Combining parametric and nonparametric approaches for more efficient time series prediction

We introduce a two-step procedure for more efficient nonparametric prediction of a strictly stationary process admitting an ARMA representation. The procedure is based on the estimation of the ARMA representation, followed by a nonparametric regression where the ARMA residuals are used as explanatory variables. Compared to standard nonparametric regression methods, the number of explanatory variables can be reduced because our approach exploits the linear dependence of the process. We establish consistency and asymptotic normality results for our estimator. A Monte Carlo study and an empirical application on stock market indices suggest that significant gains can be achieved with our approach.ARMA representation; noisy data; Nonparametric regression; optimal prediction

Régression et prédiction non-paramétrique spatiale

International audienceNous nous intéressons à l'estimation de la fonction de régression r(x)=E\left(Y_{\mathbfu}|X_{\mathbfu}=x\right) à partir d'observations d'un processus \left\{ Z_{\mathbfi}=\left(X_{\mathbfi},\ Y_{\mathbfi}\right),\,\mathbfi\in\mathbbZ^N\right\}. On suppose que les variables Z_{\mathbfi}$sont de même distribution que$Z=(X,Y)$, où$Y$est une variable réelle, intégrable et$X$un vecteur aléatoire à valeurs dans un espace séparable$\mathcalE$muni (éventuellement de dimension infinie). Dans ce travail, la convergence nos estimateurs est étudiée sous conditions de mélange à partir d'observations dans une région rectangulaire de$\mathbbZ^N$. Nous illustrerons nos résultats par des simulations. L'application de nos méthodes à la prédiction spatiale sera également abordée

k-nearest neighbors prediction and classification for spatial data

We propose a nonparametric predictor and a supervised classification based on the regression function estimate of a spatial real variable using k-nearest neighbors method (k-NN). Under some assumptions, we establish almost complete or sure convergence of the proposed estimates which incorporate a spatial proximity between observations. Numerical results on simulated and real fish data illustrate the behavior of the given predictor and classification method

Spatial mode estimation for functional random fields with application to bioturbation problem

This work provides a useful tool to study the effects of bioturbation on the distribution of oxygen within sediments. We propose here heterogeneity measurements based on functional spatial mode. To obtain the mode, one usually needs to estimate the spatial probability density. The approach considered here consists in looking each observation as a curve that represents the history of the oxygen concentration at a fixed pixel

KERNEL SPATIAL DENSITY ESTIMATION IN INFINITE DIMENSION SPACE

In this paper, we propose a nonparametric estimation of the spatial density of a functional stationary random field. This later is with values in some infinite dimensional space and admitted a density with respect to some reference measure. The weak and strong consistencies of the estimator are shown and rates of convergence are given. Special attention is paid to the links between the probabilities of small balls in the concerned infinite dimensional space and the rates of convergence. The practical use and the behavior of the estimator are illustrated through some simulations and a real data application

KERNEL REGRESSION ESTIMATION FOR SPATIAL FUNCTIONAL RANDOM VARIABLES

Given a spatial random process (Xi; Yi) 2 E R; i 2 ZN , we investigate a nonparametric estimate of the conditional expectation of the real random variable Yi given the functional random field Xi valued in a semi-metric space E. The weak and strong consistencies of the estimate are shown and almost sure rates of convergence are given. Special attention is paid to apply the regression estimate introduced to spatial prediction problems

Uncovering Data Across Continua: An Introduction to Functional Data Analysis

In a world increasingly awash with data, the need to extract meaningful insights from data has never been more crucial. Functional Data Analysis (FDA) goes beyond traditional data points, treating data as dynamic, continuous functions, capturing ever-changing phenomena nuances. This article introduces FDA, merging statistics with real-world complexity, ideal for those with mathematical skills but no FDA background

Fusion regression methods with repeated functional data

Linear regression and classification methods with repeated functional data are considered. For each statistical unit in the sample, a real-valued parameter is observed over time under different conditions. Two regression methods based on fusion penalties are presented. The first one is a generalization of the variable fusion methodology based on the 1-nearest neighbor. The second one, called group fusion lasso, assumes some grouping structure of conditions and allows for homogeneity among the regression coefficient functions within groups. A finite sample numerical simulation and an application on EEG data are presented