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In this paper, we propose an R package, called RKHSMetaMod, that implements a procedure for estimating a meta-model of a complex model. The meta-model approximates the Hoeffding decomposition of the complex model and allows us to perform sensitivity analysis on it. It belongs to a reproducing kernel Hilbert space that is constructed as a direct sum of Hilbert spaces. The estimator of the meta-model is the solution of a penalized empirical least-squares minimization with the sum of the Hilbert norm and the empirical L^2-norm. This procedure, called RKHS ridge group sparse, allows both to select and estimate the terms in the Hoeffding decomposition, and therefore, to select and estimate the Sobol indices that are non-zero. The RKHSMetaMod package provides an interface from R statistical computing environment to the C++ libraries Eigen and GSL. In order to speed up the execution time and optimize the storage memory, except for a function that is written in R, all of the functions of this package are written using the efficient C++ libraries through RcppEigen and RcppGSL packages. These functions are then interfaced in the R environment in order to propose a user-friendly package., Comment: arXiv admin note: text overlap with arXiv:1701.04671

Mathematical and numerical models are increasingly used in microbial ecology to model the fate of microbial communities in their ecosystem. These models allow to connect in a mechanistic framework species-level informations, such as the microbial genomes, with macro-scale features, such as species spatial distributions or metabolite gradients. Numerous models are built upon specieslevel metabolic models that predict the metabolic behaviour of a microbe by solving an optimization problem knowing its genome and its nutritional environment. However, screening the community dynamics with these metabolic models implies to solve such an optimization problem by species at each time step, leading to a significant computational load further increased by several orders of magnitude when spatial dimensions are added. In this paper, we propose a statistical framework based on Reproducing Kernel Hilbert Space (RKHS) metamodels that are used to provide fast approximations of the original metabolic model. The metamodel can replace the optimization step in the system dynamics, providing comparable outputs at a much lower computational cost. We will first build a system dynamics model of a simplified gut microbiota composed of a unique commensal bacterial strain in interaction with the host and challenged by a Salmonella infection. Then, the machine learning method will be introduced, and particularly the ANOVA-RKHS that will be exploited to achieve variable selection and model parsimony. A training dataset will be constructed with the original system dynamics model and hyper-parameters will be carefully chosen to provide fast and accurate approximations of the original model. Finally, the accuracy of the trained metamodels will be assessed, in particular by comparing the system dynamics outputs when the original model is replaced by its metamodel. The metamodel allows an overall relative error of 4.71% but reducing the computational load by a speed-up factor higher than 45, while correctly reproducing the complex behaviour occurring during Salmonella infection. These results provide a proof-of-concept of the potentiality of machine learning methods to give fast approximations of metabolic model outputs and pave the way towards PDEbased spatio-temporal models of microbial communities and host-microbiota-pathogen interactions.

In the context of the Gaussian regression model, the package RKHSMetaMod allows to estimate a meta model by solving the ridge group sparse optimization problem based on the Reproducing Kernel Hilbert Spaces (RKHS). The obtained estimator is an additive model that satisfies the properties of the Hoeffding decomposition, and its terms estimate the terms in the Hoeffding decomposition of the unknown regression function. The estimators of the Sobol indices are deduced from the estimated meta model. This package provides an interface from R statistical computing environment to the C++ libraries Eigen and GSL. In order to speed up the execution time, almost all of the functions of the RKHSMetaMod package are written using the efficient C++ libraries through RcppEigen and RcppGSL packages. These functions are then interfaced in the R environment in order to propose an user friendly package.

oral communication, texte intégral; Modelling complex systems such as agroecosystems often requires the quantification of a large number of input factors. Sensitivity analyses are useful to fix the appropriate spatial and temporal resolution of models and to reduce the number of input factors to be measured or estimated accurately. Comprehensive spatial and dynamic sensitivity analyses were applied to the Nitroscape model, a deterministic spatially distributed model describing nitrogen transfers and transformations in a rural landscape. Simulations were led on a virtual landscape that represented five years of farm management in an intensive rural area of 3 km². Cluster analyses were applied to summarize the results of the sensitivity analysis on the ensemble of model outcomes. The 29 studied output variables were split into five different clusters that grouped outcomes with similar response to input factors. Among the 11 studied factors, model outcomes were mainly sensitive to inputs characterizing the lateral transmissivity of soil. The horizontal resolution of the model was a significant factor driving ammonium and nitrate mineralisation, and uptake by plants. The vertical resolution of the model had the highest impact on the cumulate emissions of nitrous oxides. The interactions between the amount of nitrogen used in fertilization and the lateral transmissivity of soil was the most important factorial effect driving the amount of nitrogen in the catchment discharge.

