Linear models for multivariate functional data
Issam-Ali Moindjié · theses.fr (ABES) · 2023
In this thesis, we were interested in the problem of predicting a real or categorical variable using multivariate functional variables. In the existing literature, the proposed methods often assume the case of a single domain. This means that each dimension of the multivariate functional variable has the same domain of definition. This assumption limits their use to a limited number of application fields. Indeed, technological advances in data collection and storage have made it possible to observe several functional characteristics, sometimes of different natures, for the same statistical individual. To solve the prediction problem with this type of variables, we proposed two methods inspired by the PLS regression: MFPLS and TMFPLS. The first one is an extension of the PLS algorithm to the case of explanatory multivariate functional data, where the dimensions are potentially defined on different domains. This method can be used for regression and binary classification. The second method: TMFPLS, is a decision tree which can be used for more complex classification tasks (non-linear relationship between the target variable and the predictors, multiclass classification). These methods can be used for a wide range of applications; however, their interpretation becomes difficult when the predictors have numerous dimensions. This is typically the case when many sensors are used to measure a functional variable in several locations. Or more generally, when it comes to repeated functional data. In this case, we presented parsimonious methods based on the fusion penalty, to obtain more interpretable models. Applications on simulated data and real data (EEG, ECG, etc.) have demonstrated the good performance of our methods.