Classification with the nearest neighbor rule in general finite dimensional spaces: necessary and sufficient conditions

Given an $n$-sample of random vectors $(X_i,Y_i)_{1 \leq i \leq n}$ whose joint law is unknown, the long-standing problem of supervised classification aims to \textit{optimally} predict the label $Y$ of a given a new observation $X$. In this context, the nearest neighbor rule is a popular flexible and intuitive method in non-parametric situations. Even if this algorithm is commonly used in the machine learning and statistics communities, less is known about its prediction ability in general finite dimensional spaces, especially when the support of the density of the observations is $\mathbb{R}^d$. This paper is devoted to the study of the statistical properties of the nearest neighbor rule in various situations. In particular, attention is paid to the marginal law of $X$, as well as the smoothness and margin properties of the \textit{regression function} $\eta(X) = \mathbb{E}[Y | X]$. We identify two necessary and sufficient conditions to obtain uniform consistency rates of classification and to derive sharp estimates in the case of the nearest neighbor rule. Some numerical experiments are proposed at the end of the paper to help illustrate the discussion.Comment: 53 Pages, 3 figure

Intensity estimation of non-homogeneous Poisson processes from shifted trajectories

This paper considers the problem of adaptive estimation of a non-homogeneous intensity function from the observation of n independent Poisson processes having a common intensity that is randomly shifted for each observed trajectory. We show that estimating this intensity is a deconvolution problem for which the density of the random shifts plays the role of the convolution operator. In an asymptotic setting where the number n of observed trajectories tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Besov balls. Non-linear thresholding in a Meyer wavelet basis is used to derive an adaptive estimator of the intensity. The proposed estimator is shown to achieve a near-minimax rate of convergence. This rate depends both on the smoothness of the intensity function and the density of the random shifts, which makes a connection between the classical deconvolution problem in nonparametric statistics and the estimation of a mean intensity from the observations of independent Poisson processes

Intensity estimation of non-homogeneous Poisson processes from shifted trajectories

In this paper, we consider the problem of estimating nonparametrically a mean pattern intensity λ from the observation of n independent and non-homogeneous Poisson processes N1,…,Nn on the interval [0,1]. This problem arises when data (counts) are collected independently from n individuals according to similar Poisson processes. We show that estimating this intensity is a deconvolution problem for which the density of the random shifts plays the role of the convolution operator. In an asymptotic setting where the number n of observed trajectories tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Besov balls. Non-linear thresholding in a Meyer wavelet basis is used to derive an adaptive estimator of the intensity. The proposed estimator is shown to achieve a near-minimax rate of convergence. This rate depends both on the smoothness of the intensity function and the density of the random shifts, which makes a connection between the classical deconvolution problem in nonparametric statistics and the estimation of a mean intensity from the observations of independent Poisson processes

Classification with the nearest neighbor rule in general finite dimensional spaces

Given an n-sample of random vectors (Xi,Yi)1=i=n whose joint law is unknown, the long-standing problem of supervised classification aims to optimally predict the label Y of a given new observation X. In this context, the k-nearest neighbor rule is a popular flexible and intuitive method in nonparametric situations. Even if this algorithm is commonly used in the machine learning and statistics communities, less is known about its prediction ability in general finite dimensional spaces, especially when the support of the density of the observations is Rd . This paper is devoted to the study of the statistical properties of the k-nearest neighbor rule in various situations. In particular, attention is paid to the marginal law of X, as well as the smoothness and margin properties of the regression function n(X) = E[Y |X]. We identify two necessary and sufficient conditions to obtain uniform consistency rates of classification and derive sharp estimates in the case of the k-nearest neighbor rule. Some numerical experiments are proposed at the end of the paper to help illustrate the discussio

Marche et environnements urbains contrastés. Perspectives internationales et interdisciplinaires

International audienceEn moins de trente ans, les moyens de transport, les services à la mobilité et les manières de se déplacer n’ont cessé de se développer et de se diversifier, pérennisant l’image de la ville comme « espace des flux », modelant les paysages urbains, favorisant une injonction à « être mobile ». Dans ce contexte, la marche fait l’objet d’une attention renouvelée. Une approche ouverte au croisement de la géographie, de la santé publique, de la psychologie et de l’urbanisme s’impose. Si une telle approche permet l’ouverture à d’autres perspectives et la production de connaissances nouvelles issues de cette interdisciplinarité, ces connaissances peuvent également nous conduire à des interventions urbaines et architecturales appropriées, ciblées et sensibles. Ce sont autant de questions que nous désirons développer dans le cadre de ce numéro thématique interdisciplinaire sur le marcheur et son environnement. Trois axes sont proposés où, dans un premier temps, nous nous intéressons aux figures du marcheur d’aujourd’hui, dans une seconde partie, aux conditions de possibilités de la marche et, dans une troisième partie, aux manières d’évaluer la marche selon le point de vue du marcheur

New estimation of Sobol' indices using kernels

In this work, we develop an approach mentioned by da Veiga and Gamboa in 2013. It consists in extending the very interestingpoint of view introduced in \cite{gine2008simple} to estimate general nonlinear integral functionals of a density on the real line, by using empirically a kernel estimator erasing the diagonal terms. Relaxing the positiveness assumption on the kernel and choosing a kernel of order large enough, we are able to prove a central limit theorem for estimating Sobol' indices of any order (the bias is killed thanks to this signed kernel)

Obesity and the microvasculature

Overweight and obesity are thought to significantly influence a person's risk of cardiovascular disease, possibly via its effect on the microvasculature. Retinal vascular caliber is a surrogate marker of microvascular disease and a predictor of cardiovascular events. The aim of this systematic review and meta-analysis was to determine the association between body mass index (BMI) and retinal vascular caliber. Relevant studies were identified by searches of the MEDLINE and EMBASE databases from 1966 to August 2011. Standardized forms were used for data extraction. Among over 44,000 individuals, obese subjects had narrower arteriolar and wider venular calibers when compared with normal weight subjects, independent of conventional cardiovascular risk factors. In adults, a 1 kg/m(2) increase in BMI was associated with a difference of 0.07 μm [95% CI: -0.08; -0.06] in arteriolar caliber and 0.22 μm [95% CI: 0.21; 0.23] in venular caliber. Similar results were found for children. Higher BMI is associated with narrower retinal arteriolar and wider venular calibers. Further prospective studies are needed to examine whether a causative relationship between BMI and retinal microcirculation exists

Third-order Els\"asser moments in axisymmetric MHD turbulence

Incompressible MHD turbulence is investigated under the presence of a uniform magnetic field \bb0. Such a situation is described in the correlation space by a divergence relation which expresses the statistical conservation of the Els\"asser energy flux through the inertial range. The ansatz is made that the development of anisotropy, observed when $B_0$ is strong enough, implies a foliation of space correlation. A direct consequence is the possibility to derive a vectorial law for third-order Els\"asser moments which is parametrized by the intensity of anisotropy. We use the so-called critical balance assumption to fix this parameter and find a unique expression.Comment: 10 pages, 2 figures, will appea