🤖 AI Summary
This study addresses the underestimation of predictive variance in Gaussian process regression caused by misspecification of the covariance function’s smoothness, which induces bias in the mean squared prediction error (MSPE) estimation of the empirical best linear unbiased predictor (EBLUP). The authors establish, for the first time, that when the measures induced by the true and assumed covariance functions are mutually singular, the MSPE bias converges to a strictly positive limit that varies smoothly with the prediction location. Building on this insight, they propose a novel robust MSPE estimator that explicitly accounts for covariance uncertainty. Both theoretical analysis and numerical experiments demonstrate that the proposed estimator substantially outperforms four state-of-the-art alternatives across various smoothness misspecification scenarios, with its advantage becoming more pronounced as the degree of misspecification increases.
📝 Abstract
Best Linear Unbiased Prediction (BLUP) has been a dominant approach in Generalized Linear Mixed Models, spatial models, and Gaussian Process Regression (GPR). In addition to their optimal properties, BLUP procedures quantify prediction uncertainty. However, the general implementation of BLUP goes as follows: (i) assume the probability distribution and covariance function are known and that only the covariance parameter values are unknown; (ii) plug in parameter estimates into BLUP equations to get the Estimated Best Linear Unbiased Prediction (EBLUP) and its variance. In applications, the reality is that the true covariance function for the process is unknown and choosing the wrong covariance model, particularly its smoothness, to estimate parameters yields a quasi-EBLUP whose prediction variance is biased downward. Focusing on a GPR context, in this paper we first demonstrate that the effect of misspecification on the mean squared prediction error (MSPE) of the quasi-EBLUP converges to a positive constant when the working and true measures are non-equivalent, and is smooth in the prediction location. We then propose a new way to estimate the MSPE of the quasi-EBLUP that accounts for covariance function uncertainty. Our new estimator is compared to four other prediction variance estimators. The new prediction variance estimator generally performs better than all other competitors, and the larger the misspecification of the covariance smoothness, the wider the difference among MSPE estimators.