Singular-value decomposition and embedding dimension (original) (raw)
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Statistical dimension estimation in singular spectrum analysis
Journal of the Italian Statistical Society, 1996
Singular spectrum analysis has been proposed in the field of nonlinear dynamical systems as filtering method. In this paper a criterion to choose the number of components which leads to the best filtering is proposed. The selection is made by minimizing the prediction elTor.
In this paper we propose a method of using nonlinear generalization of Singular Value Decomposition (SVD) to arrive at an upper bound for the dimension of a manifold which is embedded in some RN. We have assumed that the data about its co-ordinates is available. We would also assume that there exists at least one small neighborhood with sufficient number of data points. Given these conditions, we show a method to compute the dimension of a manifold. We begin by looking at the simple case when the manifold is in the form of a lower dimensional affine subspace. In this case, we show that the well known technique of SVD can be used to (i) calculate the dimension of the manifold and (ii) to get the equations which define the subspace. For the more general case, we have applied a nonlinear generalization of the SVD (i) to search for an upper bound for the dimension of the manifold and (ii) to find the equations for the local charts of the manifold. We have included a brief discussion about how this method would be highly useful in the context of the Takens’ embedding which is used in the analysis of a time series data from a dynamical system. We show a specific problem that has recently been found out when applying this method. One very effective solution is to develop a model which is based on local charts and for this purpose a good estimate of the underlying dimension of an embedded data is required.
A Tutorial On The Singular Value Decomposition
1996
An m y, n real matm A can be factored as ITWV , where V and V are orthonormal, and W is upper left diagonal. This factorization is called Singular Vaiue Decomposilion (SVD). The matrlces U, W, and V are useful in characterizing ihe malrix A. in this manuscript geometric characterizations are emphasized. Geometric characierizations are anaiyzed in terms ofsubspaces, matrix scaling, cmd norms. We also preseni a numerical viewpoint for SVD m orcfer to keep the maierial setf-contained. in the last section we ireat a special problem where action ofihe matrix A is restncted to a given subspace.
Estimating the embedding dimension
Physica D: Nonlinear Phenomena, 1991
Well reconstructed dynamics should be described as a mapping. We propose to search fl)r the lowest embedding dimension for which the property of a continuous mapping is satisfied: the images of close points are close. We demonstrate the effectiveness of the approach by finding the lowest embedding dimension of several chaotic systems embedded by the method of delays.
Singular value decomposition and spectral analysis
Conference on Decision and Control, 1981
Linear-Prediction-based (LP) methods for fitting multiple-sinusoid signal models to observed data, such as the forward-backward (FBLP) method of Nuttall (5) and Ulrych and Clayton (6), are very ill-conditioned. The locations of estimated spectral peaks can be greatly affected by a small amount of additive noise. LP estimation of frequencies can be greatly improved by singular value decomposition of the LP
The Restricted Singular Value Decomposition: Properties and Applications
SIAM Journal on Matrix Analysis and Applications, 1991
The restricted singular value dccomporition (RSVD) is the factorization of a given matrix, relative to two other given matrices. It can be interpreted as the ordinarjr singular ualuc decomporition with different inner products in row and column spaces. Its properties and structure are investigated in detail as well as its connection to generalized eigenvdue problems, canonicd correlation andysiz and other generdizationz of the singular value decompozition. Applications that are discussed include the analysis of the extended shorted operator, unitarily invariant norm minimization with rank constraints, rank minimization in matrix ballz, the analysis and solution of linear matrix equations, rank minimization of a partitioned matrix and the connection with generalized Schur complements, constrained linear and total linear least squares problems, with mixed exact and noisy data, including a generalized Gauss-Markov estimation scheme. Two constructive proofs of the RSVD in terms of other generalizations of the ordinary singular value decomposition are provided as well.
2014
In this paper we propose a method of using nonlinear generalization of Singular Value Decomposition (SVD) to arrive at an upper bound for the dimension of a manifold which is embedded in some RN. We have assumed that the data about its co-ordinates is available. We would also assume that there exists at least one small neighborhood with sufficient number of data points. Given these conditions, we show a method to compute the dimension of a manifold. We begin by looking at the simple case when the manifold is in the form of a lower dimensional affine subspace. In this case, we show that the well known technique of SVD can be used to (i) calculate the dimension of the manifold and (ii) to get the equations which define the subspace. For the more general case, we have applied a nonlinear generalization of the SVD (i) to search for an upper bound for the dimension of the manifold and (ii) to find the equations for the local charts of the manifold. We have included a brief discussion abo...
Singular-value decomposition approach to time series modelling
IEE Proceedings F Communications, Radar and Signal Processing, 1983
In various signal processing applications, as exemplified by spectral analysis, deconvolution and adaptive filtering, the parameters of a linear recursive model are to be selected so that the model is 'most' representative of a given set of time series observations. For many of these applications, the parameters are known to satisfy a theoretical recursive relationship involving the time series' autocorrelation lags. Conceptually, one may then use this recursive relationship, with appropriate autocorrelation lag estimates substituted, to effect estimates for the operator's parameters. A procedure for carrying out this parameter estimation is given which makes use of the singular-value decomposition (SVD) of an extended-order autocorrelation matrix associated with the given time series. Unlike other SVD modelling methods, however, the approach developed does not require a full-order SVD determination. Only a small subset of the matrix's singular values and associated characteristic vectors need be computed. This feature can significantly alleviate an otherwise overwhelming computational burden that is necessitated when generating a full-order SVD. Furthermore, the modelling performance of this new method has been found empirically to excel that of a near maximum-likelihood SVD method as well as several other more traditional modelling methods. 'The symbol [«,, n 2 ] denotes the set of integers satisfying n x < n < n2 while [ n,, °°) specifies the set of integers satisfying n > n,.
Determination of the dimension of a signal subspace from short data records
IEEE Transactions on Signal Processing, 1994
A nested sequence of constant false alarm rate (CFAR) hypothesis tests is presented for rank determination over short &la records. The procedure is based on the interpretation of sum of squares of singular values as energy in a particular subspace. The CFAR thresholds are set up based on distributions derived from matrix perturbation ideas. Expressions for probability of overestimation and underestimation are presented.
Singular Value Decomposition for Multidimensional Matrices
Singular Value Decomposition (SVD) is of great significance in theory development of mathematics and statistics. In this paper we propose the SVD for 3-dimensional (3-D) matrices and extend it to the general Multidimensional Matrices (MM). We use the basic operations associated with MM introduced by Solo to define some additional aspects of MM. We achieve SVD for 3-D matrix through these MM operations. The proposed SVD has similar characterizations as for 2-D matrices. Further we summarize various characterizations of singular values obtained through the SVD of MM. We demonstrate our results with an example and compare them with the existing method. We also develop Matlab functions to perform SVD of MM and some related MM operations.