schmidt, modified Gram-Schmidt, Householder transformation, Givens method, and singular value decomposition are reviewed. The classical Gram-Schmidt, modified Gram-Schmidt, and Householder transformation algorithms are then extended to combine structure determination, or which terms to include in the model, and parameter estimation in a very simple and efficient manner for a class of multivariate discrete-time non-linear stochastic systems which are linear in the parameters.

On- and off-line identification of linear state-space models

Abstract: A geometrically inspired matrix algorithm is derived for the identification of statespace models for multivariable linear time-invariant systems using (possibly noisy) input-output (I/O) measurements only. As opposed to other (mostly stochastic) identification schemes, no variance-covariance information whatever is involved, and only a limited number of I/O-data are required for the determination of the system matrices. Hence, the algorithm can be best described and understood in the matrix formalism, and consists of the following two steps. First, a state vector sequence is realized as the intersection of the row spaces of two block Hankel matrices, constructed with I/O-data. Then the system matrices are obtained at once from the least-squares solution of a set of linear equations. When dealing with noisy data, this algorithm draws its excellent performance from repeated use of the numerically stable and accurate singular value decomposition. Also, the algorithm is easily applied to slowly time-...

Inversion Of Imaging Spectrometry Data Using Singular Value Decomposition

Abstract: The use of imaging spectrometers, which acquire data that are both spectrally contiguous images and spatially contiguous spectra, for quantitative remote sensing of the earth is addressed. Such data sets cannot be analyzed fully using either existing spectroscopic or image techniques. Singular value decomposition (SVD) is used here for spectral unmixing and determination of the spatial scales of mixing. It is shown that when it is used to invert the mixing endmember library, SVD allows more insight into library characteristics and more control of the inversion process than other commonly used matrix inversion techniques.

Analysis and properties of the generalized total least squares problem AX≈B when some or all columns in A are subject to error

Abstract: The Total Least Squares (TLS) method has been devised as a more global fitting technique than the ordinary least squares technique for solving overdetermined sets of linear equations $AX \approx B$ when errors occur in all data. This method, introduced into numerical analysis by Golub and Van Loan, is strongly based on the Singular Value Decomposition (SVD). If the errors in the measurements A and B are uncorrelated with zero mean and equal variance, TLS is able to compute a strongly consistent estimate of the true solution of the corresponding unperturbed set $A_0 X = B_0 $. In the statistical literature, these coefficients are called the parameters of a classical errors-in-variables model.In this paper, the TLS problem, as well as the TLS computations, are generalized in order to maintain consistency of the parameter estimates in a general errors-in-variables model; i.e., some of the columns of A may be known exactly and the covariance matrix of the errors in the rows of the remaining data matrix may be...

The singular value decomposition: computation and applications to robotics

Abstract: The singular value decomposition has been extensively used for the analysis of the kinematic and dynamic characteristics of robotic manipulators. Due to a reputation for being nu merically expensive to compute, however, it has not been used for real-time applications. This work illustrates a for mulation for the singular value decomposition that takes advantage of the nature of robotics matrix calculations to ob tain a computationally feasible algorithm. Several applica tions, including the control of redundant manipulators and the optimization of dexterity, are discussed. A detailed illus tration of the use of the singular value decomposition to deal with the general problem of singularities is also presented.

Incomplete data problems in X-ray computerized tomography

Abstract: In the present paper we study truncated projections for the fanbeam geometry in computerized tomography First we derive consistency conditions for the divergent beam transform Then we study a singular value decomposition for the case where only the interior rays in the fan are provided, as for example in region-of-interest tomography We show that the high angular frequency components of the searched-for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition

Block reduction of matrices to condensed forms for eigenvalue computations

Abstract: In this paper we described block algorithms for the reduction of a real symmetric matrix to tridiagonal form and for the reduction of a general real matrix to either bidiagonal or Hessenberg form using Householder transformations. The approach is to aggregate the transformations and to apply them in a blocked fashion, thus achieving algorithms that are rich in matrix-matrix operations. These reductions to condensed form typically comprise a preliminary step in the computation of eigenvalues or singular values. With this in mind, we also demonstrate how the initial reduction to tridiagonal or bidiagonal form may be pipelined with the divide and conquer technique for computing the eigensystem of a symmetric matrix or the singular value decomposition of a general matrix to achieve algorithms which are load balanced and rich in matrix-matrix operations.

