Simultaneous Spatial Autoregression Object

DESCRIPTION:

An object of class cov.family containing functions for fitting simultaneous autoregression models when used as input to the spatial linear models function, slm. See cov.family.object for a discussion of the attributes contained in the SAR object. The discussion here centers around the covariance matrix model which the SAR object supports.

DETAILS:

Let N denote a neighbor matrix obtained from an object of class "spatial.neighbor" , and let S denote the covariance matrix of a vector y of spatially correlated dependent variables. Finally, let W denote a diagonal matrix of weights. Then a simultaneous autoregression model assumes that

S = ((I-rho N)^t W^(-1) (I - rho N))^(-1) sigma^2

where rho is a scalar parameter to be estimated, and sigma is a scale parameter which is also to be estimated. Here ^ is the exponentiation operator, and ^t denotes matrix transpose. This model for the covariance matrix can be generalized to multiple matrices N[i] using multiple parameters rho[i] as follows:

S = (sum (I - rho[i] N[i]))^t W^(-1) (sum (I - rho[i] N[i]))^(-1) sigma^2

where the N[i] are specified through component matrix of the "spatial.neighbor" object (see routine spatial.neighbor). The "regression" aspect of a spatial regression fits the multivariate normal mean vector

mu = E(y|x) = x*beta

for unknown parameters beta. The multivariate normal likelihood is expressed in terms of the unknowns rho, sigma, and beta. The SAR object assumes that a profile likelihood for rho is fit.

The SAR model can be expressed as an autoregressive model for the spatial parameters as follows:

y = X*beta + rho*N(y - X*beta) + W^(1/2) * epsilon

This allows one to decompose the sum of squares in y into three components (see Haining, 1990, page 258): 1) the trend, X*beta; 2) the noise, W^(1/2) *epsilon = (I - rho*N)(y - X*beta); and 3) the signal, y - X*beta - W^(1/2) *epsilon. Function residual.fun of the SAR object computes epsilon, the standardized residuals, and routine slm returns these in component residuals. The estimated trend, X*beta, is returned by routine slm as the fitted values.

Two functions are required to compute the profile likelihood: 1) a function for computing the determinant `|S|', and 2) a function for computing the vector S^(-1) z for arbitrary vector z. When the single neighbor matrix N is symmetric, the determinant can be expressed and efficiently computed as a function of the eigenvalues of N . If N is not symmetric, or if the dimension of N is large (over 150), then sparse matrix routines by Kundert (1988) are used to compute the determinant of S. Because the covariance matrix is parameterized in terms of its inverse, the computation of S^(-1) z is particularly simple and is carried using (sparse) matrix multiplication. See routine spatial.multiply.

REFERENCES:

Haining, R. (1990). Spatial Data Analysis in the Social and Environmental Sciences. Cambridge University Press. Cambridge.

Kundert, Kenneth S. and Sangiovanni-Vincentelli, Alberto (1988). A Sparse Linear Equation Solver. Department of EE and CS, University of California, Berkeley.

SEE ALSO:

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