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Fabrice Gamboa

Publications and source records attributed to Fabrice Gamboa.

At least 73 records · Page 4Linked to original sources
arXiv2015

Sum rules via large deviations

In this paper, a sum rule means a relationship between a functional defined on a subset of all probability measures on $\mathbb{R}$ involving the reverse Kullback-Leibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction. The first sum rule is the Szegő-Verblunsky theorem (see Simon (2011) Theorem 1.8.6 p. 29) and concerns the case of trigonometrical polynomials on the torus. More recently, Killip and Simon (2003) have given a revival interest to this subject by showing a quite surprising sum rule for measures dominating the semicircular distribution. Indeed, this sum rule includes a contribution of the atomic part of the measure away from the support of the circular law. In this paper, we recover this sum rules by using only large deviations tools on random matrices. Furthermore, we obtain new (up to our knowledge) magic sum rules for the reverse Kullback-Leibler divergence with respect to the Pastur-Marchenko or Kesten-McKay distributions. As in the semicircular case, these formulas generally include a contribution of the atomic part appearing away from the support of the reference measure. From the point of view of large deviations, we obtain these formulas by showing a large deviation principle for the point mass measure concentrated on random eigenvalues of classical ensembles and combine it with a large deviation principle due to Gamboa Rouault (2011) for the recursion coefficients.

math.PR↗
arXiv2014

A unified framework for the study of the PLS estimator's properties

In this paper we propose a new approach to study the properties of the Partial Least Squares (PLS) estimator. This approach relies on the link between PLS and discrete orthogonal polynomials. Indeed many important PLS objects can be expressed in terms of some specific discrete orthogonal polynomials, called the residual polynomials. Based on the explicit analytical expression we have stated for these polynomials in terms of signal and noise, we provide a new framework for the study of PLS. Furthermore, we show that this new approach allows to simplify and retreive independent proofs of many classical results (proved earlier by different authors using various approaches and tools). This general and unifying approach also sheds new light on PLS and helps to gain insight on its properties.

math.ST↗
arXiv2014

PLS: a new statistical insight through the prism of orthogonal polynomials

Partial Least Square (PLS) is a dimension reduction method used to remove multicollinearities in a regression model. However contrary to Principal Components Analysis (PCA) the PLS components are also choosen to be optimal for predicting the response $Y$. In this paper we provide a new and explicit formula for the residuals. We show that the residuals are completely determined by the spectrum of the design matrix and by the noise on the observations. Because few are known on the behaviour of the PLS components we also investigate their statistical properties in a regression context. New results on regression and prediction error for PLS are stated under the assumption of a low variance of the noise.

math.ST↗
arXiv2014

Generalized Sobol sensitivity indices for dependent variables: numerical methods

The 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.

stat.AP↗
arXiv2014

Density modification based reliability sensitivity analysis

Sensitivity analysis of a numerical model, for instance simulating physical phenomena, is useful to quantify the influence of the inputs on the model responses. This paper proposes a new sensitivity index, based upon the modification of the probability density function (pdf) of the random inputs, when the quantity of interest is a failure probability (probability that a model output exceeds a given threshold). An input is considered influential if the input pdf modification leads to a broad change in the failure probability. These sensitivity indices can be computed using the sole set of simulations that has already been used to estimate the failure probability, thus limiting the number of calls to the numerical model. In the case of a Monte Carlo sample, asymptotical properties of the indices are derived. Based on Kullback-Leibler divergence, several types of input perturbations are introduced. The relevance of this new sensitivity analysis method is analysed through three case studies.

math.ST↗
arXiv2014

Spike detection from inaccurate samplings

This article investigates the support detection problem using the LASSO estimator in the space of measures. More precisely, we study the recovery of a discrete measure (spike train) from few noisy observations (Fourier samples, moments...) using an $\ell_{1}$-regularization procedure. In particular, we provide an explicit quantitative localization of the spikes.