SVD and signal processing: algorithms, applications and architectures

Abstract: Parts: I. Tutorials. 1. Singular value decomposition: an introduction (P. Dewilde, E.F. Deprettere). 2. A variety of applications of singular value decomposition in identification and signal processing (J. Vandewalle, B. De Moor). 3. Eigen and singular value decomposition techniques for the solution of harmonic retrieval problems (M. Bouvet, H. Clergeot). 4. Advances in principal component signal processing (R.J. Vaccaro et al.). II: Model Reduction and Identification. 5. An overview of Hankel norm model reduction (A.C.M. Ran). 6. Identification of linear state space models with singular value decomposition using canonical correlation concepts (B. De Moor et al.). 7. Detection of multiple sinusoids in white noise: a signal enhancement approach (J.A. Cadzow et al.). III: Total Least Squares and GSVD. 8. The total least squares technique: computation, properties and applications (S. van Huffel, J. Vandewalle). 9. Oriented energy and oriented signal-to-signal ratio concepts in the analysis of vector sequences and time series (B. De Moor et al.). 10. ESPRIT - Estimation of signal parameters via rotational invariance techniques (R. Roy, T. Kailath). IV: Real-Time, Adaptive and Acceleration Algorithms. 11. On-line algorithm for signal separation based on SVD (D. Callaerts et al.). 12. A family of rank-one subspace updating methods (R.D. DeGroat, R.A. Roberts). 13. An array processing technique using the first principal component (P. Comon). 14. A novel method for reducing the computational load of SVD-based high discrimination algorithms (J.L. Mather). 15. Singular value decomposition of Frobenius Matrices for approximate and multi-objective signal processing tasks (E.A. Trachtenberg). V: Algorithms and Architectures. 16. On block Kogbetliantz methods for computation of the SVD (K.V. Fernando, S.J. Hammarling). 17. Reducing the number of sweeps in Hestenes' Method (P.C. Hansen). 18. Computational arrays for cyclic-by-rows Jacobi-algorithms (L. Thiele). 19. The symmetric tridiagonal eigenproblem on a custom linear array and hypercubes (E. de Doncker et al.). 20. Computing the singular value decomposition on the connection machine (L.M. Ewerbring, F.T. Luk). 21. Singular value decomposition on warp (M. Annaratone). 22. Execution of linear algebra operations on the SPRINT (A.J. De Groot et al.). VI. Resolution Limits, Enhancements and Questions. 23. An SVD analysis of resolution limits for harmonic retrieval problems (J.R. Casar, G. Cybenko). 24. A new application of SVD to harmonic retrieval (S. Mayrargue, J.P. Jouveau). 25. Retrieval of significant parameters from magnetic resonance signals via singular value decomposition (R. de Beer et al.).

Robust eigensystem assignment for flexible structures

Abstract: An improved method is developed for eigenvalues and eigenvectors placement of a closed-loop control system using either state or output feedback. The method basically consists of three steps. First, the singular value of QR decomposition is used to generate an orthonormal basis that spans admissible eigenvector space corresponding to each assigned eigenvalue. Secondly, given a unitary matrix, the eigenvector set which best approximates the given matrix in the least-square sense and still satisfy eigenvalue cosntraints is determined. Thirdly, a unitary matrix is sought to minimize the error between the unitary matrix and the assignable eigenvector matrix. For use as the desired eigenvector set, two matrices, namely, the open-loop eigenvector matrix and its closest unitary matrix are proposed. The latter matrix generally encourages both minimum conditioning and control gains. In addition, the algorithm is formulated in real arithmetic for efficient implementation. To illustrate the basic concepts, numerical examples are included.

