# GPpenalty Implements maximum likelihood estimation for Gaussian processes, supporting both isotropic and separable models with predictive capabilities. Includes penalized likelihood estimation, with cross-validation guided by decorrelated prediction error (DPE) metric. DPE metric, motivated by Mahalanobis distance, serves as evaluation criteria that accounts for predictive uncertainty in tuning parameter selection. Designed specifically for small datasets. ## Functions * `mle_gp`: The function computes maximum likelihood estimates for the lengthscale, scale, mu, and nugget (g) parameters using `optim`, with options to fix or assume zero for certain parameters. * `predict_gp`: Computes the posterior mean and covariance matrix for a given set of input locations based on a fitted model. * `gp_cv`: Performs cross-validation to select an optimal tuning parameter for penalized MLE of the lengthscale parameter in Gaussian processes. * `mle_penalty`: Computes penalized maximum likelihood estimates for the lengthscale parameter using `optim`. * `score`: Calculates a score value. Higher score values indicate better fits. * `dpe`: Calculates a decorrelated prediction error value. Lower dpe values indicate better fits. * `kernel`: Compute the squared exponential kernel defined as $k = \exp(-\theta (x - x')^2) + g$ , where $\theta$ is the lengthscale parameter and $g$ is a jitter term. Both isotropic and separable kernels are supported. ## Examples ```r #### define function ### f_x <- function(x) { return(sin(2*pi*x) + x^2) } ### x and y ### x <- runif(8, min=0, max=1) y <- f_x(x) x.test <- runif(100, min=0, max=1) y.test <- f_x(x.test) ### no penalization ### # fit fit <- mle_gp(y, x) # prediction pred <- predict_gp(fit, x.test) # obtain kernel function cov_function <- kernel(x1=x, theta=fit$theta) # evaluate the predictive performance with score score_value <- score(y.test, pred$mup, pred$Sigmap) ### penalization ### # leave-one-out cross validation loocv.lambda <- gp_cv(y, x) # fit fit.loocv <- mle_penalty(loocv.lambda) # prediction pred.loocv <- predict_gp(fit.loocv, x.test) # k-fold cross validation with the dpe metric kfold.dpe <- gp_cv(y, x, k=4) # fit fit.kfold.dpe <- mle_penalty(kfold.dpe) # prediction pred.kfold.dpe <- predict_gp(fit.kfold.dpe, x.test) # k-fold cross validation with the mse metric kfold.mse <- gp_cv(y, x, k=4, metric="mse") # fit fit.kfold.mse <- mle_penalty(kfold.mse) # prediction pred.kfold.mse <- predict_gp(fit.kfold.mse, x.test) ```

GPpenalty-package