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Céline Lévy-Leduc

Publications and source records attributed to Céline Lévy-Leduc.

At least 19 recordsLinked to original sources
arXiv2026

Optimal minimization of an unknown function in a nonparametric multivariate regression model thanks to a dimension reduction approach

In this paper, we propose a novel approach for estimating the minimum of a smooth function and its location from observations corresponding to a multivariate regression function depending a priori on d variables but actually only on r < d active variables and corrupted by some additional noise. Our method consists of two steps: The rst one is a variable selection approach which is used for identifying the r active variables on which f depends and the second one consists in estimating the minimum of the function and its location. The estimation of the minimizers is obtained by using a projected gradient descent where the gradient is estimated using a local polynomial approximation of the regression function limited to its active variables obtained in the rst step. The estimation of the minimum is obtained by evaluating the estimator of the regression function using a local polynomial approach at the estimator of one of the minimizers previously obtained. We establish non asymptotic upper bounds for the quadratic risk of the estimators of the minimizers and of the minimum and prove that they reach the optimal rate that could be expected as if the active variables were known beforehand up to a factor smaller than a power of a logarithmic term.

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arXiv2024

A novel approach for estimating functions in the multivariate setting based on an adaptive knot selection for B-splines with an application to a chemical system used in geoscience

In this paper, we will outline a novel data-driven method for estimating functions in a multivariate nonparametric regression model based on an adaptive knot selection for B-splines. The underlying idea of our approach for selecting knots is to apply the generalized lasso, since the knots of the B-spline basis can be seen as changes in the derivatives of the function to be estimated. This method was then extended to functions depending on several variables by processing each dimension independently, thus reducing the problem to a univariate setting. The regularization parameters were chosen by means of a criterion based on EBIC. The nonparametric estimator was obtained using a multivariate B-spline regression with the corresponding selected knots. Our procedure was validated through numerical experiments by varying the number of observations and the level of noise to investigate its robustness. The influence of observation sampling was also assessed and our method was applied to a chemical system commonly used in geoscience. For each different framework considered in this paper, our approach performed better than state-of-the-art methods. Our completely data-driven method is implemented in the glober R package which is available on the Comprehensive R Archive Network (CRAN).

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arXiv2022

Variable selection in sparse GLARMA models

In this paper, we propose a novel and efficient two-stage variable selection approach for sparse GLARMA models, which are pervasive for modeling discrete-valued time series. Our approach consists in iteratively combining the estimation of the autoregressive moving average (ARMA) coefficients of GLARMA models with regularized methods designed for performing variable selection in regression coefficients of Generalized Linear Models (GLM). We first establish the consistency of the ARMA part coefficient estimators in a specific case. Then, we explain how to efficiently implement our approach. Finally, we assess the performance of our methodology using synthetic data, compare it with alternative methods and illustrate it on an example of real-world application. Our approach, which is implemented in the GlarmaVarSel R package and available on the CRAN, is very attractive since it benefits from a low computational load and is able to outperform the other methods in terms of coefficient estimation, particularly in recovering the non null regression coefficients.

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arXiv2022

Variable selection in high-dimensional logistic regression models using a whitening approach

In bioinformatics, the rapid development of sequencing technology has enabled us to collect an increasing amount of omics data. Classification based on omics data is one of the central problems in biomedical research. However, omics data usually has a limited sample size but high feature dimensions, and it is assumed that only a few features (biomarkers) are active, i.e. informative to discriminate between different categories (cancer subtypes, responder/non-responder to treatment, for example). Identifying active biomarkers for classification has therefore become fundamental for omics data analysis. Focusing on binary classification, we propose an innovative feature selection method aiming at dealing with the high correlations between the biomarkers. Various research has shown the notorious influence of correlated biomarkers and the difficulty of accurately identifying active ones. Our method, WLogit, consists in whitening the design matrix to remove the correlations between biomarkers, then using a penalized criterion adapted to the logistic regression model to select features. The performance of WLogit is assessed using synthetic data in several scenarios and compared with other approaches. The results suggest that WLogit can identify almost all active biomarkers even in the cases where the biomarkers are highly correlated, while the other methods fail, which consequently leads to higher classification accuracy. The performance is also evaluated on the classification of two Lymphoma subtypes, and the obtained classifier also outperformed other methods. Our method is implemented in the \texttt{WLogit} R package available from the Comprehensive R Archive Network (CRAN).

