Conditional quantile estimation through optimal quantization

In this paper, we use quantization to construct a nonparametric estimator of conditional quantiles of a scalar response $Y$ given a d-dimensional vector of covariates $X$. First we focus on the population level and show how optimal quantization of $X$, which consists in discretizing $X$ by projecting it on an appropriate grid of $N$ points, allows to approximate conditional quantiles of $Y$ given $X$. We show that this is approximation is arbitrarily good as $N$ goes to infinity and provide a rate of convergence for the approximation error. Then we turn to the sample case and define an estimator of conditional quantiles based on quantization ideas. We prove that this estimator is consistent for its fixed-$N$ population counterpart. The results are illustrated on a numerical example. Dominance of our estimators over local constant/linear ones and nearest neighbor ones is demonstrated through extensive simulations in the companion paper Charlier et al.(2014b)

ClustGeo: an R package for hierarchical clustering with spatial constraints

In this paper, we propose a Ward-like hierarchical clustering algorithm including spatial/geographical constraints. Two dissimilarity matrices $D_0$ and $D_1$ are inputted, along with a mixing parameter $\alpha \in [0,1]$. The dissimilarities can be non-Euclidean and the weights of the observations can be non-uniform. The first matrix gives the dissimilarities in the "feature space" and the second matrix gives the dissimilarities in the "constraint space". The criterion minimized at each stage is a convex combination of the homogeneity criterion calculated with $D_0$ and the homogeneity criterion calculated with $D_1$. The idea is then to determine a value of $\alpha$ which increases the spatial contiguity without deteriorating too much the quality of the solution based on the variables of interest i.e. those of the feature space. This procedure is illustrated on a real dataset using the R package ClustGeo

Multivariate Analysis of Mixed Data: The R Package PCAmixdata

Mixed data arise when observations are described by a mixture of numerical and categorical variables. The R package PCAmixdata extends standard multivariate analysis methods to incorporate this type of data. The key techniques/methods included in the package are principal component analysis for mixed data (PCAmix), varimax-like orthogonal rotation for PCAmix, and multiple factor analysis for mixed multi-table data. This paper gives a synthetic presentation of the three algorithms with details to help the user understand graphical and numerical outputs of the corresponding R functions. The three main methods are illustrated on a real dataset composed of four data tables characterizing living conditions in different municipalities in the Gironde region of southwest France

Conditional Spatial Quantile: Characterization and Nonparametric Estimation

Conditional quantiles are required in various economic, biomedical or industrial problems. Lack of objective basis for ordering multivariate observations is a major problem in extending the notion of quantiles or conditional quantiles (also called regression quantiles) in a multidimensional setting. We first recall some characterizations of the unconditional spatial quantiles and the corresponding estimators. Then, we consider the conditional case. In this work, we focus our study on the geometric (or spatial) notion of quantiles introduced by Chaudhuri (1992a, 1996). We generalize, in the conditional framework, the Theorem 2.1.2 of Chaudhuri (1996), and we present algorithms allowing the calculation of the unconditional and conditional spatial quantile estimators. Finally, these various concepts are illustrated using simulated data.Conditional Spatial Quantile, Contours, Kernel Estimators, Spatial Quantile

Estimation récursive en régression inverse par tranche (sliced inverse regression)

International audienceDans cette communication, nous nous intéressons à la méthode SIR (Sliced Inverse Regression, que l'on peut traduire par régression inverse par tranches) qui permet d'estimer le paramètre $\theta$ dans un modèle semi-paramétrique de régression du type $y=f(x'\theta,\varepsilon)$ sans avoir à estimer le paramètre fonctionnel $f$ ni à spécifier la loi de l'erreur $\varepsilon$. Nous proposons un estimateur récursif de la direction de $\theta$ dans le cas particulier où l'on considère $H=2$ tranches. Nous donnons des propriétés asymptotiques de cet estimateur (convergence et normalité asymptotique). Nous illustrons aussi sur des simulations le bon comportement numérique de la méthode proposée

Une solution analytique pour la rotation planaire en Analyse Factorielle des Correspondances Multiples

International audienceL'Analyse en Composantes Principales (ACP) et l'Analyse Factorielle des Correspondances Multiples (AFCM) sont respectivement deux méthodes de description statistique multidimensionnelle de données quantitatives et qualitatives. Une rotation peut ensuite être appliquée à la matrice des scores des composantes principales. La définition d'un critère de rotation permet alors d'obtenir une structure simple, facilitant ainsi l'interprétation des résultats. Une solution analytique en deux dimensions a été proposée pour le critère varimax en ACP. Nous proposons ici une solution analytique en deux dimensions pour la rotation en AFCM utilisant un critère inspiré de varimax et basé sur la notion de rapport de corrélation

Conditional Quantile Estimation based on Optimal Quantization: from Theory to Practice

International audienceSmall-sample properties of a nonparametric estimator of conditional quantiles based on optimal quantization, that was recently introduced (J. Statist. Plann. Inference, 156, 14–30, 2015), are investigated. More precisely, (i) the practical implementation of this estimator is discussed (by proposing in particular a method to properly select the corresponding smoothing parameter, namely the number of quantizers) and (ii) its finite- sample performances are compared to those of classical competitors. Monte Carlo studies reveal that the quantization-based estimator competes well in all cases and sometimes dominates its competitors, particularly when the regression function is quite complex. A real data set is also treated. While the main focus is on the case of a univariate covariate, simulations are also conducted in the bivariate case

Mechanical loss in state-of-the-art amorphous optical coatings

We present the results of mechanical characterizations of many different high-quality optical coatings made of ion-beam-sputtered titania-doped tantala and silica, developed originally for interferometric gravitational-wave detectors. Our data show that in multi-layer stacks (like high-reflection Bragg mirrors, for example) the measured coating dissipation is systematically higher than the expectation and is correlated with the stress condition in the sample. This has a particular relevance for the noise budget of current advanced gravitational-wave interferometers, and, more generally, for any experiment involving thermal-noise limited optical cavities.Comment: 31 pages, 14 figure