Wilhelm, Matthieu

2018-1-10, Tillé, Yves, Qualité, Lionel, Wilhelm, Matthieu

We present a new family of sampling designs in finite population based on the use of chain processes and of multivariate discrete distributions. In Bernoulli sampling, the number of non-selected units between two selected units has a geometric distribution, while, in simple random sampling, it has a negative hypergeometric distribution. We propose to replace these distributions by more general ones, which enables us to include a tuning parameter for the joint inclusion probabilities that have a relatively simple form. An effect of repulsion or attraction can then be added in the selection of the units in such a way that a large set of new designs are defined that include Bernoulli sampling, simple random sampling and systematic sampling. A set of simulations show the interest of the method.

2017, Wilhelm, Matthieu, Qualité, Lionel, Tillé, Yves

A specific family of point processes are introduced that allow to select samples for the purpose of estimating the mean or the integral of a function of a real variable. These processes, called quasi-systematic processes, depend on a tuning parameter $r>0$ that permits to control the likeliness of jointly selecting neighbor units in a same sample. When $r$ is large, units that are close tend to not be selected together and samples are well spread. When $r$ tends to infinity, the sampling design is close to systematic sampling. For all $r > 0$, the first and second-order unit inclusion densities are positive, allowing for unbiased estimators of variance. Algorithms to generate these sampling processes for any positive real value of $r$ are presented. When $r$ is large, the estimator of variance is unstable. It follows that $r$ must be chosen by the practitioner as a trade-off between an accurate estimation of the target parameter and an accurate estimation of the variance of the parameter estimator. The method's advantages are illustrated with a set of simulations.

2015-12-12, Wilhelm, Matthieu, Dedè, Luca, Sangalli, Laura M., Wilhelm, Pierre

We propose a new approach for smoothing on surfaces. More precisely, we aim at estimating functions lying on a surface represented by NURBS, which are geometrical representations commonly used in industrial applications. The estimation is based on the minimization of a penalized least-square functional which is equivalent to solve a 4th-order Partial Differential Equation (PDE). We use Isogeometric Analysis (IGA), which is a method for numerically solve PDE, for the numerical approximation of such surface PDE, leading to an IsoGeometric Smoothing (IGS) method. IGA has the great advantage to use the exact geometrical representation of the surface in the analysis, thus avoiding complex meshing procedures. IGA also provides at least globally $C^1-$continuous basis functions with compact support. We show the performance of the proposed IGS method by means of numerical simulations and we apply it to the estimation of the pressure coefficient, and associated aerodynamic force on a winglet of the SOAR space shuttle.

2014-8-19, Wilhelm, Matthieu

2017-12-20, Tillé, Yves, Wilhelm, Matthieu

In this paper, we first aim to formalize the choice of the sampling design for a particular estimation problem. Next several principles are proposed: randomization, over-representation and restriction. These principles are fundamental to assist in the determination of the most appropriate design. A priori knowledge of the population can be also formalized by modelling the population, which can be helpful when choosing the sampling design. We present a list of sampling designs by specifying their corresponding models and the principles used to derive them. Emphasis is placed on new spatial sampling methods and their related models. A simulation shows the advantages of the different methods.

2016-5-10, Wilhelm, Matthieu, Sangalli, Laura M.

We propose a method for the analysis of data scattered over a spatial irregularly shaped domain and having a distribution within the exponential family. This is a generalized additive model for spatially distributed data. The model is fitted by maximizing a penalized log-likelihood function with a roughness penalty term that involves a differential operator of the spatial field over the domain of interest. Efficient spatial field estimation is achieved resorting to the finite element method, which provides a basis for piecewise polynomial surfaces. The method is illustrated by an application to the study of criminality in the city of Portland, Oregon, USA.

2015-6-25, Wilhelm, Matthieu, Tillé, Yves

We present a family of sampling designs depending on a integer parameter r. Then, simple random sampling is a particular case of this sampling design, namely when r = 1 and the systematic sampling design is the limiting case when r tends to the infinity. For every r > 0, this sampling design has the important property to have first and second order densities which are tractable and positive. Thus, the Horvitz-Thompson estimator is unbiased and the estimator of the variance of the Horvitz-Thompson estimator is also unbiased. This family of sampling design can be used in finite population or on any finite interval of R.