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arXiv2022

Identification of prognostic and predictive biomarkers in high-dimensional data with PPLasso

In clinical trials, identification of prognostic and predictive biomarkers is essential to precision medicine. Prognostic biomarkers can be useful for the prevention of the occurrence of the disease, and predictive biomarkers can be used to identify patients with potential benefit from the treatment. Previous researches were mainly focused on clinical characteristics, and the use of genomic data in such an area is hardly studied. A new method is required to simultaneously select prognostic and predictive biomarkers in high dimensional genomic data where biomarkers are highly correlated. We propose a novel approach called PPLasso (Prognostic Predictive Lasso) integrating prognostic and predictive effects into one statistical model. PPLasso also takes into account the correlations between biomarkers that can alter the biomarker selection accuracy. Our method consists in transforming the design matrix to remove the correlations between the biomarkers before applying the generalized Lasso. In a comprehensive numerical evaluation, we show that PPLasso outperforms the traditional Lasso approach on both prognostic and predictive biomarker identification in various scenarios. Finally, our method is applied to publicly available transcriptomic data from clinical trial RV144. Our method is implemented in the PPLasso R package available from the Comprehensive R Archive Network (CRAN).

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arXiv2021

An active learning approach for improving the performance of equilibrium based chemical simulations

In this paper, we propose a novel sequential data-driven method for dealing with equilibrium based chemical simulations, which can be seen as a specific machine learning approach called active learning. The underlying idea of our approach is to consider the function to estimate as a sample of a Gaussian process which allows us to compute the global uncertainty on the function estimation. Thanks to this estimation and with almost no parameter to tune, the proposed method sequentially chooses the most relevant input data at which the function to estimate has to be evaluated to build a surrogate model. Hence, the number of evaluations of the function to estimate is dramatically limited. Our active learning method is validated through numerical experiments and applied to a complex chemical system commonly used in geoscience.

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arXiv2021

Sign Consistency of the Generalized Elastic Net Estimator

In this paper, we propose a novel variable selection approach in the framework of high-dimensional linear models where the columns of the design matrix are highly correlated. It consists in rewriting the initial high-dimensional linear model to remove the correlation between the columns of the design matrix and in applying a generalized Elastic Net criterion since it can be seen as an extension of the generalized Lasso. The properties of our approach called gEN (generalized Elastic Net) are investigated both from a theoretical and a numerical point of view. More precisely, we provide a new condition called GIC (Generalized Irrepresentable Condition) which generalizes the EIC (Elastic Net Irrepresentable Condition) of Jia and Yu (2010) under which we prove that our estimator can recover the positions of the null and non null entries of the coefficients when the sample size tends to infinity. We also assess the performance of our methodology using synthetic data and compare it with alternative approaches. Our numerical experiments show that our approach improves the variable selection performance in many cases.

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arXiv2020

A variable selection approach for highly correlated predictors in high-dimensional genomic data

In genomic studies, identifying biomarkers associated with a variable of interest is a major concern in biomedical research. Regularized approaches are classically used to perform variable selection in high-dimensional linear models. However, these methods can fail in highly correlated settings. We propose a novel variable selection approach called WLasso, taking these correlations into account. It consists in rewriting the initial high-dimensional linear model to remove the correlation between the biomarkers (predictors) and in applying the generalized Lasso criterion. The performance of WLasso is assessed using synthetic data in several scenarios and compared with recent alternative approaches. The results show that when the biomarkers are highly correlated, WLasso outperforms the other approaches in sparse high-dimensional frameworks. The method is also successfully illustrated on publicly available gene expression data in breast cancer. Our method is implemented in the WLasso R package which is available from the Comprehensive R Archive Network.

