## Section: New Results

### Model selection in Regression and Classification

Participants : Gilles Celeux, Serge Cohen, Jairo Cugliari, Tim Van Erwen, Clément Levrard, Erwan Le Pennec, Pascal Massart, Nelo Molter Magalhaes, Lucie Montuelle, Mohammed Sedki.

Erwan Le Pennec is still working with Serge Cohen (IPANEMA Soleil) on hyperspectral image segmentation based on a spatialized Gaussian Mixture Model. Their scheme is supported by some theoretical investigation and have been applied in pratice with an efficient minimization algorithm combining EM algorithm, dynamic programming and model selection implemented with MIXMOD. Lucie Montuelle is studying extensions of this model that comprise parametric logistic weights and regression mixtures.

Unsupervised segmentation is an issue similar to unsupervised classification with an added spatial aspect. Functional data is acquired on points in a spatial domain and the goal is to segment the domain in homogeneous domain. The range of applications includes hyperspectral images in conservation sciences, fMRi data and all spatialized functional data. Erwan Le Pennec and Lucie Montuelle are focusing on the questions of the way to handle the spatial component from both the theoretical and the practical point of views. They study in particular the choice of the number of clusters. Furthermore, as functional data require heavy computation, they are required to propose numerically efficient algorithms. They have also extend the model to regression mixture.

Lucie Montuelle focused on conditional density estimation by Gaussian mixtures with logistic weights. Using maximum likelihood estimators, a model selection procedure has been applied, supported by a theoretical guarantee. Numerical experiments have been conducted for regression mixtures with parametric logistic weights, using EM and Newton algorithms. This work is available in the research report and a submitted article.

In collaboration with Lucien Birgé (Université Paris 6), Pascal Massart and Nelo Molter Magalhaes define for the algorithm selection problem a new general cross validation procedure based on robust tests, which is an extension of the hold-out defined by Birgé. They get an original procedure based on the Hellinger distance. This procedure is the unique procedure which does not use any contrast function since it does not estimate the risk. They provide theoretical results showing that, under some weak assumptions on the considered statistical methods, the selected estimator satisfies an oracle type inequality. And, they prove that their robust method can be implemented with a sub-quadratic complexity. Simulations show that their estimator performs generally well for estimating a density with different sample sizes and can handle well-known problems, such as histogram or bandwidth selection.

In collaboration with Gérard Biau (Université Paris 6), Clément Levrard and Pascal Massart provide intuitive conditions have been derived for the $k$-means clustering algorithm to achieve its optimal rate of convergence. They can be thought of as margin conditions such as ones introduced by Mammen and Tsybakov in the statistical learning framework. These conditions can be checked in many cases, such as Gaussian mixtures with a known number of components and do not require the underlying distribution to have a density, on the contrary to the previous fast rates conditions introduced in this domain. Moreover, It allows to derive non-asymptotic bounds on the mean squared distortion of the $k$-mean estimator, emphasizing the role played by several other parameters of the quantization issue, such as the smallest distance between optimal codepoints or the excess risk of local minimizers. The influence of these parameters is still in discussion, but some previous results show that some of them are crucial for the minimax results obtained in quantization theory.

Tim van Erven is studying model selection for the long term. When a model selection procedure forms an integrated part of a company’s day-to-day activities, its performance should be measured not on a single day, but on average over a longer period, like for example a year. Taking this long-term perspective, it is possible to aggregate model predictions optimally even when the data probability distribution is so irregular that no statistical guarantees can be given for any individual day separately. He studies the relation between model selection for individual days and for the long term, and how the geometry of the models affects both. This work has potential applications in model aggregation for the forecasting of electrical load consumption at EDF. Together with Jairo Cugliari it has also been applied to improve regional forecasts of electrical load consumption using the fact that the consumption of all regions together must add up to the total consumption over the whole country.

The well-documented and consistent variable selection procedure in model-based cluster analysis and classification, that Cathy Maugis (INSA Toulouse) has designed during her PhD. thesis in select , makes use of stepwise algorithms which are painfully slow in high dimensions. In order to circumvent this drawback, Gilles Celeux and Mohammed Sedki, in collaboration with Cathy Maugis, proposed to sort the variables using a lasso-like penalization adapted to the Gaussian mixture model context. Using this rank to select the variables they avoid the combinatory problem of stepwise procedures. Their algorithm is now tested on several challenging simulated and real data sets, showing encouraging performances.

In collaboration with Jean-Michel Marin (Université de Montpellier) and Olivier Gascuel (LIRMM), Gilles Celeux has started a research aiming to select a short list of models rather a single model. This short list of models is declared to be compatible with the data using a $p$-value derived from the Kullback-Leibler distance between the model and the empirical distribution. And, the Kullback-Leibler distances at hand are estimated trough parametric bootstrap procedures.