2017, Wilhelm, Matthieu, Tillé, Yves

Cette thèse vise à explorer le concept de répulsion et à l’exploiter afin de développer de nouveaux plans de sondages.
Le premier chapitre est une revue de la littérature qui porte sur l’échantillonnage en population finie. Il est constitué de deux parties distinctes. Dans la première, trois principes visant à guider les praticiens dans leur conception d’un plan de sondage sont préconisés : la maximisation de l’entropie, la surreprésentation et la restriction. Dans la seconde partie, on fait une revue de la littérature concernant l’échantillonnage équilibré ainsi que de ses différentes variantes spatiales.
Le deuxième chapitre de cette thèse porte sur l’échantillonnage en population finie. En cherchant à modéliser l’écart entre deux unités sélectionnées, on obtient une famille de plans de sondage dont on peut ajuster la répulsion. Les cas de tailles fixes et aléatoires sont traités séparément et dans les deux cas, on propose des plans dont les probabilités d’inclusion d’ordre un sont constantes et celles d’ordre deux sont connues sous forme close.
Le troisième chapitre traite un problème similaire mais dans le cas d’un intervalle de la droite réelle. Le plan d’échantillonnage est en fait un processus ponctuel que l’on désire stationnaire et on modélise les distances entre deux occurrences du processus. Lorsque les distances entre deux unités successives sont indépendantes, le processus est de renouvellement et la taille de l’échantillon est en général aléatoire. On peut imposer la taille fixe et, dans ce cas, les intervalles entre deux unités ne sont plus indépendants mais seulement échangeables. Dans les deux cas, de taille fixe et de taille aléatoire, la famille de processus dévelopée englobe les processus les plus basiques, respectivement de Poisson et binomial ainsi que le tirage systématique comme cas limite. Ainsi, on dispose d’une famille de processus dépendant d’un paramètre qui permet de faire varier continument la répulsion.
Finalement, on traite du problème de l’échantillonnage spatial à intensité variable. En particulier, on développe un processus ponctuel répulsif basé sur les processus déterminantaux, que l’on peut échantillonner sur un domaine géométriquement complexe et dont on connaît analytiquement les intensités jointes d’ordre un et deux. L’estimateur d’Horvitz-Thompson est sans biais et on peut estimer sans biais sa variance., In this thesis, we explore the concept of repulsion and exploit it to develop new sampling designs.
The first chapter is a review of the literature about sampling in finite population and consists of two distinct parts. In the first one, we suggest three principles as guidelines for designing a survey: randomization, overrepresentation and restriction. In the second part, we review the literature about balanced sampling and its spatial versions. We intend to give the reader insights about some models and the corresponding optimal sampling design.
The second chapter of this thesis relates to sampling in finite population. By modelling the distribution of the number of units between two selected units, we obtain a family of sampling designs with a tunable repulsion. The cases of fixed and random sample size are considered separately. In both cases, the first order inclusion probabilities are equal and the second order inclusion probabilities are known under closed-form.
The third chapter is about repulsive sampling in an interval of the real line. In continuous spaces, a sampling design is a point process. We aim at imposing the process to be stationary which is similar to imposing equal first order inclusion probabilities in finite population. We focus on the intervals between two successive occurrences of the process, called spacings. If spacings between units are independent, the point process is a renewal process and the sample size is random. We can impose a fixed size sample but the spacings are only exchangeable and not independent. In both cases, the proposed family of point processes encompasses the most basic processes, that is, binomial in the case of fixed sample size and Poisson in the case of binomial sample size; it also includes the systematic sampling as a limiting case. Thus we obtain a family of processes depending on a parameter that allows us to continuously tune the repulsion between selections.
Finally, we consider the problem of developing a repulsive spatial point process with varying prescribed intensity. In particular, we propose a repulsive point process based on Determinantal processes from which we can easily sample on a geometrically complex domain. The first and second order inclusion densities are known under closed-form and are strictly positive. The Horvitz-Thompson estimator is unbiased and we can unbiasedly estimate its variance.

2016-1-14, Wilhelm, Matthieu, Dedè, Luca, Sangalli, Laura M., Wilhelm, Pierre

We propose an Isogeometric approach for smoothing on surfaces, namely estimating a function starting from noisy and discrete measurements. More precisely, we aim at estimating functions lying on a surface represented by NURBS, which are geometrical representations commonly used in industrial applications. The estimation is based on the minimization of a penalized least-square functional. The latter is equivalent to solve a 4th-order Partial Differential Equation (PDE). In this context, we use Isogeometric Analysis (IGA) for the numerical approximation of such surface PDE, leading to an IsoGeometric Smoothing (IGS) method for fitting data spatially distributed on a surface. Indeed, IGA facilitates encapsulating the exact geometrical representation of the surface in the analysis and also allows the use of at least globally C^1-Continuous NURBS basis functions for which the 4th-order PDE can be solved using the standard Galerkin method. We show the performance of the proposed IGS method by means of numerical simulations and we apply it to the estimation of the pressure coefficient, and associated aerodynamic force on a winglet of the SOAR space shuttle.

2014-11-20, Wilhelm, Matthieu