Laurent Gardes

On kernel smoothing for extremal quantile regression

Nonparametric regression quantiles obtained by inverting a kernel estimator of the conditional distribution of the response are long established in statistics. Attention has been, however, restricted to ordinary quantiles staying away from the tails of the conditional distribution. The purpose of this paper is to extend their asymptotic theory far enough into the tails. We focus on extremal quantile regression estimators of a response variable given a vector of covariates in the general setting, whether the conditional extreme-value index is positive, negative, or zero. Specifically, we elucidate their limit distributions when they are located in the range of the data or near and even beyond the sample boundary, under technical conditions that link the speed of convergence of their (intermediate or extreme) order with the oscillations of the quantile function and a von-Mises property of the conditional distribution. A simulation experiment and an illustration on real data were proposed. The real data are the American electric data where the estimation of conditional extremes is found to be of genuine interest.

Retrieval of Mars surface physical properties from OMEGA hyperspectral images using Regularized Sliced Inverse Regression

Hyperspectral remote sensing, also known as imaging spectroscopy, is a promising space technology regularly selected by agencies with regard to the exploration and observation of planets, to earths geology or to the monitoring of the environment. It allows to collect for each pixel of a scene, the intensity of light energy reflected from planets as it varies across different wavelengths. More than one hundred spectels in the visible and near infra-red are typically recorded, making it possible to observe a continuous spectrum for each image cell. Usually, in space exploration, the analysis of these spectral signatures allows to retrieve the physical, chemical or mineralogical properties of surfaces and of atmospheres that may help to understand the geological and climatological history of planets. We propose in this paper a statistical method to evaluate the physical properties of surface materials on Mars from hyperspectral images collected by the OMEGA instrument aboard the Mars express spacecraft. The approach we develop is based on the estimation of the functional relationship F between some physical parameters and observed spectra. For this purpose, a database of synthetic spectra is generated by a physical radiative transfer model and used to estimate F. The high dimension of spectra is reduced by using Gaussian regularized sliced inverse regression (GRSIR) to overcome the curse of dimensionality and consequently the sensitivity of the inversion to noise (ill-conditioned problems). Compared with a naive spectrum matching approach such as the k-nearest neighbors algorithm, estimates are more accurate and realistic.

Gaussian Regularized Sliced Inverse Regression

Sliced Inverse Regression (SIR) is an effective method for dimension reduction in high-dimensional regression problems. The original method, however, requires the inversion of the predictors covariance matrix. In case of collinearity between these predictors or small sample sizes compared to the dimension, the inversion is not possible and a regularization technique has to be used. Our approach is based on a Fisher Lecture given by R.D. Cook where it is shown that SIR axes can be interpreted as solutions of an inverse regression problem. We propose to introduce a Gaussian prior distribution on the unknown parameters of the inverse regression problem in order to regularize their estimation. We show that some existing SIR regularizations can enter our framework, which permits a global understanding of these methods. Three new priors are proposed leading to new regularizations of the SIR method. A comparison on simulated data as well as an application to the estimation of Mars surface physical properties from hyperspectral images are provided.

Estimating Extreme Quantiles of Weibull Tail Distributions

ABSTRACT We present a new estimator of extreme quantiles dedicated to Weibull tail distributions. This estimate is based on a consistent estimator of the Weibull tail coefficient. This parameter is defined as the regular variation coefficient of the inverse cumulative hazard function. We give conditions in order to obtain the weak consistency and the asymptotic distribution of the extreme quantiles estimator. Its asymptotic as well as its finite sample performances are compared to classical ones.

Functional kernel estimators of large conditional quantiles

We address the estimation of conditional quantiles when the covariate is functional and when the order of the quantiles converges to one as the sample size increases. In a first time, we investigate to what extent these large conditional quantiles can still be estimated through a functional kernel estimator of the conditional survival function. Sufficient conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed estimators. In a second time, basing on these result, a functional Weissman estimator is derived, permitting to estimate large conditional quantiles of arbitrary large order. These results are illustrated on finite sample situations.

A Note on Sliced Inverse Regression with Regularizations

In Li and Yin (2008, Biometrics 64, 124-131), a ridge SIR estimator is introduced as the solution of a minimization problem and computed thanks to an alternating least-squares algorithm. This methodology reveals good performance in practice. In this note, we focus on the theoretical properties of the estimator. It is shown that the minimization problem is degenerated in the sense that only two situations can occur: Either the ridge SIR estimator does not exist or it is zero.

On the estimation of the functional Weibull tail-coefficient

We present a nonparametric family of estimators for the tail index of a Weibull tail-distribution when functional covariate is available. Our estimators are based on a kernel estimator of extreme conditional quantiles. Asymptotic normality of the estimators is proved under mild regularity conditions. Their finite sample performances are illustrated both on simulated and real data.

Nadaraya’s Estimates for Large Quantiles and Free Disposal Support Curves

A new characterization of partial boundaries of a free disposal multivariate support, lying near the true support curve, is introduced by making use of large quantiles of a simple transformation of the underlying multivariate distribution. Pointwise empirical and smoothed estimators of the full and partial support curves are built as extreme sample and smoothed quantiles. The extreme-value theory holds then automatically for the empirical frontiers and we show that some fundamental properties of extreme order statistics carry over to Nadaraya’s estimates of upper quantile-based frontiers. The benefits of the new class of partial boundaries are illustrated through simulated examples and a real data set, and both empirical and smoothed estimates are compared via Monte Carlo experiments. When the transformed distribution is attracted to the Weibull extreme-value type distribution, the smoothed estimator of the full frontier outperforms frankly the sample estimator in terms of both bias and Mean-Squared Error, under optimal bandwidth. In this domain of attraction, Nadaraya’s estimates of extreme quantiles might be superior to the sample versions in terms of MSE although they have a higher bias. However, smoothing seems to be useless in the heavy tailed case.

Functional Kernel Estimators of Conditional Extreme Quantiles

We address the estimation of “extreme” conditional quantiles i.e. when their order converges to one as the sample size increases. Conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed kernel estimators. A Weissman-type estimator and kernel estimators of the conditional tail-index are derived, permitting to estimate extreme conditional quantiles of arbitrary order.

Machine learning techniques for the inversion of planetary hyperspectral images

In this paper, the physical analysis of planetary hyperspectral images is addressed. To deal with high dimensional spaces (image cubes present 256 bands), two methods are proposed. The first method is the support vectors machines regression (SVM-R) which applies the structural risk minimization to perform a non-linear regression. Several kernels are investigated in this work. The second method is the Gaussian regularized sliced inverse regression (GRSIR). It is a two step strategy; the data are map onto a lower dimensional vector space where the regression is performed. Experimental results on simulated data sets have showed that the SVM-R is the most accurate method. However, when dealing with real data sets, the GRSIR gives the most interpretable results.