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

L2 Boosting on generalized Hoeffding decomposition for dependent variables. Application to Sensitivity Analysis

This paper is dedicated to the study of an estimator of the generalized Hoeffding decomposition. We build such an estimator using an empirical Gram-Schmidt approach and derive a consistency rate in a large dimensional settings. Then, we apply a greedy algorithm with these previous estimators to Sensitivity Analysis. We also establish the consistency of this $\mathbb L_2$-boosting up to sparsity assumptions on the signal to analyse. We end the paper with numerical experiments, which demonstrates the low computational cost of our method as well as its efficiency on standard benchmark of Sensitivity Analysis.Comment: 48 pages, 7 Figure

Meta-Analysis on grain yield effects of cereals-legume intercropping

Meta-Analysis on grain yield effects of cereals-legume intercroppin

L2-Boosting for sensitivity analysis with dependent inputs

International audienceThis paper is dedicated to the study of an estimator of the generalized Hoeffding decomposition. We build such an estimator using an empirical Gram-Schmidt approach and derive a consistency rate in a large dimensional setting. We then apply a greedy algorithm with these previous estimators to a sensitivity analysis. We also establish the consistency of this L2-boosting under sparsity assumptions of the signal to be analyzed. The paper concludes with numerical experiments, that demonstrate the low computational cost of our method, as well as its efficiency on the standard benchmark of sensitivity analysis

Meta-Analysis on grain yield effects of cereal-legume intercropping. Second Year’s Training Report. June-August 2009

In intercropping, more than one crop species are cultivated simultaneously on the same piece of land, to promote interaction between species. To determine the effects on yield performance of intercropping of legumes and cereals, a meta analysis has been set up based on data from publications on this kind of field trials as found from an extensive search on Web-of Science. The effect of intercropping has been assessed by comparing yield in pure stand and in the intercrop of each species. Two measures were considered : i) The LER (Land Equivalent Ratio), which is a simple way to compare the pure yield with the intercropping yield by a sum of ratios and ii) the overall effect decomposed into sum of a ”selection effect” and a ”complementary effect” (Loreau and Hector, Partitioning selection and complementary in biodiversity experiments, Nature, 2001). A large selection effect predicts a dominance of one of the species on the other, whereas a large complementary effect means cultures stimulate each other. From the literature search, 8 papers were found to fulfil the requirements for a meta-analysis on yield effects of cereal/legume intercropping. For the estimation, a fixed-effects and a mixed-effects analysis of variance was used on 80 observations extracted from 8 papers. At first, an excess yield from intercropping was observed in all analysis, however, the effect varied depending on the cultivation practices, and the way in which the data were analysed in the papers. Our analysis based on LER values revealed a significant yield effect of intercropping, but predicted also an important publication bias, implying that sometimes negative results have not been published. The simulataneous analysis of selection and complementary effects suggested this benefit was due to dominance of one of the two species, whereas the competition was inefficient; this do not support the general principle of intercropping. However, one study had a strong influence in the results and was responsible for a large negative complementary effect. When omitting this study, we got an important positive complementary effect and the selection effect remained positive but lower. The specific paper was the only one to involve oat+pea without chemical fertilizers and at a location with low annual rainfall. In conclusion, meta-analysis was a valuable method for combining results from different studies. Unfortunately, many publications did not present enough details for been possible to include them in the analysis. Finally, the decomposed measure seems to give additional information about the mechanisms of the interactions observed

Indices de Sobol généralisés pour variables dépendantes

A mathematical model aims at characterizing a complex system or process that is too expensive to experiment. However, in this model, often strongly non linear, input parameters can be affected by a large uncertainty including errors of measurement of lack of information. Global sensitivity analysis is a stochastic approach whose objective is to identify and to rank the input variables that drive the uncertainty of the model output. Through this analysis, it is then possible to reduce the model dimension and the variation in the output of the model. To reach this objective, the Sobol indices are commonly used. Based on the functional ANOVA decomposition of the output, also called Hoeffding decomposition, they stand on the assumption that the incomes are independent. Our contribution is on the extension of Sobol indices for models with non independent inputs. In one hand, we propose a generalized functional decomposition, where its components is subject to specific orthogonal constraints. This decomposition leads to the definition of generalized sensitivity indices able to quantify the dependent inputs' contribution to the model variability. On the other hand, we propose two numerical methods to estimate these constructed indices. The first one is well-fitted to models with independent pairs of dependent input variables. The method is performed by solving linear system involving suitable projection operators. The second method can be applied to more general models. It relies on the recursive construction of functional systems satisfying the orthogonality properties of summands of the generalized decomposition. In parallel, we illustrate the two methods on numerical examples to test the efficiency of the techniques.Dans un modèle qui peut s'avérer complexe et fortement non linéaire, les paramètres d'entrée, parfois en très grand nombre, peuvent être à l'origine d'une importante variabilité de la sortie. L'analyse de sensibilité globale est une approche stochastique permettant de repérer les principales sources d'incertitude du modèle, c'est-à-dire d'identifier et de hiérarchiser les variables d'entrée les plus influentes. De cette manière, il est possible de réduire la dimension d'un problème, et de diminuer l'incertitude des entrées. Les indices de Sobol, dont la construction repose sur une décomposition de la variance globale du modèle, sont des mesures très fréquemment utilisées pour atteindre de tels objectifs. Néanmoins, ces indices se basent sur la décomposition fonctionnelle de la sortie, aussi connue sous le nom de décomposition de Hoeffding. Mais cette décomposition n'est unique que si les variables d'entrée sont supposées indépendantes. Dans cette thèse, nous nous intéressons à l'extension des indices de Sobol pour des modèles à variables d'entrée dépendantes. Dans un premier temps, nous proposons une généralisation de la décomposition de Hoeffding au cas où la forme de la distribution des entrées est plus générale qu'une distribution produit. De cette décomposition généralisée aux contraintes d'orthogonalité spécifiques, il en découle la construction d'indices de sensibilité généralisés capable de mesurer la variabilité d'un ou plusieurs facteurs corrélés dans le modèle. Dans un second temps, nous proposons deux méthodes d'estimation de ces indices. La première est adaptée à des modèles à entrées dépendantes par paires. Elle repose sur la résolution numérique d'un système linéaire fonctionnel qui met en jeu des opérateurs de projection. La seconde méthode, qui peut s'appliquer à des modèles beaucoup plus généraux, repose sur la construction récursive d'un système de fonctions qui satisfont les contraintes d'orthogonalité liées à la décomposition généralisée. En parallèle, nous mettons en pratique ces méthodes sur différents cas tests