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Theses

Régression isotonique itérée

Abstract : This thesis is part of non parametric univariate regression. Assume that the regression function is of bounded variation then the Jordan’s decomposition ensures that it can be written as the sum of an increasing function and a decreasing function. We propose and analyse a novel estimator which combines the isotonic regression related to the estimation of monotonefunctions and the backfitting algorithm devoted to the estimation of additive models. The first chapter provides an overview of the references related to isotonic regression and additive models. The next chapter is devoted to the theoretical study of iterative isotonic regression. As a first step we show that increasing the number of iterations tends to reproduce the data. Moreover, we manage to identify the individual limits by making a connexion with the general property of isotonicity of projection onto convex cones and deriving another equivalent algorithm based on iterative bias reduction. Finally, we establish the consistency of the estimator.The third chapter is devoted to the practical study of the estimator. As increasing the number of iterations leads to overfitting, it is not desirable to iterate the procedure until convergence. We examine stopping criteria based on adaptations of criteria usually used in the context of linear smoothing methods (AIC, BIC, ...) as well as criteria assuming the knowledge of thenumber of modes of the regression function. As it is observed an interesting behavior of the method when the regression function has breakpoints, we apply the algorithm to CGH-array data where breakopoints detections are of crucial interest. Finally, an application to the estimation of unimodal functions is proposed
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Nicolas Jégou. Régression isotonique itérée. Autre [cs.OH]. Université Rennes 2, 2012. Français. ⟨NNT : 2012REN20048⟩. ⟨tel-00776627⟩

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