vecchia_profbeta_loglik_grad_info {GpGp}

Vecchia's loglikelihood, gradient, and Fisher information

Description

This function returns Vecchia's (1988) approximation to the Gaussian loglikelihood, profiling out the regression coefficients, and returning the gradient and Fisher information. Vecchia's approximation modifies the ordered conditional specification of the joint density; rather than each term in the product conditioning on all previous observations, each term conditions on a small subset of previous observations.

Usage

vecchia_profbeta_loglik_grad_info(covparms, covfun_name, y, X, locs, NNarray)

Arguments

covparms

A vector of covariance parameters appropriate for the specified covariance function

covfun_name

See GpGp for information about covariance functions.

y

vector of response values

X

Design matrix of covariates. Row i of X contains the covariates for the observation at row i of locs.

locs

matrix of locations. Row i of locs specifies the location of element i of y, and so the length of y should equal the number of rows of locs.

NNarray

A matrix of indices, usually the output from find_ordered_nn. Row i contains the indices of the observations that observation i conditions on. By convention, the first element of row i is i.

Value

A list containing

• loglik: the loglikelihood

• grad: gradient with respect to covariance parameters

• info: Fisher information for covariance parameters

• betahat: profile likelihood estimate of regression coefs

• betainfo: information matrix for betahat.

The covariance matrix for $betahat is the inverse of $betainfo.

Examples

n1 <- 20
n2 <- 20
n <- n1*n2
locs <- as.matrix( expand.grid( (1:n1)/n1, (1:n2)/n2 ) )
X <- cbind(rep(1,n),locs[,2])
covparms <- c(2, 0.2, 0.75, 0)
y <- X %*% c(1,2) + fast_Gp_sim(covparms, "matern_isotropic", locs, 50 )
ord <- order_maxmin(locs)
NNarray <- find_ordered_nn(locs,20)
#loglik <- vecchia_profbeta_loglik_grad_info( covparms, "matern_isotropic", 
#    y, X, locs, NNarray )