Maximum likelihood estimates
mle.Rd
Maximum likelihood estimates of a simple spatial model
Usage
neg2loglikelihood.spam(y, X, distmat, Covariance,
beta, theta, Rstruct = NULL, cov.args = NULL, ...)
neg2loglikelihood(y, X, distmat, Covariance,
beta, theta, cov.args = NULL, ...)
neg2loglikelihood.nomean(y, distmat, Covariance,
theta, cov.args = NULL, ...)
mle.spam(y, X, distmat, Covariance,
beta0, theta0, thetalower, thetaupper, optim.control=NULL,
Rstruct = NULL, hessian = FALSE, cov.args = NULL, ...)
mle(y, X, distmat, Covariance,
beta0, theta0, thetalower, thetaupper, optim.control=NULL,
hessian = FALSE, cov.args = NULL, ...)
mle.nomean.spam(y, distmat, Covariance,
theta0, thetalower, thetaupper, optim.control=NULL,
Rstruct = NULL, hessian = FALSE, cov.args = NULL, ...)
mle.nomean(y, distmat, Covariance,
theta0, thetalower, thetaupper, optim.control=NULL,
hessian = FALSE, cov.args = NULL, ...)
Arguments
- y
data vector of length n.
- X
the design matrix of dimension n x p.
- distmat
a distance matrix. Usually the result of a call to
nearest.dist
.- Covariance
function defining the covariance. See example.
- beta
parameters of the trend (fixed effects).
- theta
parameters of the covariance structure.
- Rstruct
the Cholesky structure of the covariance matrix.
- beta0,theta0
inital values.
- thetalower,thetaupper
lower and upper bounds of the parameter
theta
.- optim.control
arguments passed to
optim
.- hessian
Logical. Should a numerically differentiated Hessian matrix be returned?
- cov.args
additional arguments passed to
Covariance
.- ...
additional arguments passed to
chol
.
Details
We provide functions to calculate the negative-2-log-likelihood and maximum likelihood estimates for the model
y ~ N_n( X beta, Sigma(h;theta) )
in the case of a sparse or ordinary covariance matrices.
In the case of the *.spam
versions, the covariance function has
to return a spam
object. In the other case, the methods are
correctly overloaded and work either way, slightly slower than the
*.spam
counterparts though.
When working on the sphere, the distance matrix has to be transformed by
h -> R / 2 sin(h/2)
where R is the radius of the sphere.
The covariance function requires that the first argument is the distance
matrix and the second the parameters. One can image cases in which the
covariance function does not take the entire distance matrix but only
some partial information thereof. (An example is the use of a kronecker
type covariance structure.) In case of a sparse covariance construction
where the argument Rstruct
is not given, the first parameter
element needs to be the range parameter. (This results from the fact,
that a sparse structure is constructed that is independent of the
parameter values to exploit the fast Choleski decomposition.)
In the zero-mean case, the neg2loglikelihood
is calculated by setting
the parameters X
or beta
to zero.
Examples
# True parameter values:
truebeta <- c(1,2,.2) # beta = (intercept, linear in x, linear in y)
truetheta <- c(.5,2,.02) # theta = (range, sill, nugget)
# We now define a grid, distance matrix, and a sample:
x <- seq(0,1,l=5)
locs <- expand.grid( x, x)
X <- as.matrix( cbind(1,locs)) # design matrix
distmat <- nearest.dist( locs, upper=NULL) # distance matrix
Sigma <- cov.sph( distmat, truetheta) # true covariance matrix
set.seed(15)
y <- c(rmvnorm.spam(1,X %*% truebeta,Sigma)) # construct sample
# Here is the negative 2 log likelihood:
neg2loglikelihood.spam( y, X, distmat, cov.sph,
truebeta, truetheta)
#> [1] 79.26271
# We pass now to the mle:
res <- mle.spam(y, X, distmat, cov.sph,
truebeta, truetheta,thetalower=c(0,0,0),thetaupper=c(1,Inf,Inf))
# Similar parameter estimates here, of course:
mle.nomean.spam(y-X%*%res$par[1:3], distmat, cov.sph,
truetheta, thetalower=c(0,0,0), thetaupper=c(1,Inf,Inf))
#> $par
#> [1] 0.8057742 1.6038407 0.1376179
#>
#> $value
#> [1] 73.76311
#>
#> $counts
#> function gradient
#> 15 15
#>
#> $convergence
#> [1] 0
#>
#> $message
#> [1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"
#>