An Inverse‐regression Method of Dependent Variable Transformation for Dimension Reduction with Non‐linear Confounding

Many model‐free dimension reduction methods have been developed for high‐dimensional regression data but have not paid much attention on problems with non‐linear confounding. In this paper, we propose an inverse‐regression method of dependent variable transformation for detecting the presence of non‐linear confounding. The benefit of using geometrical information from our method is highlighted. A ratio estimation strategy is incorporated in our approach to enhance the interpretation of variable selection. This approach can be implemented not only in principal Hessian directions (PHD) but also in other recently developed dimension reduction methods. Several simulation examples that are reported for illustration and comparisons are made with sliced inverse regression and PHD in ignorance of non‐linear confounding. An illustrative application to one real data is also presented.     

Gaussian Regularized Sliced Inverse Regression

Sliced Inverse Regression (SIR) is an effective method for dimension reduction in high-dimensional regression problems. The original method, however, requires the inversion of the predictors covariance matrix. In case of collinearity between these predictors or small sample sizes compared to the dimension, the inversion is not possible and a regularization technique has to be used. Our approach is based on a Fisher Lecture given by R.D. Cook where it is shown that SIR axes can be interpreted as solutions of an inverse regression problem. We propose to introduce a Gaussian prior distribution on the unknown parameters of the inverse regression problem in order to regularize their estimation. We show that some existing SIR regularizations can enter our framework, which permits a global understanding of these methods. Three new priors are proposed leading to new regularizations of the SIR method. A comparison on simulated data as well as an application to the estimation of Mars surface physical properties from hyperspectral images are provided.     

Shrinkage inverse regression estimation for model-free variable selection

Summary.  The family of inverse regression estimators that was recently proposed by Cook and Ni has proven effective in dimension reduction by transforming the high dimensional predictor vector to its low dimensional projections. We propose a general shrinkage estimation strategy for the entire inverse regression estimation family that is capable of simultaneous dimension reduction and variable selection. We demonstrate that the new estimators achieve consistency in variable selection without requiring any traditional model, meanwhile retaining the root n estimation consistency of the dimension reduction basis. We also show the effectiveness of the new estimators through both simulation and real data analysis.     

Sufficient dimension reduction in regressions through cumulative Hessian directions

To reduce the predictors dimension without loss of information on the regression, we develop in this paper a sufficient dimension reduction method which we term cumulative Hessian directions. Unlike many other existing sufficient dimension reduction methods, the estimation of our proposal avoids completely selecting the tuning parameters such as the number of slices in slicing estimation or the bandwidth in kernel smoothing. We also investigate the asymptotic properties of our proposal when the predictors dimension diverges. Illustrations through simulations and an application are presented to evidence the efficacy of our proposal and to compare it with existing methods.     

A mixture model for dimension reduction

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. In this paper, we propose 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.     

Application of the Bootstrap Approach to the Choice of Dimension and the α Parameter in the SIRα Method

To reduce the dimensionality of regression problems, sliced inverse regression approaches make it possible to determine linear combinations of a set of explanatory variables X related to the response variable Y in general semiparametric regression context. From a practical point of view, the determination of a suitable dimension (number of the linear combination of X) is important. In the literature, statistical tests based on the nullity of some eigenvalues have been proposed. Another approach is to consider the quality of the estimation of the effective dimension reduction (EDR) space. The square trace correlation between the true EDR space and its estimate can be used as goodness of estimation. In this article, we focus on the SIR_α method and propose a naïve bootstrap estimation of the square trace correlation criterion. Moreover, this criterion could also select the α parameter in the SIR_α method. We indicate how it can be used in practice. A simulation study is performed to illustrate the behavior of this approach.     

Adaptive nonparametric estimation in the presence of dependence

We consider nonparametric estimation problems in the presence of dependent data, notably nonparametric regression with random design and nonparametric density estimation. The proposed estimation procedure is based on a dimension reduction. The minimax optimal rate of convergence of the estimator is derived assuming a sufficiently weak dependence characterised by fast decreasing mixing coefficients. We illustrate these results by considering classical smoothness assumptions. However, the proposed estimator requires an optimal choice of a dimension parameter depending on certain characteristics of the function of interest, which are not known in practice. The main issue addressed in our work is an adaptive choice of this dimension parameter combining model selection and Lepski's method. It is inspired by the recent work of Goldenshluger and Lepski [(2011), ‘Bandwidth Selection in Kernel Density Estimation: Oracle Inequalities and Adaptive Minimax Optimality’, The Annals of Statistics, 39, 1608–1632]. We show that this data-driven estimator can attain the lower risk bound up to a constant provided a fast decay of the mixing coefficients.     

