Optimal Designs for Kriging Models with Multiple Responses
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Exact optimal designs for efficient prediction in simple and ordinary bivariate kriging models with one dimensional inputs are determined in this article. Two families of stationary covariance structures, namely the generalized Markov type and proportional covariances are investigated. These designs are found by minimizing the integrated and maximum prediction variance. For simple cokriging models with known covariance parameters, the equispaced design is shown to be optimal for both criterion functions. The more realistic scenario of unknown covariance parameters is addressed by assuming prior distributions on the parameter vector. The prior information is incorporated into both criterion functions by integrating it over the prior distribution, thus adopting a pseudo-Bayesian approach to the design problem. The equispaced design is proved to be optimal in the pseudo Bayesian sense for both criterion functions. For ordinary bivariate kriging models, it is shown theoretically that the equispaced design minimizes the maximum prediction variance, irrespective of known or unknown covariance parameters. The proposed work is motivated by a water quality study from a river in South India, where the interest is in designing an optimal river monitoring system. To the best of knowledge of the authors, these are the first explicit results on exact optimal designs for bivariate kriging models.
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