gls-methods {kergp}

Generalized Least Squares Estimation with a Given Covariance Kernel

Description

Generalized Least Squares (GLS) estimation for a linear model with a covariance given by the covariance kernel object. The method gives auxiliary variables as needed in many algebraic computations.

Usage

## S4 method for signature 'covAll'
gls(object,
    y, X, F = NULL, varNoise = NULL, 
    beta = NULL, checkNames = TRUE,
    ...)

Arguments

object

An object with "covAll" class.

y

The response vector with length n.

X

The input (or spatial design) matrix with n rows and d columns. This matrix must be compatible with the given covariance object, see checkX,covAll,matrix-method.

F

A trend design matrix with n rows and p columns. When F is NULL no trend is used and the response y is simply a realization of a centered Gaussian Process with covariance kernel given by object.

varNoise

A known noise variance. When provided, must be a positive numeric value.

beta

A known vector of trend parameters. Default is NULL indicating that the trend parameters must be estimated.

checkNames

Logical. If TRUE (default), check the compatibility of X with object, see checkX.

...

not used yet.

Details

There are two options: for unknown trend, this is the usual GLS estimation with given covariance kernel; for a known trend, it returns the corresponding auxiliary variables (see value below).

Value

A list with several elements.

betaHat

Vector \widehat{\boldsymbol{\beta}} of length p containing the estimated coefficients if beta = NULL, or the known coefficients \boldsymbol{\beta} either.

L

The (lower) Cholesky root matrix \mathbf{L} of the covariance matrix \mathbf{C}. This matrix has n rows and n columns and \mathbf{C} = \mathbf{L} \mathbf{L}^\top.

eStar

Vector of length n: \mathbf{e}^\star = \mathbf{L}^{-1}(\mathbf{y} - \mathbf{X} \widehat{\boldsymbol{\beta}}).

Fstar

Matrix n \times p: \mathbf{F}^\star := \mathbf{L}^{-1}\mathbf{F}.

sseStar

Sum of squared errors: {\mathbf{e}^\star}^\top\mathbf{e}^\star.

RStar

The 'R' upper triangular p \times p matrix in the QR decomposition of FStar: \mathbf{F}^\star = \mathbf{Q}\mathbf{R}^\star.

All objects having length p or having one of their dimension equal to p will be NULL when F is NULL, meaning that p = 0.

Author(s)

Y. Deville, O. Roustant

References

Kenneth Lange (2010), Numerical Analysis for Statisticians 2nd ed. pp. 102-103. Springer-Verlag,

Examples

## a possible 'covTS'
myCov <- covTS(inputs = c("Temp", "Humid"),
               kernel = "k1Matern5_2",
               dep = c(range = "input"),
               value = c(range = 0.4))
d <- myCov@d; n <- 100;
X <- matrix(runif(n*d), nrow = n, ncol = d)
colnames(X) <- inputNames(myCov)
## generate the 'GP part'  
C <- covMat(myCov, X = X)
L <- t(chol(C))
zeta <- L %*% rnorm(n)
## trend matrix 'F' for Ordinary Kriging
F <- matrix(1, nrow = n, ncol = 1)
varNoise <- 0.5
epsilon <- rnorm(n, sd = sqrt(varNoise))
beta <- 10
y <- F %*% beta + zeta + epsilon
fit <- gls(myCov, X = X, y = y, F = F, varNoise = varNoise)