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arXiv2019

Estimation of large block structured covariance matrices: Application to "multi-omic" approaches to study seed quality

Motivated by an application in high-throughput genomics and metabolomics, we propose a novel, efficient and fully data-driven approach for estimating large block structured sparse covariance matrices in the case where the number of variables is much larger than the number of samples without limiting ourselves to block diagonal matrices. Our approach consists in approximating such a covariance matrix by the sum of a low-rank sparse matrix and a diagonal matrix. Our methodology also can deal with matrices for which the block structure appears only if the columns and rows are permuted according to an unknown permutation. Our technique is implemented in the R package \texttt{BlockCov} which is available from the Comprehensive R Archive Network (CRAN) and from GitHub. In order to illustrate the statistical and numerical performance of our package some numerical experiments are provided as well as a thorough comparison with alternative methods. Finally, our approach is applied to the use of "multi-omic" approaches for studying seed quality.

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arXiv2019

Variable selection in sparse high-dimensional GLARMA models

In this paper, we propose a novel variable selection approach in the framework of sparse high-dimensional GLARMA models. It consists in combining the estimation of the autoregressive moving average (ARMA) coefficients of these models with regularized methods designed for Generalized Linear Models (GLM). The properties of our approach are investigated both from a theoretical and a numerical point of view. More precisely, we establish in a specific case the consistency of the ARMA part coefficient estimators. We explain how to implement our approach and we show that it is very attractive since it benefits from a low computational load. We also assess the performance of our methodology using synthetic data and compare it with alternative approaches. Our numerical experiments show that combining the estimation of the ARMA part coefficients with regularized methods designed for GLM dramatically improves the variable selection performance.

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arXiv2017

Variable selection in multivariate linear models with high-dimensional covariance matrix estimation

In this paper, we propose a novel variable selection approach in the framework of multivariate linear models taking into account the dependence that may exist between the responses. It consists in estimating beforehand the covariance matrix of the responses and to plug this estimator in a Lasso criterion, in order to obtain a sparse estimator of the coefficient matrix. The properties of our approach are investigated both from a theoretical and a numerical point of view. More precisely, we give general conditions that the estimators of the covariance matrix and its inverse have to satisfy in order to recover the positions of the null and non null entries of the coefficient matrix when the size of the covariance matrix is not fixed and can tend to infinity. We prove that these conditions are satisfied in the particular case of some Toeplitz matrices. Our approach is implemented in the R package MultiVarSel available from the Comprehensive R Archive Network (CRAN) and is very attractive since it benefits from a low computational load. We also assess the performance of our methodology using synthetic data and compare it with alternative approaches. Our numerical experiments show that including the estimation of the covariance matrix in the Lasso criterion dramatically improves the variable selection performance in many cases.

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arXiv2016

Improving heritability estimation by a variable selection approach in sparse high dimensional linear mixed models

Motivated by applications in neuroanatomy, we propose a novel methodology for estimating the heritability which corresponds to the proportion of phenotypic variance which can be explained by genetic factors. Estimating this quantity for neuroanatomical features is a fundamental challenge in psychiatric disease research. Since the phenotypic variations may only be due to a small fraction of the available genetic information, we propose an estimator of the heritability that can be used in high dimensional sparse linear mixed models. Our method consists of three steps. Firstly, a variable selection stage is performed in order to recover the support of the genetic effects -- also called causal variants -- that is to find the genetic effects which really explain the phenotypic variations. Secondly, we propose a maximum likelihood strategy for estimating the heritability which only takes into account the causal genetic effects found in the first step. Thirdly, we compute the standard error and the 95% confidence interval associated to our heritability estimator thanks to a nonparametric bootsrap approach. Our main contribution consists in providing an estimation of the heritability with standard errors substantially smaller than methods without variable selection when the genetic effects are very sparse. Since the real genetic architecture is in general unknown in practice, we also propose an empirical criterion which allows the user to decide whether it is relevant to apply a variable selection based approach or not. We illustrate the performance of our methodology on synthetic and real neuroanatomic data coming from the Imagen project. We also show that our approach has a very low computational burden and is very efficient from a statistical point of view.