A proof of convergence for two parallel Jacobi SVD algorithms

Abstract: The authors consider two parallel Jacobi algorithms, due to R.P. Brent et al. (J. VLSI Comput. Syst., vol.1, p.242-70, 1985) and F.T. Luk (1986 J. Lin. Alg. Applic., vol.77, p.259-73), for computing the singular value decomposition of an n*n matrix. By relating the algorithms to the cyclic-by-rows Jacobi method, they prove convergence of the former for odd n and of the latter for any n. The authors also give a nonconvergence example for the former method for all even n>or=4. >

Constrained reconstruction: a superresolution, optimal signal-to-noise alternative to the Fourier transform in magnetic resonance imaging.

Abstract: Many problems in physics involve imaging objects with high spatial frequency content in a limited amount of time. The limitation of available experimental data leads to the infamous problem of diffraction limited data which manifests itself by causing ringing in the image. This ringing is due to the interference phenomena in optics and is known as the Gibbs phenomenon in engineering. Present tehniques to cope with this problem include filtering and regularization schemes based on minimum norm or maximum entropy constraints. In this paper, a new technique based on object modeling and estimation is developed to achieve superresolutionreconstruction from partial Fourier transform data. The nonlinear parameters of the object model are obtained using the singular value decomposition (SVD)‐based all‐pole model framework, and the linear parameters are determined using a standard least squares estimation method. This technique is capable, in principle, of unlimited resolution and is robust with respect to Gaussian white noiseperturbation to the measured data and with respect to systematic modeling errors. Reconstruction results from simulated data and real magnetic resonance data are presented to illustrate the performance of the proposed method.

Robust Principal Components

Abstract: This paper proposes a new algorithm to obtain an eigenvalue decomposition for the sample covariance matrix of a multivariate dataset. The algorithm is based on the rotation technique employed by Ammann and Van Ness (1988a,b) to obtain a robust solution to an errors-in-variables problem. When this rotation technique is combined with an iterative reweighting of the data, a robust eigenvalue decomposition is obtained. This robust eigenvalue decomposition has important applications to principal component analysis. Monte Carlo simulations are performed to compare ordinary principal component analysis using the standard eigenvalue decomposition with this algorithm, referred to as ROPRC. It is seen that ROPRC is reasonably efficient compared to an eigenvalue decomposition when Gaussian data is available, and that ROPRC is much better than the eigenvalue decomposition if outliers are present or if the data has a heavy-tailed distribution. The algorithm returns useful numerical diagnostic information in the form o...

Matrix analysis of metamorphic mineral assemblages and reactions

Abstract: Assemblage diagrams are widely used in interpreting metamorphic mineral assemblages. In simple systems, they can help to identify assemblages which may represent equilibrium states; to determine whether differences between assemblages reflect changes in metamorphic grade or variations in bulk composition; and to characterize isograd reactions. In multicomponent assemblages these questions can be approached by investigating the rank, composition space (range) and reaction space (null-space) of a matrix representing the compositions of the phases involved. Singular value decomposition (SVD) provides an elegant way of (1) finding the rank of a matrix and detemining orthonormal bases for both the composition space and the reaction space needed to represent an assemblage or pair of assemblages; and (2) finding a model matrix of specified rank which is closest in a least squares sense to an observed assemblage. Although closely related to least squares techniques, the SVD approach has the advantages that it tolerates errors in all observations and is computationally simpler and more stable than non-linear least squares algorithms. Models of this sort can be used to interpret multicomponent mineral assemblages by straightforward generalizations of the methods used to interpret assemblage diagrams in simpler systems. SVD analysis of mineral assemblages described by Lang and Rice (1985) demonstrates the utility of the approach.

