Modèle linéaire de prédiction fonctionnelle sur données environnementales : choix de modélisation

Bayle, Séverine; Monestiez, Pascal; Nerini, David

Journal de la société française de statistique, Tome 155 (2014) no. 2, pp. 121-137.

L’analyse de données fonctionnelles est devenue ces dernières années un champ d’étude important en statistiques, car de plus en plus de données observées dans différents domaines se trouvent sous forme de courbes (météorologie, économie, ...). Un des outils de l’analyse de données fonctionnelles est le modèle linéaire « pleinement » fonctionnel, qui est utilisé dans le cas où la variable à prédire et la variable prédictive sont toutes les deux des courbes. Ce modèle a fait l’objet de recherches théoriques approfondies, mais les applications l’utilisant restent peu nombreuses à ce jour. Nous proposons dans cet article une démarche méthodologique à travers un exemple d’application de ce modèle sur des profils océanographiques de lumière et de Chlorophylle a. Il est utilisé ici pour prédire des profils de Chlorophylle a à partir des dérivées des profils de luminosité. La démarche méthodologique permet de clarifier les choix de modélisation que nous avons eu à faire pour traiter les profils océanographiques. Les questionnements à travers notre étude de cas concernent entre autres le choix du type et du nombre de fonctions de base à utiliser, le choix de la valeur du paramètre de lissage, ainsi que le critère pour évaluer la qualité de l’ajustement. Nous montrons que l’utilisation du modèle linéaire fonctionnel permet d’obtenir une bonne qualité de reconstruction pour accéder aux variations hautes fréquences des profils de Chlorophylle a à fine échelle.

Functional data analysis (FDA) has become in recent years an important field in statistics, because more data observed in different domains are in the shape of curves (meteorology, economics, linguistics, ...). One tool in FDA is the fully functional linear model, which is used in the particular case where the variable to be predicted and the predictor are both curves. This model has been the subject of extensive theoretical research, but applications using it are few in number to date. We propose in this paper a methodological approach through an application of this model on light and Chlorophyll a oceanographic profiles. It is used here to predict Chlorophyll a profiles from derivatives of light data. The methodological approach helps to clarify modeling choices necessary to treat oceanographic profiles. Questions through our case study include the choice of the type and the number of basis functions to use, the choice of the value of the smoothing parameter and the goodness of fit criterion. We show that the utilisation of the functional linear model provides a good quality of reconstruction to access high frequency variations of Chlorophyll a profiles at fine scale.

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