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.