A one-sided Jacobi algorithm for computing the singular value decomposition on avector computer

Abstract: An old algorithm for computing the singular value decomposition, which was first mentioned by Hestenes [SIAM J. Appl. Math., 6 (1958), pp. 51–90], has gained renewed interest because of its properties of parallelism and vectorizability. Some computational modifications are given and a comparison with the well-known Golub–Reinsch algorithm is made. Comparative experiments on a CYBER 205 are reported.

Matrix pencil of closed-loop descriptor systems: infinite-eigenvalue assignment

Abstract: The regular matrix pencil [sE – A – BK] is considered that characterizes a closed-loop descriptor system , u(t) = Kx(t) + ũ(t). A carefully contrived and efficient method is given for the determination of a set of matrices K such that the determinant ∣sE – A – BK∣ is a constant value independent of s. The problem is formulated as an infinite-eigenvalue assignment by finite-gain descriptor-variable feedback via a singular value decomposition of the matrix E. The result is interesting in its own right and finds application in controller and observer design.

Phillips–Tikhonov regularization of plasma image reconstruction with the generalized cross validation

Abstract: The use of the Phillips–Tikhonov regularization is proposed for numerically stabilizing the ill‐conditioned plasma image reconstructions. An objective function to be minimized leads to a linear estimator of the image intensity distribution and, with the aid of the singular value decomposition, makes it possible to use the generalized cross validation for optimizing a regularization parameter. An excellent behavior of the estimator with computational facility is obtained on the Hα emission computerized tomography of a toroidal plasma.

Singular value decomposition of integral equations of EM and applications to the cavity resonance problem

Abstract: The use of the method of moments (MM) based on the electric-field integral equation is considered for conducting bodies with closed regions. For such a calculation, the scattered field is theoretically well defined, but numerical problems have been found near resonant frequencies. To study these problems, the use of the singular value decomposition (SVD) in MM calculations is developed, and the reason the SVD is the appropriate tool for the numerical analysis of any MM calculation (even when the matrix equation is to be solved by an iterative technique) is shown. In particular, the SVD casts a MM calculation into a diagonal form, which allows a careful numerical analysis. The problems near resonance are found to be due to the creation of an approximate impedance matrix Z, which has the wrong resonant frequency (i.e., it is not consistent with that of the radiation problem). One numerical method that avoids these difficulties by using the SVD is discussed, and other more efficient ways of avoiding the usual numerical difficulties are suggested. >

Frequency domain direct parameter identification for modal analysis : state space formulation

Abstract: Using a state space formulation for the equations of motion for a mechanical structure, an algorithm is developed to identify its state transition matrix from measured multiple input/multiple output relations. From the identified matrix quantities, it is possible to derive a modal model of natural frequencies, damping values, mode shapes and modal participation factors

Generalized singular value decompositions: a proposal for a standardized nomenclature

Abstract: An alphabetic and mnemonic system of names for several matrix decompositions related to the singular value decomposition is proposed: the OSVD, PSVD, QSVD, RSVD, SSVD, TSVD. The main purpose of this note is to propose a standardization of the nomenclature and the structure of these matrix decompositions.

Deriving source-time functions using principal component analysis

Abstract: Factors such as source complexity, microseismic noise, and lateral heterogeneity all introduce nonuniqueness into the source-time function. The technique of principal component analysis is used to factor the moment tensor into a set of orthogonal source-time functions. This is accomplished through the singular value decomposition of the time-varying moment tensor. The adequacy of assuming a single source-time function may then be examined through the singular values of the decomposition. The F test can also be used to assess the significance of the various principal component basis functions. The set of significant basis functions can be used to test models of the source-time functions, including multiple sources. Application of this technique to the Harzer nuclear explosion indicated that a single source-time function was found to adequately explain the moment tensor. It consists of a single pulse appearing on the diagonal elements of the moment-rate tensor. The decomposition of the moment tensor for a deep teleseism in the Bonin Islands revealed three basis functions associated with relatively large singular values. The F test indicated that only two of the principal components were significant. The principal component associated with the largest singular value consists of a large pulse followed 16-sec later by a diminished pulse. The second principal component, a long-period oscillation, appears to be a manifestation of the poor resolution of the moment-rate tensor at low frequencies.

