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
CAR object. The
discussion here centers around the covariance matrix model which the
CAR
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 conditional
autoregression model assumes that
S = (I - rho N)^(-1)*W*sigma^2
where
rho and and
sigma are scale parameters 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]))^(-1)*W*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
CAR
object assumes that a profile likelihood for
rho is fit.
The CAR 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
For the CAR model, the residuals,
epsilon, defined here are not
independent. The formula for
y 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
CAR object computes
epsilon, the
standardized residuals, and routine
slm returns these in component
residuals
. The estimated trend, 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 simple and is carried out 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.