logdetAinvUpdate {estimateW}

Efficient update of the log-determinant and the matrix inverse

Description

While updating the elements of the spatial weight matrix in SAR and SDM type spatial models in a MCMC sampler, the log-determinant term has to be regularly updated, too. When the binary elements of the adjacency matrix are treated unknown, the Matrix Determinant Lemma and the Sherman-Morrison formula are used for computationally efficient updates.

Usage

logdetAinvUpdate(ch_ind, diff, AI, logdet)

Arguments

ch_ind

vector of non-negative integers, between 1 and n. Denotes which rows of A should be updated.

diff

a numeric length(ch_ind) by n matrix. This value will be added to the corresponding rows of A.

AI

numeric n by n matrix that is the inverse of A = (I_n - \rho W). This inverse will be updated using the Sherman-Morrison formula.

logdet

single number that is the log-determinant of the matrix A. This log-determinant will be updated through the Matrix Determinant Lemma.

Details

Let A = (I_n - \rho W) be an invertible n by n matrix. v is an n by 1 column vector of real numbers and u is a binary vector containing a single one and zeros otherwise. Then the Matrix Determinant Lemma states that:

A + uv' = (1 + v'A^{-1}u)det(A)

.

This provides an update to the determinant, but the inverse of A has to be updated as well. The Sherman-Morrison formula proves useful:

(A + uv')^{-1} = A^{-1} \frac{A^{-1}uv'A^{-1}}{1 + v'A^{-1}u}

.

Using these two formulas, an efficient update of the spatial projection matrix determinant can be achieved.

Value

A list containing the updated n by n matrix A^{-1}, as well as the updated log determinant of A

References

Sherman, J., and Morrison, W. J. (1950) Adjustment of an inverse matrix corresponding to a change in one element of a given matrix. The Annals of Mathematical Statistics, 21(1), 124-127.

Harville, D. A. (1998) Matrix algebra from a statistician's perspective. Taylor & Francis.