A new sliced inverse regression method for multivariate response

Coudret, Raphaël; Girard, Sabine; Saracco, Jérôme

doi:10.1016/j.csda.2014.03.006

Cited by 20 publications

(18 citation statements)

References 59 publications

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“…In our experiments, the algorithm has shown fast convergence being a promising alternative to SIR since nowadays most datasets include outliers. Future work would be to extend this setting to a multivariate response following the lead of [34,35].…”

Section: Discussionmentioning

confidence: 99%

Student Sliced Inverse Regression

Chiancone

Forbes

Girard

2017

Computational Statistics & Data Analysis

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International audienceSliced Inverse Regression (SIR) has been extensively used to reduce the dimension of the predictor space before performing regression. SIR is originally a model free method but it has been shown to actually correspond to the maximum likelihood of an inverse regression model with Gaussian errors. This intrinsic Gaussianity of standard SIR may explain its high sensitivity to outliers as observed in a number of studies. To improve robustness, the inverse regression formulation of SIR is therefore extended to non-Gaussian errors with heavy-tailed distributions. Considering Student distributed errors it is shown that the inverse regression remains tractable via an Expectation- Maximization (EM) algorithm. The algorithm is outlined and tested in the presence of outliers, both in simulated and real data, showing improved results in comparison to a number of other existing approaches

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Section: Discussionmentioning

confidence: 99%

Student Sliced Inverse Regression

Chiancone

Forbes

Girard

2017

Computational Statistics & Data Analysis

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“…This dataset is described in Cook [7] and concerns the performance of students in n = 63 Minneapolis schools along with some various social and economic variables. It is studied, among others, in Yin and Bura [34] or more recently in Coudret et al [11]. We follow these authors and consider a q = 4 dimensional response variable Y that consists of the percentages of students in a school scoring above and below average on standardized fourth and sixth grade reading com- We first look at the dimension of the DR subspace.…”

Section: Real Datamentioning

confidence: 99%

“…Yin and Bura [34] propose a moment based dimension reduction for multivariate data. More recently, Coudret et al [11] present another extension of SIR that clusters components of a multivariate response variable Y that are related to the same DR subspace.…”

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confidence: 99%

A mixture model for dimension reduction

Dortet-Bernadet

Gardès

2017

Communications in Statistics - Theory and Methods

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The existence of a Dimension Reduction (DR) subspace is a common assumption in regression analysis when dealing with high-dimensional predictors. The estimation of such a DR subspace has received considerable attention in the past few years, the most popular method being undoubtedly the Sliced Inverse Regression. We propose in this paper a new estimation procedure of the DR subspace by assuming that the joint distribution of the predictor and the response variables is a finite mixture of distributions. The new method is compared through a simulation study to some classical methods.Regression analysis concerns inference on the conditional distribution of a response variable Y ∈ R q given the value X = x of a vector of predictors X ∈ R p . For instance, a classical problem is the nonparametric estimation of the conditional mean function E(Y |X) for which a popular estimator, when the dimension p is not too large, has been proposed by Nadaraya [22] and Watson [30].When the dimension p becomes large, the so-called "curse of dimensionality" problem arises and 1 inference on the conditional distribution of Y given X = x becomes difficult. A common procedure when dealing with a high-dimensional predictor X is to determine a subspace S ⊂ R p , with dim(S) = d ≤ p, that carries all the information that X has about Y . Such a subspace S is called a Dimension Reduction (DR) subspace. It is spanned by the columns of a full rank matrix Γ ∈ R p×d such that X and Y are conditionally independent given Γ t X. A DR subspace always exists since the trivial choice Γ = I p is possible, but does not produce a reduction of dimension. Under minor conditions (see Cook [6]), the intersection of two DR subspaces is still a DR subspace and the intersection of all DR subspaces is called the central subspace. As seen in Li [19], a regression model admitting a central subspace is given by Y = g(Γ t X, ε), where ε is a random value independent of X and g : R d+1 → R q is an arbitrary function.One of the earliest method to estimate the central subspace (i.e. a matrix Γ) is the Sliced Inverse Regression (SIR) procedure introduced by Li [19]. This method is based on the estimation of Var(E(X|Y )) using a set {S h , h = 1, . . . , H} of non-overlapping slices that cover the range of Y .The asymptotic properties of SIR and related methods are derived for instance by Saracco [24,25].The SIR central subspace estimator is motivated in Li [19] by a geometric property of the covariance matrix Var(E(X|Y )). Another way to understand the SIR method is proposed in Cook [8] where Γ is interpreted as a parameter of an inverse regression model. This model is equivalent to assume that, for all h = 1, . . . , H, the conditional distribution of X given Y ∈ S h is a multivariate Gaussian distribution. Considering n independent replications of the random vector (X, Y ), and Szretter and Yohai [28] show that the maximum likelihood estimator of Γ corresponds to the SIR estimator of the central subspace. This inverse regression model is also used by to propose a ...

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“…,b c . Let us note that a similar procedure has been used in [12], Subsection 3.6 to cluster the components of the multivariate response variable Y related to the same e.d.r. spaces.…”

Section: Sample Version: D Unknown Z Is Observed and F Knownmentioning

confidence: 99%

Collaborative sliced inverse regression

Chiancone

Girard

Chanussot

2016

Communications in Statistics - Theory and Methods

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Sliced Inverse Regression (SIR) is an effective method for dimensionality reduction in high-dimensional regression problems. However, the method has requirements on the distribution of the predictors that are hard to check since they depend on unobserved variables. It has been shown that, if the distribution of the predictors is elliptical, then these requirements are satisfied. In case of mixture models, the ellipticity is violated and in addition there is no assurance of a single underlying regression model among the different components. Our approach clusterizes the predictors space to force the condition to hold on each cluster and includes a merging technique to look for different underlying models in the data. A study on simulated data as well as two real applications are provided. It appears that SIR, unsurprisingly, is not capable of dealing with a mixture of Gaussians involving different underlying models whereas our approach is able to correctly investigate the mixture.

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A new sliced inverse regression method for multivariate response

Cited by 20 publications

References 59 publications

Student Sliced Inverse Regression

Student Sliced Inverse Regression

A mixture model for dimension reduction

Collaborative sliced inverse regression

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