Designs for Spatial Random Fields

DESIRE addresses the problem of defining efficient designs (location of observations points, sensors, etc.) for observation of spatial fields within a statistical framework. While our objectives cover numerous industrial and scientific applications, the work of the project stays at a fundamental level, with the aim of identifying appropriate modeling frameworks for distinct specific instances of the spatial observation problem, as well as expedite algorithms to compute associated efficient designs. To demonstrate the relevance of the results obtained, and in cooperation with external collaborators of the institutions involved, the project also compares the techniques developed to existing state-of-the-art approaches in a small number of real problems. When setting up an experiment to observe a spatial phenomenon, the common practice is to use space-filling designs that spread out the inputs more or less uniformly across the available space. This has become a standard approach in particular in computer experiments, where physical (costly) experiments are replaced by (sometimes time-consuming but cheaper) numerical simulations. In industrial contexts, this is known as virtual prototyping. The same sampling approach is also currently used for the deployment of the majority of existing sensor networks. At the same time, the uncertainty due to the estimation of the stochastic characteristics of the random field is usually ignored. The project aims at (i) taking this uncertainty into account in the definition of design objectives related to the precision of prediction/interpolation, (ii) defining simplified design criteria that can be efficiently optimized through specific algorithms, (iii) constructing optimal designs sequentially, taking dynamical constraints into account in order to consider applications to the deployment of mobile sensors. This last point forms the ultimate objective of the project. It corresponds to the sequential construction of optimal placements of mobile sensors, which forms a (stochastic) control problem, and can be reformulated as an attempt to fill the gap between optimal control and statistical inference. In terms of modeling, the field will be considered as being the superposition of a deterministic component and a realization of a (spatially correlated) stochastic process. Compared to more classical (parametric) approaches, it has the important added values of flexibility and reduced sensitivity with respect to the choice of the parametric families on which the field is described. The prediction at unsampled locations is then obtained (explicitly) by kriging (Krige, 1951), a method that is now rather standard in spatial statistics, including applications to computer (i.e., simulated) experiments since the pioneering work of Sacks et al. (1989). Although this prediction technique is now rather usual, we intend to improve current results in several directions: (i) a more precise accounting of the added uncertainty on predictions due to estimation of the stochastic characteristics of the field from the same data set as that used to construct predictions, (ii) the extension of this corrected "prediction of uncertainty on predictions" to localized objectives, such as the construction of a level set (input space) for a response of interest (observation space) or the optimization of a response (to be contrasted with global objectives, where one aims at constructing precise predictions of the response over the whole input space), (iii) the investigation of an alternative modeling framework for extended dependence structures based on copulas.

Efficient observation of spatial fields has applications in many distinct areas. Two equally important and radically different examples concern the optimization of industrial processes, where it is the geometry of the response surface with respect to a set of design variables that is of interest, and the observation or prediction of distinctive features of environmental fields, such as fronts, local maxima, etc., on the other hand. In both cases, the design space (the set of possible values of the controlled explanatory variables) is huge and the cost associated to each collected observation (the execution of numerical computer models in the first case and the placement of sensing equipment in a given site in the second one) large. The construction of adaptive experimental design strategies, that incorporate acquired information as soon as it is observed and avoid redundancy by directing future observations to the most relevant areas of the design space, is thus especially important. This requires the construction of criteria for measuring the precision of field reconstruction and the definition of methodologies for optimizing those criteria with respect to design variables. It is customary to perform field reconstruction via kriging prediction and to measure precision through Mean-Squared Prediction Error (MSPE). (i) Accounting for uncertainty on statistical characteristics of the random field (on parameters of its covariance function) then requires the introduction of an additional term in the MSPE criterion and the development of specific strategies for design optimization. Our approach is derived from an equivalence property in the more usual context of parametric models with independent errors, which states that optimal designs for parameter estimation are also optimal for prediction (Kiefer-Wolfowitz equivalence theorem). (ii) Whereas the MSPE integrated over a given domain of interest is an appealing global measure of precision, it is scarcely used due to its high computational cost. By a careful decomposition of the calculations, we manage to perform most calculations off-line, independently of the design, and then to optimize designs at reasonable cost. Alternatively, a spectral (Karhunen-Loève) expansion allows us to use all the machinery of convex optimization, classical in approximate-design theory, for the construction of designs minimizing the MSPE. Main resultsA design strategy accounting for prior uncertainty on statistical characteristics of the random field. Reduction of the computation cost of the evaluation of the Integrated Mean-Squared Prediction Error which allows design optimization; construction of Bayesian linear models alternative to random fields which allow optimization via convex programming.Multivariate dependencies that may differ from those in usual Gaussian processes can be flexibly handled by employing copula modeling and respective designs. Construction of efficient screening designs.Measures of spatial dispersion (of design points) based on simplicial volumes.A Bayesian local kriging approach to face non-stationarity issues (see the illustration below).A new computational technique (ABCD) well suited for a variety of complex and otherwise untreatable design problems.