Canonical correlations and generalized SVD: Applications and new algorithms

Abstract: In this paper we consider canonical correlations and a generalization of the singular value decomposition (SVD) that involves three matrices. We show how the two matrix problems are related and how they can be used in important applications such as weighted least squares and optimal prediction. We present two new computational procedures for the problems based on implicit SVD methods for triple matrixproducts. Our algorithms are well suited for parallel implementation.

The hyperbolic singular value decomposition and applications

Abstract: A new generalization of singular value decomposition (SVD), the hyperbolic SVD, is advanced, and its existence is established under mild restrictions. Two algorithms for effecting this decomposition are discussed. The new decomposition has applications in downdating in problems where the solution depends on the eigenstructure of the normal equations and in the covariance differencing algorithm for bearing estimation in sensor arrays. Numerical examples demonstrate that, like its conventional counterpart, the hyperbolic SVD exhibits superior numerical behavior relative to explicit formation and solution of the normal equations. (However, unlike ordinary SVD, it is applicable to eigenanalysis of covariances arising from a difference of outer products). >

Updating singular value decompositions : A parallel implementation

Abstract: In this paper, we give an overview of a few recently obtained results regarding al­ gorithms and systolic arrays for updating singular value decompositions. The Ordinary SVD as well as the Product SVD and the Quotient SVD will be discussed.The updating algorithms consist in an interlacing of Q^-upd&tings and a Jacobi-type SVD-algorithm applied to the triangular factor(s). At any time step an approximate decomposition is computed from a previous approximation, with a limited number of operations (O(n2 )). When combined with exponential weighting, these algorithms are seen to be highly applicable to tracking problems. Furthermore, they can elegantly be mapped onto systolic arrays, making use of slight modifications of well known systolic implementations for the matrix-vector product, the Q^-updating and the SVD. 1 Introduction Let A be an m x n matrix, with (m > n). The Ordinary Singular Value Decomposition (OSVD) is a factorization of A into a product of three matriceswhere £/"*[/ = I, V*V = I, and S is a diagonal matrix, with the singular values on the main The updating problem considered here, consists in computing the OSVD of the modified matrix[ \.A ] _ ^ ^ a1 \-U(

Perturbation bounds for discrete Tikhonov regularisation

Abstract: A highly regarded method for solving discrete ill-posed problems is the regularisation method due to Tikhonov (1963) The author uses the generalised SVD to derive perturbation bounds for the regularised solution when both the matrix and the right-hand side are perturbed

Bayesian updating revisited

Abstract: Bayesian updating methods provide an alternate philosophy to the characterization of the input variables of a stochastic mathematical model. Here, a priori values of statistical parameters are assumed on subjective grounds or by analysis of a data base from a geologically similar area. As measurements become available during site investigations, updated estimates of parameters characterizing spatial variability are generated. However, in solving the traditional updating equations, an updated covariance matrix may be generated that is not positive-definite, particularly when observed data errors are small. In addition, measurements may indicate that initial estimates of the statistical parameters are poor. The traditional procedure does not have a facility to revise the parameter estimates before the update is carried out. alternatively, Bayesian updating can be viewed as a linear inverse problem that minimizes a weighted combination of solution simplicity and data misfit. Depending on the weight given to the a priori information, a different solution is generated. A Bayesian updating procedure for log-conductivity interpolation that uses a singular value decomposition (SVD) is presented. An efficient and stable algorithm is outlined that computes the updated log-conductivity field and the a posteriori covariance of the estimated values (estimation errors). In addition, an information density matrix is constructed that indicates how well predicted data match observations. Analysis of this matrix indicates the relative importance of the observed data. The SVD updating procedure is used to interpolate the log-conductivity fields of a series of hypothetical aquifers to demonstrate pitfalls and possibilities of the method.