Title: Functional data and nonparametric modelling: theoretical/methodological/practical aspects

*Prof. Frédéric Ferraty, Toulouse Jean Jaures University, France.*

Summary: High-tech advances (automated medical monitoring system, connected objects, remote sensing, etc) provide data containing some continuum. The continuum may concerns the time but not only. For instance, in Chemometrics, spectrometers produce data where the continuum feature comes from the wavelength dimension. Consequently, a statistical unit may be a curve, a surface or any more complex mathematical object presenting at least one continuum feature. Such data are called functional data, where the word "functional" is the natural mathematical concept for handling continuum. The challenge is simple: extracting relevant information from datasets containing collections of curves, or surfaces or any other complex objects, most of the time combined with standard multivariate variables. All methodologies dealing with such functional data are gathered under the terminology Functional Data Analysis (FDA). The success of this exciting modern area of Statistics is mainly due to its ability to solve important theoretical problems while proposing original and relevant answers to practical issues coming from topical fields of applications (environmetrics, remote sensing, 3D-4D medical imaging, neuroscience, quantitative genetics, particle physics, astronomy, econometrics, etc). In this tutorial we propose mainly to focus on situations when one observes a response (scalar or functional variable) and functional predictor(s). The natural statistical question is very simple: are we able to predict correctly the response from the functional predictor(s) when one has no idea on the relationship between the response and functional predictor(s)? A suitable answer to this important statistical issue is the "functional nonparametric regression". The word "nonparametric" stands for any model requiring very few assumptions with respect to the relationship between the response and the predictor(s); the word "functional" reminds that the model has to handle functional data. So, the aim of this tutorial is to give an extensive overview on this statistical topic. In addition of some theoretical and practical key developments, real datasets complete the presentation in order to illustrate our purpose (benchmark datasets, hyperspectral image, forensic entomology in the context of criminology, etc).

Title: An online application for Functional Data Analysis Based on R

*Prof. Manuel Escabias, University of Granada, Spain.*

Summary: The Functional Data Analysis encompasses a great variety of statistical methods for the analysis of curves, surfaces or any other function that varies continuously. In most cases we have a sample of curves that measure the time evolution of a variable such as temperature or stock market price, but they can also be functions that depend on other magnitudes, as in chemometrics where the spectrum of chemical substances depend on wavelength or in sports sciences where human movement curves are functions of the percentage of a movement cycle. In practice, it is technically impossible to record complete curves and discrete observations are available instead. The analysis of functional data makes a functional treatment of the curves taking into account its continuity, smoothness, etc. Different statistical methods have been adapted to the analysis of functional data: functional principal component analysis, functional regression models (linear and non-linear, generalized, etc.), functional classification methods or functional discriminant analysis, among others.

Most of functional data analysis methods have been programmed in R in the 'fda' package developed by J.O. Ramsay, H. Wickham, S. Graves and G. Hooker in the McGill University of Montreal (Canada) as a result of the book Functional Data Analysis written by J.O. Ramsay and B.W. Silverman in 1997 (Springer), where these statistical methods are described.

Recently, our research group "Modeling and Prediction with functional data" of the Department of Statistics and O.R. of the University of Granada (FQM307) have developed Statfda, an online application for the use of some of the functional data methods based on the 'fda package 'de R. This tool is the result of the research project of Junta de Andalucía "Statistical methods of functional data analysis. Development of a WEB interface for its application (P11-FQM-8068)" . Despite of using functions of R, the application is programmed for the use of functional data analysis methods without the need of knowing the R programming.

This talk aims to show users this application and how it works. The index of the presentation is

- Introduction to functional data analysis.

- Information management: different possibilities of data files.

- Basis expansion of sample curves.

- Functional Data Analysis: Exploratory analysis, Functional PCA, Functional principal component linear regression, Functional principal component logistic regression.

- Results of the analysis: Display by screen (graphics ) and download of results (text).