Moving Average Spatial Regression Object

DESCRIPTION:

An object of class cov.family containing functions for fitting conditional 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 MA object. The discussion here centers around the covariance matrix model which the MA 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 moving average spatial regression model assumes that

S = (I + rho N) W ((I + rho N)^t 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. 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])) W (sum (I + rho[i] N[i])^t 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 MA object assumes that a profile likelihood for rho is fit.

The MA model can be expressed as a moving average model for the spatial parameters as follows:

y = X*beta + rho*N*W^(1/2)*epsilon + 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 MA 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 weight 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 too large (over 150), then sparse matrix routines by Kundert (1988) are used to compute the determinant of S. Unlike the SAR, CAR, and other spatial regression models, for moving average models the covariance matrix is not parameterized in terms of its inverse. The matrix N will usually be sparse, however, so S^(-1) z can be efficiently computed using the Kundert (1988) algorithms. See routine spatial.cg.solve for details.

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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