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R: Generalized Least Squares estimation with a given covariance...

gls-methods

R Documentation

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 of length p containing the estimated coefficients if `beta = NULL`, or the known coefficients beta either.
`L`	The (lower) Cholesky root matrix L of the covariance matrix C. This matrix has n rows and n columns and C = L %% t(L)*.
`eStar`	Vector of length n: eStar = inv(L) %% (y - X betaHat)*.
`Fstar`	Matrix (n, p): FStar = inv(L) %% F*.
`sseStar`	Sum of squared errors: t(eStar) %% eStar*.
`RStar`	The 'R' upper triangular (p, p) matrix in the QR decomposition of `FStar`: FStar = Q %% RStar*.

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)