Philippe Vieu

Nonparametric curve estimation from time series

Because of the sheer size and scope of the plastics industry, the title Developments in Plastics Technology now covers an incredibly wide range of subjects or topics. No single volume can survey the whole field in any depth and what follows is, therefore, a series of chapters on selected topics. The topics were selected by us, the editors, because of their immediate relevance to the plastics industry. When one considers the advancements of the plastics processing machinery (in terms of its speed of operation and conciseness of control), it was felt that several chapters should be included which related to the types of control systems used and the correct usage of hydraulics. The importance of using cellular, rubber-modified and engineering-type plastics has had a major impact on the plastics industry and therefore a chapter on each of these subjects has been included. The two remaining chapters are on the characterisation and behaviour of polymer structures, both subjects again being of current academic or industrial interest. Each of the contributions was written by a specialist in that field and to them all, we, the editors, extend our heartfelt thanks, as writing a contribution for a book such as this, while doing a full-time job, is no easy task.

Curves discrimination: a nonparametric functional approach

Abstract A new nonparametric tool for studying the relationship between a curve, considered as a functional predictor, and a categorical response is proposed. This is typically a problem of discrimination, also known as supervised classification, but applied to a sample of curves. Starting from a food industry context and a speech recognition problem, we nonparametrically estimate the posterior probability that an incoming curve is of a given class. A consistent kernel estimator is introduced and its practical performance is pointed out by means of a simulation study. Finally, this method is applied to the above-mentioned data sets.

Parametric modelling of growth curve data: An overview

In the past two decades a parametric multivariate regression modelling approach for analyzing growth curve data has achieved prominence. The approach, which has several advantages over classical analysis-of-variance and general multivariate approaches, consists of postulating, fitting, evaluating, and comparing parametric models for the datas mean structure and covariance structure. This article provides an overview of the approach, using unified terminology and notation. Well-established models and some developed more recently are described, with emphasis given to those models that allow for nonstationarity and for measurement times that differ across subjects and are unequally spaced. Graphical diagnostics that can assist with model postulation and evaluation are discussed, as are more formal methods for fitting and comparing models. Three examples serve to illustrate the methodology and to reveal the relative strengths and weaknesses of the various parametric models.

Editorial: Statistics for Functional Data

Functional data analysis is an active field of research in Statistics. This Special Issue on Statistics for Functional Data contains a selected set of contributions which covers a scope, as wide as possible, of this many-facetted discipline. The diversity of this field of statistics is highlighted by the wide scope of methodological problems discussed in this special issue. Also, the large set of applied scientific disciplines concerned with functional data appears through the numerous curves data set analyzed in these contributions. This introductory paper presents these contributions by emphasizing on how they are taking place in the actual development of statistical methods for analyzing functional data. A special, but not exclusive, place is given to the three more current kinds of problems: factorial analysis of functional data, regression with functional variables and curves classification. The links between functional data analysis and nonparametric statistics deserve a special attention.

Additive prediction and boosting for functional data

Additive model and estimates for regression problems involving functional data are proposed. The impact of the additive methodology for analyzing datasets involving various functional covariates is underlined by comparing its predictive power with those of standard (i.e. non additive) nonparametric functional regression methods. The comparison is made both from a theoretical point of view, and from a real environmental functional dataset. As a by-product, the method is also used for boosting nonparametric functional data analysis even in situations where a single functional covariate is observed. A second functional dataset, coming from spectrometric analysis, illustrates the interest of this functional boosting procedure.

k-Nearest Neighbour method in functional nonparametric regression

The aim of this article is to study the k-nearest neighbour (kNN) method in nonparametric functional regression. We present asymptotic properties of the kNN kernel estimator: the almost-complete convergence and its rate. Then, we illustrate the effectiveness of this method by comparing it with the traditional kernel approach first on simulated datasets and then on a real chemometrical example. We also present in this article an important technical tool which could be useful in many other situations than ours.

An introduction to recent advances in high/infinite dimensional statistics

Abstract The aim of this short contribution is to present the various papers composing this Special Issue on Statistics in HD spaces, by casting them into their bibliographical context through some necessarily short and selected discussion of the current literature.

Functional nonparametric model for time series: a fractal approach for dimension reduction

AbstracIn this paper we propose a functional nonparametric model for time series prediction. The originality of this model consists in using as predictor a continuous set of past values. This time series problem is presented in the general framework of regression estimation from dependent samples with regressor valued in some infinite dimensional semi-normed vectorial space. The curse of dimensionality induced by our approach is overridden by means of fractal dimension considerations. We give asymptotics for a kernel type nonparametric predictor linking the rates of convergence with the fractal dimension of the functional process. Finally, our method has been implemented and applied to some electricity consumption data.

Quadratic errors for nonparametric estimates under dependence

We investigate nonparametric curve estimation (including density, distribution, hazard, conditional density, and regression functions estimation) by kernel methods when the observed data satisfy a strong mixing condition. In a first attempt we show asymptotic equivalence of average square errors, integrated square errors, and mean integrated square errors. These results are extensions to dependent data of several works, in particular of those by Marron and Hardle (1986, J. Multivariate Anal. 20 91-113). Then we give precise asymptotic evaluations of these errors.

Cross-validated estimations in the single-functional index model

The functional index model consists in assuming that a functional explanatory variable acts on a scalar response only through its projection on one functional direction. The main aim of this work consists in estimating the unknown link function and the unknown functional index appearing in such a model. To this end, one focuses on an original cross-validation procedure allowing both estimations to be carried out simultaneously. This cross-validated method has several advantages: optimality property with respect to quadratic distances, ease of implementation and large flexibility for fitting and predicting purposes. One emphasizes the good behaviour in practice of the method. Finally, one discusses how such a single-functional index model can also be seen as a way of computing adaptative semi-metrics in the general frame of nonparametric functional regression.