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Christophe Giraud

Publications and source records attributed to Christophe Giraud.

32 records · Page 2Linked to original sources
arXiv2017

Estimation of species relative abundances and habitat preferences using opportunistic data

We develop a new statistical procedure to monitor, with opportunist data, relative species abundances and their respective preferences for dierent habitat types. Following Giraud et al. (2015), we combine the opportunistic data with some standardized data in order to correct the bias inherent to the opportunistic data collection. Our main contributions are (i) to tackle the bias induced by habitat selection behaviors, (ii) to handle data where the habitat type associated to each observation is unknown, (iii) to estimate probabilities of selection of habitat for the species. As an illustration, we estimate common bird species habitat preferences and abundances in the region of Aquitaine (France).

stat.AP↗
arXiv2016

PECOK: a convex optimization approach to variable clustering

The problem of variable clustering is that of grouping similar components of a $p$-dimensional vector $X=(X_{1},\ldots,X_{p})$, and estimating these groups from $n$ independent copies of $X$. When cluster similarity is defined via $G$-latent models, in which groups of $X$-variables have a common latent generator, and groups are relative to a partition $G$ of the index set $\{1, \ldots, p\}$, the most natural clustering strategy is $K$-means. We explain why this strategy cannot lead to perfect cluster recovery and offer a correction, based on semi-definite programing, that can be viewed as a penalized convex relaxation of $K$-means (PECOK). We introduce a cluster separation measure tailored to $G$-latent models, and derive its minimax lower bound for perfect cluster recovery. The clusters estimated by PECOK are shown to recover $G$ at a near minimax optimal cluster separation rate, a result that holds true even if $K$, the number of clusters, is estimated adaptively from the data. We compare PECOK with appropriate corrections of spectral clustering-type procedures, and show that the former outperforms the latter for perfect cluster recovery of minimally separated clusters.

math.ST↗
arXiv2015

Aggregation of predictors for nonstationary sub-linear processes and online adaptive forecasting of time varying autoregressive processes

In this work, we study the problem of aggregating a finite number of predictors for nonstationary sub-linear processes. We provide oracle inequalities relying essentially on three ingredients: (1) a uniform bound of the $\ell^1$ norm of the time varying sub-linear coefficients, (2) a Lipschitz assumption on the predictors and (3) moment conditions on the noise appearing in the linear representation. Two kinds of aggregations are considered giving rise to different moment conditions on the noise and more or less sharp oracle inequalities. We apply this approach for deriving an adaptive predictor for locally stationary time varying autoregressive (TVAR) processes. It is obtained by aggregating a finite number of well chosen predictors, each of them enjoying an optimal minimax convergence rate under specific smoothness conditions on the TVAR coefficients. We show that the obtained aggregated predictor achieves a minimax rate while adapting to the unknown smoothness. To prove this result, a lower bound is established for the minimax rate of the prediction risk for the TVAR process. Numerical experiments complete this study. An important feature of this approach is that the aggregated predictor can be computed recursively and is thus applicable in an online prediction context.

math.ST↗
arXiv2015

Capitalising on Opportunistic Data for Monitoring Species Relative Abundances

With the internet, a massive amount of information on species abundance can be collected under citizen science programs. However, these data are often difficult to use directly in statistical inference, as their collection is generally opportunistic, and the distribution of the sampling effort is often not known. In this paper, we develop a general statistical framework to combine such "opportunistic data" with data collected using schemes characterized by a known sampling effort. Under some structural assumptions regarding the sampling effort and detectability, our approach allows to estimate the relative abundance of several species in different sites. It can be implemented through a simple generalized linear model. We illustrate the framework with typical bird datasets from the Aquitaine region, south-western France. We show that, under some assumptions, our approach provides estimates that are more precise than the ones obtained from the dataset with a known sampling effort alone. When the opportunistic data are abundant, the gain in precision may be considerable, especially for the rare species. We also show that estimates can be obtained even for species recorded only in the opportunistic scheme. Opportunistic data combined with a relatively small amount of data collected with a known effort may thus provide access to accurate and precise estimates of quantitative changes in relative abundance over space and/or time.

stat.AP↗
arXiv2012

High-dimensional regression with unknown variance

We review recent results for high-dimensional sparse linear regression in the practical case of unknown variance. Different sparsity settings are covered, including coordinate-sparsity, group-sparsity and variation-sparsity. The emphasis is put on non-asymptotic analyses and feasible procedures. In addition, a small numerical study compares the practical performance of three schemes for tuning the Lasso estimator and some references are collected for some more general models, including multivariate regression and nonparametric regression.

math.ST↗
arXiv2012

Graph selection with GGMselect

Applications on inference of biological networks have raised a strong interest in the problem of graph estimation in high-dimensional Gaussian graphical models. To handle this problem, we propose a two-stage procedure which first builds a family of candidate graphs from the data, and then selects one graph among this family according to a dedicated criterion. This estimation procedure is shown to be consistent in a high-dimensional setting, and its risk is controlled by a non-asymptotic oracle-like inequality. The procedure is tested on a real data set concerning gene expression data, and its performances are assessed on the basis of a large numerical study. The procedure is implemented in the R-package GGMselect available on the CRAN.

