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.
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.
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.