Showing papers on "Singular value decomposition published in 1974"
5 citations
TL;DR: In this paper, a means for detecting the presence of multicollinearity and assessing the damage that such collinearities may cause estimated coefficients in the standard linear regression model is presented.
Abstract: This paper presents a means for detecting the presence of multicollinearity and for assessing the damage that such collinearity may cause estimated coefficients in the standard linear regression model. The means of analysis is the singular value decomposition, a numerical analytic device that directly exposes both the conditioning of the data matrix X and the linear dependencies that may exist among its columns. The same information is employed in the second part of the paper to determine the extent to which each regression coefficient is being adversely affected by each linear relation among the columns of X that lead to its ill conditioning.
5 citations
TL;DR: In this article, it was shown that if det A=±1, then A =±qi=1Bi, where Bi2 = I. This decomposition is used to find the Jacobian of the linear matrix transformation: Y=AX.
5 citations
TL;DR: The robustness and the computational stability of the singular value decomposition algorithm used at the NBER Computer Research Center is discussed and the effect of perturbations on input data is explored.
Abstract: This paper discusses the robustness and the computational stability of the singular value decomposition algorithm used at the NBER Computer Research Center. The effect of perturbations on input data is explored. Suggestions are made for using the algorithm to get information about the rank of a real square or rectangular matrix. The algorithm can also be used to compute the best approximate solution of linear system of equations in the least squares sense, to solve linear systems of equations with equality constraints, and to determine dependencies or near dependencies among the rows or columns of a matrix. A copy of the subroutine that is used and some examples on which it has been tested are included in the appendixes.
3 citations
Posted Content•
TL;DR: In this article, the robustness and computational stability of the singular value decomposition (SVD) algorithm used at the NBER Computer Research Center were discussed. But the effect of perturbations on input data is explored.
Abstract: This paper discusses the robustness and the computational stability of the singular value decomposition algorithm used at the NBER Computer Research Center. The effect of perturbations on input data is explored. Suggestions are made for using the algorithm to get information about the rank of a real square or rectangular matrix. The algorithm can also be used to compute the best approximate solution of linear system of equations in the least squares sense, to solve linear systems of equations with equality constraints, and to determine dependencies or near dependencies among the rows or columns of a matrix. A copy of the subroutine that is used and some examples on which it has been tested are included in the appendixes.
2 citations
Posted Content•
TL;DR: The solution of the generalized symmetric eigenproblem Ax = λBx is required in many multivariate statistical models, viz. canonical correlation, discriminant analysis, multivariate linear model, limited information maximum likelihoods.
Abstract: The solution of the generalized symmetric eigenproblem Ax = λBx is required in many multivariate statistical models, viz. canonical correlation, discriminant analysis, multivariate linear model, limited information maximum likelihoods. The problem can be solved by two efficient numerical algorithms: Cholesky decomposition or singular value decomposition. Practical considerations for implementation are also discussed.
2 citations
01 Mar 1974
TL;DR: In this article, the singular value decomposition of a matrix A is updated by deleting or adding a row or a column from the matrix A whose singular value is known, and the problem of recalculating some or all of the singular values of A is considered.
Abstract: The linear least squares problem of minimizing ||Ax~ - b~||_(2) where A is an m X n matrix, m >= n, may be solved using the singular value decomposition in approximately 2mn^(3) + 4n^(3) multiplications. In this paper the problem of solving ||A'x~ - b~||_(2) is considered where A' results from deleting or adding a column to A. This might occur when a change is made in the model of a process. Instead of computing the singular value decomposition of A' from scratch, the singular value decomposition of A is updated. Since the updating require about 6n^(3) multiplications the algorithms are useful when m >> n. The problem of recalculation some or all of the singular values of a matrix A', which is obtained by deleting or adding a row or a column from a matrix A, whose singular value decomposition is known, is also studied.
1 citations