We study spatial sampling design for prediction of stationary isotropic Gaussian processes with estimated parameters of the covariance function. The key issue is how to incorporate the parameter uncertainty into design criteria to correctly represent the uncertainty in prediction. Several possible design criteria are discussed that incorporate the parameter uncertainty. A simulated annealing algorithm is employed to search for the optimal design of small sample size and a two-step algorithm is proposed for moderately large sample sizes. Simulation results are presented for the Matérn class of covariance functions. An example of redesigning the air monitoring network in EPA Region 5 for monitoring sulfur dioxide is given to illustrate the possible differences our proposed design criterion can make in practice.

We investigate a class of partially linear functional additive models (PLFAM) that predicts a scalar response by both parametric effects of a multivariate predictor and nonparametric effects of a multivariate functional predictor. We jointly model multiple functional predictors that are cross-correlated using multivariate functional principal component analysis (mFPCA), and model the nonparametric effects of the principal component scores as additive components in the PLFAM. To address the high-dimensional nature of functional data, we let the number of mFPCA components diverge to infinity with the sample size, and adopt the component selection and smoothing operator (COSSO) penalty to select relevant components and regularize the fitting. A fundamental difference between our framework and the existing high-dimensional additive models is that the mFPCA scores are estimated with error, and the magnitude of measurement error increases with the order of mFPCA. We establish the asymptotic convergence rate for our estimator, while allowing the number of components diverge. When the number of additive components is fixed, we also establish the asymptotic distribution for the partially linear coefficients. The practical performance of the proposed methods is illustrated via simulation studies and a crop yield prediction application. Supplementary materials for this article are available online.