International audience

In the regression model with errors in variables, we observe $n$ i.i.d. copies of $(Y,Z)$ satisfying $Y=f_{\theta^0}(X)+\xi$ and $Z=X+\varepsilon$ involving independent and unobserved random variables $X,\xi,\varepsilon$ plus a regression function $f_{\theta^0}$, known up to some finite dimensional $\theta^0$. The common densities of the $X_i$'s and of the $\xi_i$'s are unknown whereas the distribution of $\varepsilon$ is completely known. We aim at estimating the parameter $\theta^0$ by using the observations $(Y_1,Z_1),\cdots, (Y_n,Z_n)$. We propose two estimation procedures based on the least square criterion $\tilde S_{\theta^0,g}(\theta)=\mathbb{E}_{\theta^0,g}[((Y-f_\theta(X))^2w(X)]$ where $w$ is some weight function, to be chosen. In the first estimation procedure, $w$ does not depend on $\theta$ and the distribution of the $\xi$'s is unknown. The second estimation procedure is based on $S_{\theta^0,g}(\theta)=\mathbb{E}_{\theta^0,g}[((Y-f_\theta(X))^2-\sigma_{\xi,2}^2)w_\theta(X)]$ where $w_\theta$ is positive weight function, to be chosen, and requires the knowledge of $\sigma_{\xi,2}^2=\mbox{Var}(\xi)$. In both cases, we propose two estimators and derive upper bounds for the risk of those estimators, depending on the smoothness of the errors density $p_\varepsilon$ and on the smoothness properties of $w(x)f_\theta(x)$ or $w_\theta(x)f_\theta(x)$ with respect to $x$. Furthermore we give sufficient conditions that ensure that the parametric rate of convergence is achieved. We provide practical recipes for the choice of $w$ or $ w_\theta$ in the case of nonlinear regressionfunctions which are smooth on pieces allowing to gain in the order of the rate of convergence, up to the parametric rate in some cases.

We consider the regression model with errors-in-variables where we observe $n$ i.i.d. copies of $(Y,Z)$ satisfying $Y=f(X)+\xi, \; Z=X+\sigma\varepsilon$, involving independent and unobserved random variables $X,\xi,\varepsilon$. The density $g$ of $X$ is unknown, whereas the density of $\sigma\varepsilon$ is completely known. Using the observations $(Y_i, Z_i)$, $i=1,\cdots,n$, we propose an estimator of the regression function $f$, built as the ratio of two penalized minimum contrast estimators of $\ell=fg$ and $g$, without any prior knowledge on their smoothness. We prove that its $\mathbb{L}_2$-risk on a compact set is bounded by the sum of the two $\mathbb{L}_2(\mathbb{R})$-risks of the estimators of $\ell$ and $g$, and give the rate of convergence of such estimators for various smoothness classes for $\ell$ and $g$, when the errors $\varepsilon$ are either ordinary smooth or super smooth. The resulting rate is optimal in a minimax sense in all cases where lower bounds are available.

Dans un modèle de régression non linéaire avec erreurs sur les variables, on suppose les variables explicatives sont des variables aléatoires réelles indépendantes, de densité inconnue, qui sont observées à une erreur additive indépendantes et gaussienne près. La fonction de régression est connue à une paramètre fini-dimensionnel près. L'objectif est d'estimer ce paramètre dans ce modèle semi-paramétrique. Nous procédons en deux étapes. Le chapitre 2 est consacrée à l'estimation de fonctionnelles linéaires intégrales d'une densité dans le modèle de convolution. En particulier nous établissons une borne inférieure du risque quadratique minimax pour l'estimation d'une densité en un point sur la classe des densités obtenues par convolution avec la densité gaussienne standard. Dans le chapitre 3, en utilisant les résultats précédents, nous proposons un critère des moindres carrés modifié, basé sur l'estimation d'une espérance conditionnelle dépendant de la densité inconnue des variables explicatives. Nous montrons que l'estimateur obtenu par minimisation du critère ainsi construit est consistant et que sa vitesse de convergence est d'autant plus rapide que la fonction de régression admet de fortes propriétés de régularité (par rapport aux variables explicatives), et qu'elle est généralement plus lente que la vitesse paramétrique tex2html_wrap_inline25 . Néanmoins elle est d'ordre tex2html_wrap_inline27 pour un certain nombre de fonctions de régressions admettant un prolongement analytique sur le plan complexe., In the non linear errors-in-variables model, we suppose that the explanatory variables are real random variables independently distributed with common unknown density and are observed with a Gaussian additive error which is supposed to be independent from the explanatory variables. The regression function is known up to a finite dimensional parameter. We aim to estimate the unknown parameter in this semi-parametric model. We proceed in two steps. The chapter 2 is devoted to the estimation of some linear functional integrals of a density in the convolution model. In particular we establish a lower bound of the minimax quadratic risk for estimating the value of a density at a point over the class of densities obtained by convolution with the standard Gaussian density. In chapter 3, using the preceeding results, we propose a modified least squares criterion based on the estimation of a conditional expectation depending on the unknown density of the explanatory variables. We show that the estimator of parameters obtained by minimization of this criterion is consistent and that its rate of convergence is strongly related to the regularity of the regression function. The more regular the regression function (with respect to the explanatory variables) the faster the rate is. This rate is generally slower than the parametric rate. Nevertheless it is very close to this parametric rate for some regression functions admitting an analytic continuation on the complex plane.

