Dimension reductio via Sliced Inverse Regression : ideas and extensions

Alessandro Chiancone 1, 2
2 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
Abstract : This thesis proposes three extensions of Sliced Inverse Regression namely: Collaborative SIR, Student SIR and Knockoff SIR.One of the weak points of SIR is the impossibility to check if the Linearity Design Condition (LDC) holds. It is known that if X follows an elliptic distribution thecondition holds true, in case of a mixture of elliptic distributions there are no guaranties that the condition is satisfied globally, but locally holds. Starting from this consideration an extension is proposed. Given the predictor variable X, Collaborative SIR performs initially a clustering. In each cluster, SIR is applied independently. The result from each component collaborates to give the final solution.Our second contribution, Student SIR, comes from the need to robustify SIR. Since SIR is based on the estimation of the covariance, and contains a PCA step, it is indeed sensitive to noise. To extend SIR, an approach based on a inverse formulation of SIR proposed by R.D. Cook has been used.Finally Knockoff SIR is an extension of SIR to perform variable selection and give sparse solution that has its foundations in a recently published paper by R. F. Barber and E. J. Candès that focuses on the false discovery rate in the regression framework. The underlying idea of this paper is to construct copies of the original variables that have some properties. It is shown that SIR is robust to this copies and a strategy is proposed to use this result for variable selection and to generate sparse solutions.
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Alessandro Chiancone. Dimension reductio via Sliced Inverse Regression : ideas and extensions. Complex Variables [math.CV]. Université Grenoble Alpes, 2016. English. ⟨NNT : 2016GREAM051⟩. ⟨tel-01571824v2⟩

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