Special Invited Paper: Dimension Reduction and Visualization in Discriminant Analysis (with discussion)

This paper discusses visualization methods for discriminant analysis. It does not address numerical methods for classification per se, but rather focuses on graphical methods that can be viewed as pre-processors, aiding the analyst's understanding of the data and the choice of a final classifier. The methods are adaptations of recent results in dimension reduction for regression, including sliced inverse regression and sliced average variance estimation. A permutation test is suggested as a means of determining dimension, and examples are given throughout the discussion.     

Rank reducible varying coefficient model

We propose in this article a novel dimension reduction method for varying coefficient models. The proposed method explores the rank reducible structure of those varying coefficients, hence, can do dimension reduction and semiparametric estimation, simultaneously. As a result, the new method not only improves estimation accuracy but also facilitates practical interpretation. To determine the structure dimension, a consistent BIC criterion is developed. Numerical experiments are also presented.     

Sufficient dimension reduction and prediction through cumulative slicing PFC

In this article, a new method named cumulative slicing principle fitted component (CUPFC) model is proposed to conduct sufficient dimension reduction and prediction in regression. Based on the classical PFC methods, the CUPFC avoids selecting some parameters such as the specific basis function form or the number of slices in slicing estimation. We develop the estimator of the central subspace in the CUPFC method under three error-term structures and establish its consistency. The simulations investigate the effectiveness of the new method in prediction and reduction estimation with other competitors. The results indicate that the new proposed method generally outperforms the existing PFC methods no matter how the predictors are truly related to the response. The application to real data also verifies the validity of the proposed method.     

An example of applying the asymptotic quasi-likelihood to dimension estimation for random spatial patterns

With reference to a specific example of a random spatial fractal and the modified box-counting method of dimension estimation, this paper aims to examine firstly the estimation of pointwise dimension via modification of the box-counting procedure, secondly the regression inspired estimation procedure, including generalised least squares and, finally, to develop a new estimation procedure – the asymptotic quasi-likelihood method – for the estimation of pointwise dimension. The main focus is on practicality – to arrive at an estimation method which is easy to use and robust.     

Analysis of Double Single Index Models

Motivated from problems in canonical correlation analysis, reduced rank regression and sufficient dimension reduction, we introduce a double dimension reduction model where a single index of the multivariate response is linked to the multivariate covariate through a single index of these covariates, hence the name double single index model. Because nonlinear association between two sets of multivariate variables can be arbitrarily complex and even intractable in general, we aim at seeking a principal one‐dimensional association structure where a response index is fully characterized by a single predictor index. The functional relation between the two single‐indices is left unspecified, allowing flexible exploration of any potential nonlinear association. We argue that such double single index association is meaningful and easy to interpret, and the rest of the multi‐dimensional dependence structure can be treated as nuisance in model estimation. We investigate the estimation and inference of both indices and the regression function, and derive the asymptotic properties of our procedure. We illustrate the numerical performance in finite samples and demonstrate the usefulness of the modelling and estimation procedure in a multi‐covariate multi‐response problem concerning concrete.     

Central quantile subspace

Quantile regression (QR) is becoming increasingly popular due to its relevance in many scientific investigations. There is a great amount of work about linear and nonlinear QR models. Specifically, nonparametric estimation of the conditional quantiles received particular attention, due to its model flexibility. However, nonparametric QR techniques are limited in the number of covariates. Dimension reduction offers a solution to this problem by considering low-dimensional smoothing without specifying any parametric or nonparametric regression relation. The existing dimension reduction techniques focus on the entire conditional distribution. We, on the other hand, turn our attention to dimension reduction techniques for conditional quantiles and introduce a new method for reducing the dimension of the predictor $$\mathbf {X}$$. The novelty of this paper is threefold. We start by considering a single index quantile regression model, which assumes that the conditional quantile depends on $$\mathbf {X}$$ through a single linear combination of the predictors, then extend to a multi-index quantile regression model, and finally, generalize the proposed methodology to any statistical functional of the conditional distribution. The performance of the methodology is demonstrated through simulation examples and real data applications. Our results suggest that this method has a good finite sample performance and often outperforms the existing methods.     

