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Given a reproducing kernel Hilbert space (H, ., . ) of real-valued functions and a suitable measure µ over the source space D ⊂ R, we decompose H as the sum of a subspace of centered functions for µ and its orthogonal in H. This decomposition leads to a special case of ANOVA kernels, for which the functional ANOVA representation of the best predictor can be elegantly derived, either in an interpolation or regularization framework. The proposed kernels appear to be particularly convenient for analyzing the effect of each (group of) variable(s) and computing sensitivity indices without recursivity.

We study the class of Azéma-Yor processes defined from a general semimartingale with a continuous running maximum process. We show that they arise as unique strong solutions of the Bachelier stochastic differential equation which we prove is equivalent to the drawdown equation. Solutions of the latter have the drawdown property: they always stay above a given function of their past maximum. We then show that any process which satisfies the drawdown property is in fact an Azéma-Yor process. The proofs exploit group structure of the set of Azéma-Yor processes, indexed by functions, which we introduce.We investigate in detail Azéma-Yor martingales defined from a nonnegative local martingale converging to zero at infinity. We establish relations between average value at risk, drawdown function, Hardy-Littlewood transform and its inverse. In particular, we construct Azéma-Yor martingales with a given terminal law and this allows us to rediscover the Azéma-Yor solution to the Skorokhod embedding problem. Finally, we characterize Azéma-Yor martingales showing they are optimal relative to the concave ordering of terminal variables among martingales whose maximum dominates stochastically a given benchmark.

International audienceKriging models have been widely used in computer experiments for the analysis of time-consuming computer codes. Based on kernels, they are flexible and can be tuned to many situations. In this paper, we construct kernels that reproduce the computer code complexity by mimicking its interaction structure. While the standard tensor-product kernel implicitly assumes that all interactions are active, the new kernels are suited for a general interaction structure, and will take advantage of the absence of interaction between some inputs. The methodology is twofold. First, the interaction structure is estimated from the data, using a first initial standard Kriging model, and represented by a so-called FANOVA graph. New FANOVA-based sensitivity indices are introduced to detect active interactions. Then this graph is used to derive the form of the kernel, and the corresponding Kriging model is estimated by maximum likelihood. The performance of the overall procedure is illustrated by several 3-dimensional and 6-dimensional simulated and real examples. A substantial improvement is observed when the computer code has a relatively high level of complexity

Équation de Burgers avec conditions initiales à accroissements indépendants et homogènes

Our goal in the present article to give an insight on some important questions to be asked when choosing a Kriging model for the analysis of numerical experiments. We are especially concerned about the cases where the size of the design of experiments is relatively small to the algebraic dimension of the inputs. We first fix the notations and recall some basic properties of Kriging. Then we expose two experimental studies on subjects that are often skipped in the field of computer simulation analysis: the lack of reliability of likelihood maximization with few data and the consequences of a trend misspecification. We finally propose an example from a porous media application, with the introduction of an original Kriging method in which a non‐linear additive model is used as an external trend. Copyright © 2009 John Wiley & Sons, Ltd.

Kriging‐based exploration strategies often rely on a single Kriging model whose parametric covariance kernel is selected a priori or on the basis of an initial data set. As choosing an unadapted kernel can radically harm the results, we wish to reduce the risk of model misspecification. Here, we consider the simultaneous use of multiple kernels within Kriging. We give the equations of discrete mixtures of Kriging and derive a multikernel version of the expected improvement optimization criterion. We finally provide an illustration of the efficient global optimization algorithm with mixed exponential and Gaussian kernels, where the parameters are estimated by maximum likelihood and the mixing weights are likelihood ratios. Copyright © 2008 John Wiley & Sons, Ltd.

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