Gaussian process regression: new hyperparameter estimation algorithm for more reliable prediction
Résumé
In the framework of the emulation of CPU-time expensive numerical simulators with Gaussian process (GP) regression, we propose in this work a new algorithm for the estimation of GP covariance parameters, referred to as GP hyperparameters. The objective is twofold: to ensure a GP as predictive as possible w.r.t. to the output of interest, but also with reliable prediction intervals, i.e. representative of GP prediction error. To achieve this, we propose a new constrained multi-objective algorithm for the hyperparameter estimation. It jointly maximizes the likelihood of the observations as well as the empirical coverage function of GP prediction intervals, under the constraint of not degrading the GP predictivity. Cross validation techniques and advantageous update GP formulas are notably used. The benefit brought by the algorithm compared to standard algorithms is illustrated on a large benchmark of analytical functions (with dimensions from 1 to 20 input variables). Different designs of experiments and different covariance models are considered. An application on a real data test case modeling an aquatic ecosystem is also proposed: GP metamodeling within a log-kriging approach is used to predict the biomass of a species at a given time. The multi-objective algorithm performs better than standard algorithms and this particular metamodeling framework shows the crucial interest of well-estimated and reliable prediction variances in GP regression.
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