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arXiv2016

Nonparametric homogeneity tests and multiple change-point estimation for analyzing large Hi-C data matrices

We propose a novel nonparametric approach for estimating the location of block boundaries (change-points) of non-overlapping blocks in a random symmetric matrix which consists of random variables having their distribution changing from one block to the other. Our method is based on a nonparametric two-sample homogeneity test for matrices that we extend to the more general case of several groups. We first provide some theoretical results for the two associated test statistics and we explain how to derive change-point location estimators. Then, some numerical experiments are given in order to support our claims. Finally, our approach is applied to Hi-C data which are used in molecular biology for better understanding the influence of the chromosomal conformation on the cells functioning.

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arXiv2016

Fast Detection of Block Boundaries in Block Wise Constant Matrices: An Application to HiC data

We propose a novel approach for estimating the location of block boundaries (change-points) in a random matrix consisting of a block wise constant matrix observed in white noise. Our method consists in rephrasing this task as a variable selection issue. We use a penalized least-squares criterion with an $\ell_1$-type penalty for dealing with this issue. We first provide some theoretical results ensuring the consistency of our change-point estimators. Then, we explain how to implement our method in a very efficient way. Finally, we provide some empirical evidence to support our claims and apply our approach to HiC data which are used in molecular biology for better understanding the influence of the chromosomal conformation on the cells functioning.

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arXiv2015

Heritability estimation in high dimensional linear mixed models

Motivated by applications in genetic fields, we propose to estimate the heritability in high dimensional sparse linear mixed models. The heritability determines how the variance is shared between the different random components of a linear mixed model. The main novelty of our approach is to consider that the random effects can be sparse, that is may contain null components, but we do not know neither their proportion nor their positions. The estimator that we consider is strongly inspired by the one proposed by Pirinen et al. (2013), and is based on a maximum likelihood approach. We also study the theoretical properties of our estimator, namely we establish that our estimator of the heritability is $\sqrt{n}$-consistent when both the number of observations $n$ and the number of random effects $N$ tend to infinity under mild assumptions. We also prove that our estimator of the heritability satisfies a central limit theorem which gives as a byproduct a confidence interval for the heritability. Some Monte-Carlo experiments are also conducted in order to show the finite sample performances of our estimator.

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arXiv2015

A robust approach for estimating change-points in the mean of an AR(1) process

We consider the problem of multiple change-point estimation in the mean of a Gaussian AR(1) process. Taking into account the dependence structure does not allow us to use the dynamic programming algorithm, which is the only algorithm giving the optimal solution in the independent case. We propose a robust estimator of the autocorrelation parameter, which is consistent and satisfies a central limit theorem. Then, we propose to follow the classical inference approach, by plugging this estimator in the criteria used for change-points estimation. We show that the asymptotic properties of these estimators are the same as those of the classical estimators in the independent framework. The same plug-in approach is then used to approximate the modified BIC and choose the number of segments. This method is implemented in the R package AR1seg and is available from the Comprehensive R Archive Network (CRAN). This package is used in the simulation section in which we show that for finite sample sizes taking into account the dependence structure improves the statistical performance of the change-point estimators and of the selection criterion.

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arXiv2012

Homogeneity and change-point detection tests for multivariate data using rank statistics

Detecting and locating changes in highly multivariate data is a major concern in several current statistical applications. In this context, the first contribution of the paper is a novel non-parametric two-sample homogeneity test for multivariate data based on the well-known Wilcoxon rank statistic. The proposed two-sample homogeneity test statistic can be extended to deal with ordinal or censored data as well as to test for the homogeneity of more than two samples. The second contribution of the paper concerns the use of the proposed test statistic to perform retrospective change-point analysis. It is first shown that the approach is computationally feasible even when looking for a large number of change-points thanks to the use of dynamic programming. Computable asymptotic $p$-values for the test are then provided in the case where a single potential change-point is to be detected. Compared to available alternatives, the proposed approach appears to be very reliable and robust. This is particularly true in situations where the data is contaminated by outliers or corrupted by noise and where the potential changes only affect subsets of the coordinates of the data.

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