math.ST↗
arXiv2014

Group Lasso for generalized linear models in high dimension

Nowadays an increasing amount of data is available and we have to deal with models in high dimension (number of covariates much larger than the sample size). Under sparsity assumption it is reasonable to hope that we can make a good estimation of the regression parameter. This sparsity assumption as well as a block structuration of the covariates into groups with similar modes of behavior is for example quite natural in genomics. A huge amount of scientific literature exists for Gaussian linear models including the Lasso estimator and also the Group Lasso estimator which promotes group sparsity under an a priori knowledge of the groups. We extend this Group Lasso procedure to generalized linear models and we study the properties of this estimator for sparse high-dimensional generalized linear models to find convergence rates. We provide oracle inequalities for the prediction and estimation error under assumptions on the covariables and under a condition on the design matrix. We show the ability of this estimator to recover good sparse approximation of the true model. At last we extend these results to the case of an Elastic net penalty and we apply them to the so-called Poisson regression case which has not been studied in this context contrary to the logistic regression.

math.ST↗
arXiv2013

Sensitivity analysis for multidimensional and functional outputs

Let $X:=(X_1, \ldots, X_p)$ be random objects (the inputs), defined on some probability space $(Ω,{\mathcal{F}}, \mathbb P)$ and valued in some measurable space $E=E_1\times\ldots \times E_p$. Further, let $Y:=Y = f(X_1, \ldots, X_p)$ be the output. Here, $f$ is a measurable function from $E$ to some Hilbert space $\mathbb{H}$ ($\mathbb{H}$ could be either of finite or infinite dimension). In this work, we give a natural generalization of the Sobol indices (that are classically defined when $Y\in\mathbb R$ ), when the output belongs to $\mathbb{H}$. These indices have very nice properties. First, they are invariant. under isometry and scaling. Further they can be, as in dimension $1$, easily estimated by using the so-called Pick and Freeze method. We investigate the asymptotic behaviour of such estimation scheme.

stat.AP↗
arXiv2013

From variable density sampling to continuous sampling using Markov chains

Since its discovery over the last decade, Compressed Sensing (CS) has been successfully applied to Magnetic Reso- nance Imaging (MRI). It has been shown to be a powerful way to reduce scanning time without sacrificing image quality. MR images are actually strongly compressible in a wavelet basis, the latter being largely incoherent with the k-space or spatial Fourier domain where acquisition is performed. Nevertheless, since its first application to MRI [1], the theoretical justification of actual k-space sampling strategies is questionable. Indeed, the vast majority of k-space sampling distributions have been heuris- tically designed (e.g., variable density) or driven by experimental feasibility considerations (e.g., random radial or spiral sampling to achieve smoothness k-space trajectory). In this paper, we try to reconcile very recent CS results with the MRI specificities (mag- netic field gradients) by enforcing the measurements, i.e. samples of k-space, to fit continuous trajectories. To this end, we propose random walk continuous sampling based on Markov chains and we compare the reconstruction quality of this scheme to the state- of-the art.

stat.AP↗
arXiv2013

Shape invariant model approach for functional data analysis in uncertainty and sensitivity studies

Dynamic simulators model systems evolving over time. Often, it operates iteratively over fixed number of time-steps. The output of such simulator can be considered as time series or discrete functional outputs. Metamodeling is an e ective method to approximate demanding computer codes. Numerous metamodeling techniques are developed for simulators with a single output. Standard approach to model a dynamic simulator uses the same method also for multi-time series outputs: the metamodel is evaluated independently at every time step. This can be computationally demanding in case of large number of time steps. In some cases, simulator outputs for di erent combinations of input parameters have quite similar behaviour. In this paper, we propose an application of shape invariant model approach to model dynamic simulators. This model assumes a common pattern shape curve and curve-specific di erences in amplitude and timing are modelled with linear transformations. We provide an e cient algorithm of transformation parameters estimation and subsequent prediction algorithm. The method was tested with a CO2 storage reservoir case using an industrial commercial simulator and compared with a standard single step approach. The method provides satisfactory predictivity and it does not depend on the number of involved time steps.