ASYMPTOTIC PROPERTIES OF SUFFICIENT DIMENSION REDUCTION WITH A DIVERGING NUMBER OF PREDICTORS

We investigate asymptotic properties of a family of sufficient dimension reduction estimators when the number of predictors p diverges to infinity with the sample size. We adopt a general formulation of dimension reduction estimation through least squares regression of a set of transformations of the response. This formulation allows us to establish the consistency of reduction projection estimation. We then introduce the SCAD max penalty, along with a difference convex optimization algorithm, to achieve variable selection. We show that the penalized estimator selects all truly relevant predictors and excludes all irrelevant ones with probability approaching one, meanwhile it maintains consistent reduction basis estimation for relevant predictors. Our work differs from most model-based selection methods in that it does not require a traditional model, and it extends existing sufficient dimension reduction and model-free variable selection approaches from the fixed p scenario to a diverging p.     

Estimating the structural dimension of regressions via parametric inverse regression

A new estimation method for the dimension of a regression at the outset of an analysis is proposed. A linear subspace spanned by projections of the regressor vector X , which contains part or all of the modelling information for the regression of a vector Y on X , and its dimension are estimated via the means of parametric inverse regression. Smooth parametric curves are fitted to the p inverse regressions via a multivariate linear model. No restrictions are placed on the distribution of the regressors. The estimate of the dimension of the regression is based on optimal estimation procedures. A simulation study shows the method to be more powerful than sliced inverse regression in some situations.     

Sliced inverse moment regression using weighted chi-squared tests for dimension reduction

We propose a new method for dimension reduction in regression using the first two inverse moments. We develop corresponding weighted chi-squared tests for the dimension of the regression. The proposed method considers linear combinations of sliced inverse regression (SIR) and the method using a new candidate matrix which is designed to recover the entire inverse second moment subspace. The optimal combination may be selected based on the p-values derived from the dimension tests. Theoretically, the proposed method, as well as sliced average variance estimate (SAVE), is more capable of recovering the complete central dimension reduction subspace than SIR and principle Hessian directions (pHd). Therefore it can substitute for SIR, pHd, SAVE, or any linear combination of them at a theoretical level. Simulation study indicates that the proposed method may have consistently greater power than SIR, pHd, and SAVE.     

Goodness-of-Fit for Longitudinal Count Data with Overdispersion

In this article, we develop a method for checking the estimation equations, which is for joint estimation of the regression parameters and the overdispersion parameters, based on one dimension projected covariate. This method is different from the general testing methods in that our proposed method can be applied to high-dimensional response while the classical testing methods can not be extended to high dimension problem simply to construct a powerful test. Furthermore, the properties of the test statistics are investigated and Nonparametric Monte Carlo Test (NMCT) is suggested to determine the critical values of the test statistics under null hypothesis.     

DETECTING INFLUENTIAL OBSERVATIONS IN SLICED INVERSE REGRESSION ANALYSIS

The detection of influential observations on the estimation of the dimension reduction subspace returned by Sliced Inverse Regression (SIR) is considered. Although there are many measures to detect influential observations in related methods such as multiple linear regression, there has been little development in this area with respect to dimension reduction. One particular influence measure for a version of SIR is examined and it is shown, via simulation and example, how this may be used to detect influential observations in practice.     

Dimension reduction in estimating equations with covariates missing at random

To estimate parameters defined by estimating equations with covariates missing at random, we consider three bias-corrected nonparametric approaches based on inverse probability weighting, regression and augmented inverse probability weighting. However, when the dimension of covariates is not low, the estimation efficiency will be affected due to the curse of dimensionality. To address this issue, we propose a two-stage estimation procedure by using the dimension-reduced kernel estimation in conjunction with bias-corrected estimating equations. We show that the resulting three estimators are asymptotically equivalent and achieve the desirable properties. The impact of dimension reduction in nonparametric estimation of parameters is also investigated. The finite-sample performance of the proposed estimators is studied through simulation, and an application to an automobile data set is also presented.     

Multiple-population shrinkage estimation via sliced inverse regression

The problem of dimension reduction in multiple regressions is investigated in this paper, in which data are from several populations that share the same variables. Assuming that the set of relevant predictors is the same across the regressions, a joint estimation and selection method is proposed, aiming to preserve the common structure, while allowing for population-specific characteristics. The new approach is based upon the relationship between sliced inverse regression and multiple linear regression, and is achieved through the lasso shrinkage penalty. A fast alternating algorithm is developed to solve the corresponding optimization problem. The performance of the proposed method is illustrated through simulated and real data examples.