Abstract
In the study of natural and artificial complex systems, responses that are not completely determined by the considered decision variables are commonly modelled probabilistically, resulting in response distributions varying across decision space. We consider cases where the spatial variation of these response distributions does not only concern their mean and/or variance but also other features including for instance shape or uni-modality versus multi-modality. Our contributions build upon a non-parametric Bayesian approach to modelling the thereby induced fields of probability distributions, and in particular to a spatial extension of the logistic Gaussian model. The considered models deliver probabilistic predictions of response distributions at candidate points, allowing for instance to perform (approximate) posterior simulations of probability density functions, to jointly predict multiple moments and other functionals of target distributions, as well as to quantify the impact of collecting new samples on the state of knowledge of the distribution field of interest. In particular, we introduce adaptive sampling strategies leveraging the potential of the considered random distribution field models to guide system evaluations in a goal-oriented way, with a view towards parsimoniously addressing calibration and related problems from non-linear (stochastic) inversion and global optimisation.
Highlights
Many problems in science and engineering boil down to studying the effect on a response of interest of varying some decision or control variables x
We repeat 24 independent instances of the optimisation process for each strategy and each application and compare the performances in term of optimality gap. This approach is favoured due to the relatively high cost of one evaluation of the EQI criterion for Spatial Logistic Gaussian process (SLGP), but we expect it to be detrimental to our Gaussian Processes (GP)-based competitors, as GPs would benefit more from having scattered observations rather than batches scattered over different points
We demonstrated how a spatial generalization of the Logistic Gaussian Process model can be used for sample-based modelling of distribution fields, and how the resulting probabilistic predictions of distribution fields can be leveraged for moderate-dimensional stochastic optimization problems under unknown heterogeneous noise distributions
Summary
Many problems in science and engineering boil down to studying the effect on a response of interest of varying some decision or control variables x. We consider the situation where responses follow probability distributions μx indexed by decision variables x ∈ D and with a common support I. Existing approaches typically assume Gaussian response distributions μx Another branch of study pertaining to geostatistics and that does not rely on Gaussian or specific distributional assumptions is the so-called distributional Kriging (Aitchison [1]; Egozcue et al [14]; Talska et al [44]) but such approaches are ill-suited in the case of moderate sample size heterogeneously scattered across space. Our approach is adapted to the case of moderate and heterogeneously scattered across space sample size It delivers a probabilistic prediction of the field of probability distributions which allows us to derive sampling acquisition schemes. The main contribution of this paper lies in the exploitation of a spatial extension of the logistic Gaussian process model for the sequential design of stochastic simulations towards optimisation and inversion
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