An Empirical Process View of Inverse Regression

Portier, François

doi:10.1111/sjos.12209

Cited by 5 publications

(9 citation statements)

References 32 publications

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“…The main idea underlying these methods is that under appropriate assumptions the inverse regression curve E [X|Y ] and its higher moment variant -the columns of the conditional covariance matrix var(X|Y ) -almost surely belong to the minimal SDR. Cumulative slicing estimation (CUME), proposed in [62] and further analysed in [45] from an empirical process viewpoint, aims at recovering the largest possible subspace of the minimal SDR. It is achieved by estimating the conditional expectation of X (and its higher moment variant), conditioning on 'slices' of the target Y , in the form of 1{Y < y}, and then integrating such conditional expectations with respect to y.…”

Section: Sufficient Dimension Reduction and Inverse Methodsmentioning

confidence: 99%

“…, U n . These facts are a known feature of rank based estimators; see for instance [28] in the copula estimation context and [45] in the standard SIR context.…”

Section: Framework and Notationsmentioning

confidence: 99%

“…We start by introducing TIREX1 and TIREX2 matrices parallel to the matrix M CUME defined in (2.3) in our framework, following the integral based methods proposed by [62] (see also [45]). For any right-continuous cumulative distribution function H (be it empirical or not), we shall denote by H − the left-continuous inverse of H,…”

Section: Framework and Notationsmentioning

confidence: 99%

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Tail inverse regression for dimension reduction with extreme response

Aghbalou¹,

Portier²,

Sabourin³

et al. 2021

Preprint

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We consider the problem of dimensionality reduction for prediction of a target Y ∈ R to be explained by a covariate vector X ∈ R p , with a particular focus on extreme values of Y which are of particular concern for risk management. The general purpose is to reduce the dimensionality of the statistical problem through an orthogonal projection on a lower dimensional subspace of the covariate space. Inspired by the sliced inverse regression (SIR) methods, we develop a novel framework (TIREX, Tail Inverse Regression for EXtreme response) relying on an appropriate notion of tail conditional independence in order to estimate an extreme sufficient dimension reduction (SDR) space of potentially smaller dimension than that of a classical SDR space. We prove the weak convergence of tail empirical processes involved in the estimation procedure and we illustrate the relevance of the proposed approach on simulated and real world data.

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Section: Sufficient Dimension Reduction and Inverse Methodsmentioning

confidence: 99%

“…, U n . These facts are a known feature of rank based estimators; see for instance [28] in the copula estimation context and [45] in the standard SIR context.…”

Section: Framework and Notationsmentioning

confidence: 99%

Section: Framework and Notationsmentioning

confidence: 99%

See 1 more Smart Citation

Tail inverse regression for dimension reduction with extreme response

Aghbalou¹,

Portier²,

Sabourin³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the practical data analysis with unknown k, the estimated eigenvalues and the variation of the eigenvectors have been used to estimate k, see for example especially in the context of SIR, see [12,17,18] and the references therein. The magnitude of the eigenvalues indicates the relevance of the corresponding source to model the response.…”

Section: Cov[e(z|y)] =mentioning

confidence: 99%

Sliced average variance estimation for multivariate time series

et al. 2019

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Supervised dimension reduction for time series is challenging as there may be temporal dependence between the response y and the predictors x. Recently a time series version of sliced inverse regression, TSIR, was suggested, which applies approximate joint diagonalization of several supervised lagged covariance matrices to consider the temporal nature of the data. In this paper we develop this concept further and propose a time series version of sliced average variance estimation, TSAVE. As both TSIR and TSAVE have their own advantages and disadvantages, we consider furthermore a hybrid version of TSIR and TSAVE.Based on examples and simulations we demonstrate and evaluate the differences between the three methods and show also that they are superior to apply their iid counterparts to when also using lagged values of the explaining variables as predictors.

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“…when X is written as a function of Y . Several extensions have been developed for PLS and SIR, see Cook et al (2013), Li et al (2007) and Chiancone et al (2017), Coudret et al (2014), Portier (2016) among others or Girard et al (2022) for a review. While the above-mentioned methods adopt the frequentist point of view, there also exist a number of works in the literature based on Bayesian approaches.…”

Section: Introductionmentioning

confidence: 99%

Shrinkage for Extreme Partial Least-Squares

Arbel,

Lorenzo,

Girard

2024

Preprint

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This work focuses on dimension-reduction techniques for modelling conditional extreme values. Specifically, we investigate the idea that extreme values of a response variable can be explained by nonlinear functions derived from linear projections of an input random vector. In this context, the estimation of projection directions is examined, as approached by the Extreme Partial Least Squares (EPLS) method--an adaptation of the original Partial Least Squares (PLS) method tailored to the extreme-value framework. Further, a novel interpretation of EPLS directions as maximum likelihood estimators is introduced, utilizing the von Mises-Fisher distribution applied to hyperballs. The dimension reduction process is enhanced through the Bayesian paradigm, enabling the incorporation of prior information into the projection direction estimation. The maximum a posteriori estimator is derived in two specific cases, elucidating it as a regularization or shrinkage of the EPLS estimator. We also establish its asymptotic behavior as the sample size approaches infinity. A simulation data study is conducted in order to assess the practical utility of our proposed method. This clearly demonstrates its effectiveness even in moderate data problems within high-dimensional settings. Furthermore, we provide an illustrative example of the method's applicability using French farm income data, highlighting its efficacy in real-world scenarios. MSC 2020 subject classification: 62G32, 62H25, 62H12, 62E20.

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An Empirical Process View of Inverse Regression

Cited by 5 publications

References 32 publications

Tail inverse regression for dimension reduction with extreme response

Tail inverse regression for dimension reduction with extreme response

Sliced average variance estimation for multivariate time series

Shrinkage for Extreme Partial Least-Squares

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