Uncertainty and sensitivity analysis of functional risk curves based on Gaussian processes

A functional risk curve gives the probability of an undesirable event as a function of the value of a critical parameter of a considered physical system. In several applicative situations, this curve is built using phenomenological numerical models which simulate complex physical phenomena. To avoid cpu-time expensive numerical models, we propose to use Gaussian process regression to build functional risk curves. An algorithm is given to provide confidence bounds due to this approximation. Two methods of global sensitivity analysis of the models' random input parameters on the functional risk curve are also studied. In particular, the PLI sensitivity indices allow to understand the effect of misjudgment on the input parameters' probability density functions

Uncertainty and sensitivity analysis of functional risk curves based on Gaussian processes

International audienceA functional risk curve gives the probability of an undesirable event as a function of the value of a critical parameter of a considered physical system. In several applicative situations, this curve is built using phenomenological numerical models which simulate complex physical phenomena. To avoid cpu-time expensive numerical models, we propose to use Gaussian process regression to build functional risk curves. An algorithm is given to provide confidence bounds due to this approximation. Two methods of global sensitivity analysis of the models' random input parameters on the functional risk curve are also studied. In particular, the PLI sensitivity indices allow to understand the effect of misjudgment on the input parameters' probability density functions

Model selection for Gaussian Process regression: an application with highlights on the model variance validation

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Stochastic simulators based optimization by Gaussian process metamodels - Application to maintenance investments planning issues

International audienceThis paper deals with the construction of a metamodel (i.e. a simplified mathematical model) for a stochastic computer code (also called stochastic numerical model or stochastic simulator), where stochastic means that the code maps the realization of a random variable. The goal is to get, for a given model input, the main information about the output probability distribution by using this metamodel and without running the computer code. In practical applications, such a metamodel enables one to have estimations of every possible random variable properties, such as the expectation, the probability of exceeding a threshold or any quantile. The present work is concentrated on the emulation of the quantile function of the stochastic simulator by interpolating well chosen basis function and metamodeling their coefficients (using the Gaussian process metamodel). This quantile function metamodel is then used to treat a simple optimization strategy maintenance problem using a stochastic code, in order to optimize the quantile of an economic indicator. Using the Gaussian process framework, an adaptive design method (called QFEI) is defined by extending in our case the well known EGO algorithm. This allows to obtain an " optimal " solution using a small number of simulator runs

Estimation d'indices de sensibilité sur les quantiles

In the context of black-box numerical codes, it is relevant to use sensitivity analysis in order to assess the influence of each random input X over the output Y. Goal-oriented sensitivity analysis states that one must first focus on a certain probability feature θ(Y) from the distribution of Y (such as its mean, quantile, or a probability of failure etc...), which would be chosen regarding a relevant strategy. The wish is to evaluate the impact of each input over θ(Y). In order to get supplementary information about sensitivity, we set that θ(Y) is the α-level quantile of Y , where α ∈]0, 1[. Throughout some examples, it has been pointed out that in some cases quantile-oriented sensitivity indices can detect some influence that Sobol indices would not. Mainly, the influence over each level of quantile displays how an input distribution entirely propagates through the output. We establish further results for the quantile-oriented indices properties in order to justify their relevancy. The main contribution of this paper comes when a statistical estimator for this index is introduced

Estimation d'indices de sensibilité sur les quantiles

In the context of black-box numerical codes, it is relevant to use sensitivity analysis in order to assess the influence of each random input X over the output Y. Goal-oriented sensitivity analysis states that one must first focus on a certain probability feature θ(Y) from the distribution of Y (such as its mean, quantile, or a probability of failure etc...), which would be chosen regarding a relevant strategy. The wish is to evaluate the impact of each input over θ(Y). In order to get supplementary information about sensitivity, we set that θ(Y) is the α-level quantile of Y , where α ∈]0, 1[. Throughout some examples, it has been pointed out that in some cases quantile-oriented sensitivity indices can detect some influence that Sobol indices would not. Mainly, the influence over each level of quantile displays how an input distribution entirely propagates through the output. We establish further results for the quantile-oriented indices properties in order to justify their relevancy. The main contribution of this paper comes when a statistical estimator for this index is introduced

Stochastic simulators based optimization by Gaussian process metamodels -Application to maintenance investments planning issues Short title: Metamodel-based optimization of stochastic simulators

International audienceThis paper deals with the optimization of industrial asset management strategies, whose profitability is characterized by the Net Present Value (NPV) indicator which is assessed by a Monte Carlo simulator. The developed method consists in building a metamodel of this stochastic simulator, allowing to get, for a given model input, the NPV probability distribution without running the simulator. The present work is concentrated on the emulation of the quantile function of the stochastic simulator by interpolating well chosen basis functions and metamodeling their coefficients (using the Gaussian process metamodel). This quantile function metamodel is then used to treat a problem of strategy maintenance optimization (four systems installed on different plants), in order to optimize an NPV quantile. Using the Gaussian process framework, an adaptive design method (called QFEI) is defined by extending in our case the well known EGO algorithm. This allows to obtain an " optimal " solution using a small number of simulator runs

Analysis and modelling of patterned wafer nano-topography using multiple linear regression on design GDS and silicon PWG data

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