# Generalized Least Squares Estimation with a Given Covariance Kernel 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. methods ## 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). ```r ## 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. ## Returns A list with several elements. - **betaHat**: Vector character(0) 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 $p = 0$. ## References Kenneth Lange (2010), **Numerical Analysis for Statisticians** 2nd ed. pp. 102-103. Springer-Verlag, ## Author(s) Y. Deville, O. Roustant ## Examples ```r ## 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) ```

gls-methods function