stat.AP↗
arXiv2013

Reliability sensitivity analysis based on probability distribution perturbation with application to CO2 storage

The objective of reliability sensitivity analysis is to determine input variables that mostly contribute to the variability of the failure probability. In this paper, we study a recently introduced method for the reliability sensitivity analysis based on a perturbation of the original probability distribution of the input variables. The objective is to determine the most influential input variables and to analyze their impact on the failure probability. We propose a moment independent sensitivity measure that is based on a perturbation of the original probability density independently for each input variable. The variables providing the highest variation of the original failure probability are settled to be more influential. These variables will need a proper characterization in terms of uncertainty. The method is intended to work in applications involving a computationally expensive simulation code for evaluating the failure probability such as the CO2 storage risk analysis. An application of the method to a synthetic CO2 storage case study is provided together with some analytical examples

math.ST↗
arXiv2013

Statistical inference for Sobol pick freeze Monte Carlo method

Many 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.

stat.AP↗
arXiv2013

Functional calibration estimation by the maximum entropy on the mean principle

We extend the problem of obtaining an estimator for the finite population mean parameter incorporating complete auxiliary information through calibration estimation in survey sampling but considering a functional data framework. The functional calibration sampling weights of the estimator are obtained by matching the calibration estimation problem with the maximum entropy on the mean principle. In particular, the calibration estimation is viewed as an infinite dimensional linear inverse problem following the structure of the maximum entropy on the mean approach. We give a precise theoretical setting and estimate the functional calibration weights assuming, as prior measures, the centered Gaussian and compound Poisson random measures. Additionally, through a simple simulation study, we show that our functional calibration estimator improves its accuracy compared with the Horvitz-Thompson estimator.

math.ST↗
arXiv2012

Modeling Weather Conditions Consequences on Road Trafficking Behaviors

We provide a model to understand how adverse weather conditions modify traffic flow dynamic. We first prove that the microscopic Free Flow Speed of the vehicles is changed and then provide a rule to model this change. For this, we consider a thresholded linear model, corresponding to an application of a MARS model to road trafficking. This model adapts itself locally to the whole road network and provides accurate unbiased forecasted speed using live or short term forecasted weather data information.

stat.ME↗
arXiv2012

Derivative-based global sensitivity measures: general links with Sobol' indices and numerical tests

The estimation of variance-based importance measures (called Sobol' indices) of the input variables of a numerical model can require a large number of model evaluations. It turns to be unacceptable for high-dimensional model involving a large number of input variables (typically more than ten). Recently, Sobol and Kucherenko have proposed the Derivative-based Global Sensitivity Measures (DGSM), defined as the integral of the squared derivatives of the model output, showing that it can help to solve the problem of dimensionality in some cases. We provide a general inequality link between DGSM and total Sobol' indices for input variables belonging to the class of Boltzmann probability measures, thus extending the previous results of Sobol and Kucherenko for uniform and normal measures. The special case of log-concave measures is also described. This link provides a DGSM-based maximal bound for the total Sobol indices. Numerical tests show the performance of the bound and its usefulness in practice.

math.ST↗
arXiv2012

Exact Reconstruction using Beurling Minimal Extrapolation

We show that measures with finite support on the real line are the unique solution to an algorithm, named generalized minimal extrapolation, involving only a finite number of generalized moments (which encompass the standard moments, the Laplace transform, the Stieltjes transformation, etc). Generalized minimal extrapolation shares related geometric properties with basis pursuit of Chen, Donoho and Saunders [CDS98]. Indeed we also extend some standard results of compressed sensing (the dual polynomial, the nullspace property) to the signed measure framework. We express exact reconstruction in terms of a simple interpolation problem. We prove that every nonnegative measure, supported by a set containing s points,can be exactly recovered from only 2s + 1 generalized moments. This result leads to a new construction of deterministic sensing matrices for compressed sensing.

math.ST↗