Gaussian process regression on nested spaces

Metamodels are widely used in industry to predict the output of an expensive computer code. As industrial computer codes involve a large amount of input variables, creating directly one big metamodel depending on the whole set of inputs may be a very challenging problem. Industrialists choose instead to proceed sequentially. They build metamodels depending on nested sets of variables (the variables that are set aside are fixed to nominal values), i.e. the dimension of the input space is progressively increased. However, at each step, the previous piece of information is lost as a new Design of Experiment (DoE) is generated to learn the new metamodel. In this paper, an alternative approach will be introduced, based on all the DoEs rather than just the last one. This metamodel uses Gaussian process regression and is called seqGPR (sequential Gaussian process regression). At each step n, the output is supposed to be the realization of the sum of two independent Gaussian processes Y_{n-1}+Z_n. The first one Y_{n-1} models the output at step n-1. It is defined on the input space of step n-1 which is a subspace of the one of step n. The second Gaussian process Z_{n} is a correction term defined on the input space of step n. It represents the additional information provided by the newly released variables. Z_{n} has the particularity of being null on the subspace where Y_{n-1} is defined so that there is a coherence between the steps. Firstly, some candidate Gaussian processes for (Z_n)_{n \geq 2} are suggested, which have the property of being null on an infinite continuous set of points. Then, an EM (Expectation-Maximization) algorithm is implemented to estimate the parameters of the processes. Finally, the metamodel seqGPR is compared to a classic kriging metamodel where the output is assumed to be the realization of one second order stationary Gaussian process. The comparison is made on two analytic examples, a first one with two steps, up to dimension 4, and a second one with three steps, up to dimension 15. The introduced methodology is also tested on an industrial example which goes from dimension 11 to dimension 15. In all these test cases, seqGPR performs better than, or at least as well as kriging