We propose to estimate a metamodel and the sensitivity indices of a complex model m in the Gaussian regression framework. Our approach combines methods for sensitivity analysis of complex models and statistical tools for sparse non-parametric estimation in multivariate Gaussian regression model. It rests on the construction of a metamodel for aproximating the Hoeffding-Sobol decomposition of m. This metamodel belongs to a reproducing kernel Hilbert space constructed as a direct sum of Hilbert spaces leading to a functional ANOVA decomposition. The estimation of the metamodel is carried out via a penalized least-squares minimization allowing to select the subsets of variables that contribute to predict the output. It allows to estimate the sensitivity indices of m. We establish an oracle-type inequality for the risk of the estimator, describe the procedure for estimating the metamodel and the sensitivity indices, and assess the performances of the procedure via a simulation study., Nous considérons l’estimation d’un méta-modèle d’un modèle complexe m à partir des observations d’un n-échantillon dans un modèle de régression gaussien. Nous en déduisons une estimation des indices de sensibilité de m. Notre approche combine les méthodes d’analyse de sensibilité de modèles complexes et les outils statistiques de l’estimation non-paramétrique en régression multivariée. Elle repose sur la construction d’un méta-modèle qui approche la décomposition de Hoeffding-Sobol de m. Ce méta-modèle appartient à un espace de Hilbert à noyau reproduisant qui est lui-même la somme directe d’espaces de Hilbert, permettant ainsi une décomposition de type ANOVA. On en déduit des estimateurs des indices de sensibilité de m. Nous établissons une inégalité de type oracle pour le risque de l’estimateur, nous décrivons la procédure pour estimer le méta-modèle et les indices de sensibilité, et évaluons les performances de notre méthode à l’aide d’une étude de simulations.

International audience; Consider an autoregressive model with measurement error: we observe $Z_i=X_i+\varepsilon_i$, where $X_i$ is a stationary solution of the equation $X_i=f_{\theta^0}(X_{i-1})+\xi_i$. The regression function $f_{\theta^0}$ is known up to a finite dimensional parameter $\theta^0$. The distributions of $X_0$ and $\xi_1$ are unknown whereas the distribution of $\varepsilon_1$ is completely known. We want to estimate the parameter $\theta^0$ by using the observations $Z_0,\ldots,Z_n$. We propose an estimation procedure based on a modified least square criterion involving a weight function $w$, to be suitably chosen. We give upper bounds for the risk of the estimator, which depend on the smoothness of the errors density $f_\varepsilon$ and on the smoothness properties of $w f_\theta$.

We consider a failure hazard function, conditional on a time-independent covariate Z, given by eta (gamma 0) (t) f (beta 0) (Z). The baseline hazard function eta (gamma 0) and the relative risk f (beta 0) both belong to parametric families with theta (0) = (beta(0), gamma(0)) (inverted perpendicular) is an element of R (m+p). The covariate Z has an unknown density and is measured with an error through an additive error model U = Z + epsilon where e is a random variable, independent from Z, with known density f (epsilon). We observe a n-sample (X(i), D(i), U(i)), i = 1, ... , n, where X (i) is the minimum between the failure time and the censoring time, and D (i) is the censoring indicator. Using least square criterion and deconvolution methods, we propose a consistent estimator of theta(0) using the observations (X(i), D(i), U(i)), i = 1, ... , n. We give an upper bound for its risk which depends on the smoothness properties of f (epsilon) and f (beta) (z) as a function of z, and we derive sufficient conditions for the root n-consistency. We give detailed examples considering various type of relative risks f (beta) and various types of error density f (epsilon). In particular, in the Cox model and in the excess risk model, the estimator of theta (0) is root n-consistent and asymptotically Gaussian regardless of the form of f (epsilon).