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.     

Chi-squared tests in kth-moment sufficient dimension reduction

We present a novel approach to sufficient dimension reduction for the conditional kth moments in regression. The approach provides a computationally feasible test for the dimension of the central kth-moment subspace. In addition, we can test predictor effects without assuming any models. All test statistics proposed in the novel approach have asymptotic chi-squared distributions.     

Independent screening in high-dimensional exponential family predictors’ space

We present a methodology for screening predictors that, given the response, follow a one-parameter exponential family distributions. Screening predictors can be an important step in regressions when the number of predictors p is excessively large or larger than n the number of observations. We consider instances where a large number of predictors are suspected irrelevant for having no information about the response. The proposed methodology helps remove these irrelevant predictors while capturing those linearly or nonlinearly related to the response.     

Extending the Scope of Inverse Regression Methods in Sufficient Dimension Reduction

In the area of sufficient dimension reduction, two structural conditions are often assumed: the linearity condition that is close to assuming ellipticity of underlying distribution of predictors, and the constant variance condition that nears multivariate normality assumption of predictors. Imposing these conditions are considered as necessary trade-off for overcoming the “curse of dimensionality”. However, it is very hard to check whether these conditions hold or not. When these conditions are violated, some methods such as marginal transformation and re-weighting are suggested so that data fulfill them approximately. In this article, we assume an independence condition between the projected predictors and their orthogonal complements which can ensure the commonly used inverse regression methods to identify the central subspace of interest. The independence condition can be checked by the gridded chi-square test. Thus, we extend the scope of many inverse regression methods and broaden their applicability in the literature. Simulation studies and an application to the car price data are presented for illustration.     

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.     

Canonical Correlation Analysis Through Linear Modeling

In this paper, we introduce linear modeling of canonical correlation analysis, which estimates canonical direction matrices by minimising a quadratic objective function. The linear modeling results in a class of estimators of canonical direction matrices, and an optimal class is derived in the sense described herein. The optimal class guarantees several of the following desirable advantages: first, its estimates of canonical direction matrices are asymptotically efficient; second, its test statistic for determining the number of canonical covariates always has a chi‐squared distribution asymptotically; third, it is straight forward to construct tests for variable selection. The standard canonical correlation analysis and other existing methods turn out to be suboptimal members of the class. Finally, we study the role of canonical variates as a means of dimension reduction for predictors and responses in multivariate regression. Numerical studies and data analysis are presented.     

Dimension-reduced empirical likelihood inference for response mean with data missing at random

To make efficient inference for mean of a response variable when the data are missing at random and the dimension of covariate is not low, we construct three bias-corrected empirical likelihood (EL) methods in conjunction with dimension-reduced kernel estimation of propensity or/and conditional mean response function. Consistency and asymptotic normality of the maximum dimension-reduced EL estimators are established. We further study the asymptotic properties of the resulting dimension-reduced EL ratio functions and the corresponding EL confidence intervals for the response mean are constructed. The finite-sample performance of the proposed estimators is studied through simulation, and an application to HIV-CD4 data set is also presented.     

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.     

A new reproducing kernel-based nonlinear dimension reduction method for survival data

Based on the theories of sliced inverse regression (SIR) and reproducing kernel Hilbert space (RKHS), a new approach RDSIR (RKHS-based Double SIR) to nonlinear dimension reduction for survival data is proposed. An isometric isomorphism is constructed based on the RKHS property, then the nonlinear function in the RKHS can be represented by the inner product of two elements that reside in the isomorphic feature space. Due to the censorship of survival data, double slicing is used to estimate the weight function to adjust for the censoring bias. The nonlinear sufficient dimension reduction (SDR) subspace is estimated by a generalized eigen-decomposition problem. The asymptotic property of the estimator is established based on the perturbation theory. Finally, the performance of RDSIR is illustrated on simulated and real data. The numerical results show that RDSIR is comparable with the linear SDR method. Most importantly, RDSIR can also effectively extract nonlinearity from survival data.     

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.     

Trace pursuit variable selection for multi-population data

Variable selection is a very important tool when dealing with high dimensional data. However, most popular variable selection methods are model based, which might provide misleading results when the model assumption is not satisfied. Sufficient dimension reduction provides a general framework for model-free variable selection methods. In this paper, we propose a model-free variable selection method via sufficient dimension reduction, which incorporates the grouping information into the selection procedure for multi-population data. Theoretical properties of our selection methods are also discussed. Simulation studies suggest that our method greatly outperforms those ignoring the grouping information.