Generalized Hoeffding-Sobol Decomposition for Dependent Variables -Application to Sensitivity Analysis

In this paper, we consider a regression model built on dependent variables. This regression modelizes an input output relationship. Under boundedness assumptions on the joint distribution function of the input variables, we show that a generalized Hoeffding-Sobol decomposition is available. This leads to new indices measuring the sensitivity of the output with respect to the input variables. We also study and discuss the estimation of these new indices

Generalized Sobol sensitivity indices for dependent variables: numerical methods

International audienceThe hierarchically orthogonal functional decomposition of any measurable function f of a random vector X=(X_1,...,X_p) consists in decomposing f(X) into a sum of increasing dimension functions depending only on a subvector of X. Even when X_1,..., X_p are assumed to be dependent, this decomposition is unique if components are hierarchically orthogonal. That is, two of the components are orthogonal whenever all the variables involved in one of the summands are a subset of the variables involved in the other. Setting Y=f(X), this decomposition leads to the definition of generalized sensitivity indices able to quantify the uncertainty of Y with respect to the dependent inputs X. In this paper, a numerical method is developed to identify the component functions of the decomposition using the hierarchical orthogonality property. Furthermore, the asymptotic properties of the components estimation is studied, as well as the numerical estimation of the generalized sensitivity indices of a toy model. Lastly, the method is applied to a model arising from a real-world problem

Conditional large and moderate deviations for sums of discrete random variables. Combinatoric applications

International audienceWe prove large and moderate deviation principles for the distribution of an empirical mean conditioned by the value of the sum of discrete i.i.d. random variables. Some applications for combinatoric problems are discussed

Exploiting nonlinear propagation in echo sounders and sonar

The 10th European Conference on Underwater Acoustics (ECUA). 2010, Istanbul, Turkey. Mainstream sonars transmit and receive signals at the same frequency. As water is a nonlinear medium, a propagating signal generates harmonics at multiples of the transmitted frequency. For sonar applications, energy transferred to higher harmonics is seen as a disturbance. To satisfy requirements for calibration of echo sounders in fishery research, input power has to be limited to avoid energy loss to harmonics generation. Can these harmonics be used in sonar imaging? The frequency dependency of target echos, and the different spatial distribution of higher harmonics can contribute to additional information on detected targets in fish classification, ocean bathymetry, or bottom classification. Our starting point was the sonar equation adapted for the second harmonic. We have simulated nonlinear propagation of sound in water, and obtained estimates of received pressure levels of harmonics for a calibration sphere, or a fish as reflector. These pressure profiles were used in the sonar equation to compare harmonics to fundamental signal budget. Our results show that a 200 kHz thermal noise limited echo sounder, with a range of 800 m will reach around 300 m for the second harmonic. This means the second harmonic is useful in many applications

Statistical inference for Sobol pick freeze Monte Carlo method

International audienceMany mathematical models involve input parameters, which are not precisely known. Global sensitivity analysis aims to identify the parameters whose uncertainty has the largest impact on the variability of a quantity of interest (output of the model). One of the statistical tools used to quantify the influence of each input variable on the output is the Sobol sensitivity index. We consider the statistical estimation of this index from a finite sample of model outputs. We study asymptotic and non-asymptotic properties of two estimators of Sobol indices. These properties are applied to significance tests and estimation by confidence intervals

New estimation of Sobol' indices using kernels

In this work, we develop an approach mentioned by da Veiga and Gamboa in 2013. It consists in extending the very interestingpoint of view introduced in \cite{gine2008simple} to estimate general nonlinear integral functionals of a density on the real line, by using empirically a kernel estimator erasing the diagonal terms. Relaxing the positiveness assumption on the kernel and choosing a kernel of order large enough, we are able to prove a central limit theorem for estimating Sobol' indices of any order (the bias is killed thanks to this signed kernel)