math.ST↗
arXiv2011

A pseudo-RIP for multivariate regression

We give a suitable RI-Property under which recent results for trace regression translate into strong risk bounds for multivariate regression. This pseudo-RIP is compatible with the setting $n < p$.

math.ST↗
arXiv2011

Estimator selection in the Gaussian setting

We consider the problem of estimating the mean $f$ of a Gaussian vector $Y$ with independent components of common unknown variance $σ^{2}$. Our estimation procedure is based on estimator selection. More precisely, we start with an arbitrary and possibly infinite collection $\FF$ of estimators of $f$ based on $Y$ and, with the same data $Y$, aim at selecting an estimator among $\FF$ with the smallest Euclidean risk. No assumptions on the estimators are made and their dependencies with respect to $Y$ may be unknown. We establish a non-asymptotic risk bound for the selected estimator. As particular cases, our approach allows to handle the problems of aggregation and model selection as well as those of choosing a window and a kernel for estimating a regression function, or tuning the parameter involved in a penalized criterion. We also derive oracle-type inequalities when $\FF$ consists of linear estimators. For illustration, we carry out two simulation studies. One aims at comparing our procedure to cross-validation for choosing a tuning parameter. The other shows how to implement our approach to solve the problem of variable selection in practice.

math.ST↗
arXiv2011

Low rank Multivariate regression

We consider in this paper the multivariate regression problem, when the target regression matrix $A$ is close to a low rank matrix. Our primary interest in on the practical case where the variance of the noise is unknown. Our main contribution is to propose in this setting a criterion to select among a family of low rank estimators and prove a non-asymptotic oracle inequality for the resulting estimator. We also investigate the easier case where the variance of the noise is known and outline that the penalties appearing in our criterions are minimal (in some sense). These penalties involve the expected value of the Ky-Fan quasi-norm of some random matrices. These quantities can be evaluated easily in practice and upper-bounds can be derived from recent results in random matrix theory.

math.ST↗
arXiv2010

Atomicity Improvement for Elliptic Curve Scalar Multiplication

In this paper we address the problem of protecting elliptic curve scalar multiplication implementations against side-channel analysis by using the atomicity principle. First of all we reexamine classical assumptions made by scalar multiplication designers and we point out that some of them are not relevant in the context of embedded devices. We then describe the state-of-the-art of atomic scalar multiplication and propose an atomic pattern improvement method. Compared to the most efficient atomic scalar multiplication published so far, our technique shows an average improvement of up to 10.6%.

cs.CR↗
arXiv2009

Gaussian model selection with an unknown variance

Let $Y$ be a Gaussian vector whose components are independent with a common unknown variance. We consider the problem of estimating the mean $μ$ of $Y$ by model selection. More precisely, we start with a collection $\mathcal{S}=\{S_m,m\in\mathcal{M}\}$ of linear subspaces of $\mathbb{R}^n$ and associate to each of these the least-squares estimator of $μ$ on $S_m$. Then, we use a data driven penalized criterion in order to select one estimator among these. Our first objective is to analyze the performance of estimators associated to classical criteria such as FPE, AIC, BIC and AMDL. Our second objective is to propose better penalties that are versatile enough to take into account both the complexity of the collection $\mathcal{S}$ and the sample size. Then we apply those to solve various statistical problems such as variable selection, change point detections and signal estimation among others. Our results are based on a nonasymptotic risk bound with respect to the Euclidean loss for the selected estimator. Some analogous results are also established for the Kullback loss.

math.ST↗
arXiv2008

Estimation of Gaussian graphs by model selection

We investigate in this paper the estimation of Gaussian graphs by model selection from a non-asymptotic point of view. We start from a n-sample of a Gaussian law P_C in R^p and focus on the disadvantageous case where n is smaller than p. To estimate the graph of conditional dependences of P_C, we introduce a collection of candidate graphs and then select one of them by minimizing a penalized empirical risk. Our main result assess the performance of the procedure in a non-asymptotic setting. We pay a special attention to the maximal degree D of the graphs that we can handle, which turns to be roughly n/(2 log p).

math.ST↗
arXiv2007

Mixing Least-Squares Estimators when the Variance is Unknown

We propose a procedure to handle the problem of Gaussian regression when the variance is unknown. We mix least-squares estimators from various models according to a procedure inspired by that of Leung and Barron (2007). We show that in some cases the resulting estimator is a simple shrinkage estimator. We then apply this procedure in various statistical settings such as linear regression or adaptive estimation in Besov spaces. Our results provide non-asymptotic risk bounds for the Euclidean risk of the estimator.

math.ST↗