Computationally efficient cholesky factorization of a covariance matrix with block toeplitz structure

Many statistical procedures require some form of matrix factorization, and so can be computationally costly to implement for problems of large dimensions. However, it is often possible to reduce significantly overall computational costs by exploiting the structure of the matrix at hand. In this respect, a general analysis of the link between the amount of structure in a matrix and factorization costs has been recently reported in the literature and a whole class of fast factorization algorithms has been derived. In this paper, these recent results are invoked to derive a fast algorithm for the Choleskyfactorization of a covariance matrix associated with a stationary spatial random field generated over a regular 2-D grid. The contribution of this paper is to derive explicitly the algorithm and show that it will run to completion in exact arithmetic. The factorization algorithm is particularly suited to parallel computation and remarkably simple to implement. In addition, with only minor modifications it ex...

[1]  M. W. Davis,et al.  Production of conditional simulations via the LU triangular decomposition of the covariance matrix , 1987, Mathematical Geology.

[2]  S. Alexander,et al.  Analysis of a recursive least squares hyperbolic rotation algorithm for signal processing , 1988 .

[3]  R. Döhler,et al.  Squared Givens Rotation , 1991 .

[4]  M. R. Osborne,et al.  Estimation of covariance parameters in kriging via restricted maximum likelihood , 1991 .

[5]  Ilse C. F. Ipsen,et al.  Parallel solution of symmetric positive definite systems with hyperbolic rotations , 1986 .

[6]  G. Cybenko,et al.  Hyperbolic Householder algorithms for factoring structured matrices , 1990 .

[7]  George Cybenko,et al.  The Numerical Stability of the Levinson-Durbin Algorithm for Toeplitz Systems of Equations , 1980 .

[8]  B. M. Davis.  Uses and abuses of cross-validation in geostatistics , 1987 .

[9]  G. Stewart.  Perturbation Bounds for the $QR$ Factorization of a Matrix , 1977 .

[10]  C. Dietrich.  Modality of the restricted likelihood for spatial Gaussian random fields , 1991 .

[11]  G. Stewart.  Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems , 1973 .

[12]  T. Kailath,et al.  Fast Parallel Algorithms for QR and Triangular Factorization , 1987 .

[13]  G. Stewart.  Introduction to matrix computations , 1973 .

[14]  Kanti V. Mardia,et al.  On multimodality of the likelihood in the spatial linear model , 1989 .

[15]  Gene H. Golub,et al.  Matrix computations , 1983 .

[16]  T. Kailath,et al.  Triangular Factorization of Structured Hermitian Matrices , 1986 .

[17]  Y. Kamp,et al.  Schur Parametrization of Positive Definite Block-Toeplitz Systems , 1979 .

[18]  James R. Bunch,et al.  Stability of Methods for Solving Toeplitz Systems of Equations , 1985 .

[19]  K. Mardia,et al.  Maximum likelihood estimation of models for residual covariance in spatial regression , 1984 .

[20]  Dale L. Zimmerman,et al.  Computationally exploitable structure of covariance matrices and generalized convariance matrices in spatial models , 1989 .

[21]  T. Kailath.  A Theorem of I. Schur and Its Impact on Modern Signal Processing , 1986 .

[22]  W. E. Gentleman.  Least Squares Computations by Givens Transformations Without Square Roots , 1973 .

[23]  S. Hammarling.  A Note on Modifications to the Givens Plane Rotation , 1974 .