Local Polynomial Estimation for Sensitivity Analysis on Models With Correlated Inputs

Abstract

Sensitivity indices when the inputs of a model are not independent are estimated by local polynomial techniques. Two original estimators based on local polynomial smoothers are proposed. Both have good theoretical properties which are exhibited and also illustrated through analytical examples. They are used to carry out a sensitivity analysis on a real case of a kinetic model with correlated parameters.KEY WORDS: Nonparametric regression; Global sensitivity indices; Conditional moments estimation.Achieving better knowledge of refining processes usually requires to build a kinetic model predicting the output components produced in a unit given the input components introduced (the "feed") and the operating conditions. Such a model is based on the choice of a reaction mechanism depending on various parameters (e.g. kinetic constants). But the complexity of the mechanism, the variability of the behavior of catalysts when they are used and the difficulty of observations and experiments imply that most often these parameters cannot be inferred from theoretical considerations and need to be estimated through practical experiments. This estimation procedure leads to consider them uncertain and this uncertainty spreads on the model predictions. This can be highly problematic in real situations. It is then essential to quantify this uncertainty and to study the influence of parameters variations on the model outputs through uncertainty and sensitivity analysis.During the last decades much effort in mathematical analysis of physical processes has focused on modeling and reasoning with uncertainty and sensitivity. Model calibration and validation are examples where sensitivity and uncertainty analysis have become essential investigative scientific tools. Roughly speaking, uncertainty analysis refers to the inherent variations of a model (e.g. a modeled physical process) and is helpful in finding the relation between some variability or probability distribution on input parameters and the variability and probability distribution of outputs, while sensitivity analysis investigates the effects of varying a model input on the outputs and ascertains how much a model depends on each or some of its inputs.Over the years several mathematical and computer-assisted methods have been developed to carry out global sensitivity analysis and the reader may refer to the book of Saltelli, Chan & Scott (2000) for a wide and thorough review. Amongst these methods a particular popular class is the one composed by "variance-based" methods which is detailed below. Let us consider a mathematical model given bywhere η : R d → R is the modeling function, Y ∈ R represents the output or prediction of the model and X = (X 1 , ..., X d ) is the d-dimensional real vector of the input factors or parameters. The vector of input parameters is treated as a random vector, which implies that the output is also a random variable. also called correlation ratio. We can also introduce sensitivity indices of higher orders to take into account input interac...

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