This paper is about variable selection with the random forests algorithm in presence of correlated predictors. In high-dimensional regression or classification frameworks, variable selection is a difficult task, that becomes even more challenging in the presence of highly correlated predictors. Firstly we provide a theoretical study of the permutation importance measure for an additive regression model. This allows us to describe how the correlation between predictors impacts the permutation importance. Our results motivate the use of the Recursive Feature Elimination (RFE) algorithm for variable selection in this context. This algorithm recursively eliminates the variables using permutation importance measure as a ranking criterion. Next various simulation experiments illustrate the efficiency of the RFE algorithm for selecting a small number of variables together with a good prediction error. Finally, this selection algorithm is tested on the Landsat Satellite data from the UCI Machine Learning Repository.

The relationship between asthma and obesity appears to be quite complex. The aim of this study was to assess the effect of excess weight on asthma control evolution in a cohort of asthmatics. A prospective database was set up, which enrolled adult asthmatics with persistent (mild, moderate or severe) asthma. The control of asthma was defined as a binary variable, acceptable or unacceptable. In order to evaluate the effect of body mass index (BMI; <25 or > or =25), data were analysed using a continuous time homogeneous Markov model in which the forces ruling the transition between the two health states were estimated. The following confounding covariates were also evaluated in the model: severity of asthma, current treatment with oral corticosteroids (OCS) and history of OCS over the year preceding inclusion. About 406 asthmatics were included who made a total of 1639 consultations; the median length of follow up was 182 days. Using a univariate model, overweight patients had a lower risk of transiting from the unacceptable to the acceptable health state (RR = 0.45; P < 0.01). The effect of weight remained significant (RR = 0.53; P < 0.01) in the multivariate model including the other covariates. Moreover, transition probabilities stabilized more rapidly for patients with BMI < 25 (200 vs 300 days). In this study, we thus demonstrated that there is an association between excess weight and transition from unacceptable to acceptable control. Because control of asthma clearly drives asthma management, this finding has consequences for defining original new strategies for managing asthma in overweight patients.

The selection of grouped variables using the random forest algorithm is considered. First a new importance measure adapted for groups of variables is proposed. Theoretical insights into this criterion are given for additive regression models. Second, an original method for selecting functional variables based on the grouped variable importance measure is developed. Using a wavelet basis, it is proposed to regroup all of the wavelet coefficients for a given functional variable and use a wrapper selection algorithm with these groups. Various other groupings which take advantage of the frequency and time localization of the wavelet basis are proposed. An extensive simulation study is performed to illustrate the use of the grouped importance measure in this context. The method is applied to a real life problem coming from aviation safety.