Sufficient dimension reduction through informative predictor subspace

The purpose of this paper is to define the central informative predictor subspace to contain the central subspace and to develop methods for estimating the former subspace. Potential advantages of the proposed methods are no requirements of linearity, constant variance and coverage conditions in methodological developments. Therefore, the central informative predictor subspace gives us the benefit of restoring the central subspace exhaustively despite failing the conditions. Numerical studies confirm the theories, and real data analyses are presented.     

Sufficient dimension reduction via distance covariance with multivariate responses

In this article, we propose a new method for sufficient dimension reduction when both response and predictor are vectors. The new method, using distance covariance, keeps the model-free advantage, and can fully recover the central subspace even when many predictors are discrete. We then extend this method to the dual central subspace, including a special case of canonical correlation analysis. We illustrated estimators through extensive simulations and real datasets, and compared to some existing methods, showing that our estimators are competitive and robust.     

A study on imbalance support vector machine algorithms for sufficient dimension reduction

Li et al. (2011 Li, B., Artemiou, A., Li, L. (2011). Principal support vector machine for linear and nonlinear sufficient dimension reduction. Ann. Stat. 39:3182–3210.[Crossref], [Web of Science ®] , [Google Scholar]) presented the novel idea of using support vector machines (SVMs) to perform sufficient dimension reduction. In this work, we investigate the potential improvement in recovering the dimension reduction subspace when one changes the SVM algorithm to treat imbalance based on several proposals in the machine learning literature. We find out that in most situations, treating the imbalanced nature of the slices will help improve the estimation. Our results are verified through simulation and real data applications.     

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.     

Dimension reduction for the conditional mean and variance functions in time series

This paper deals with the nonparametric estimation of the mean and variance functions of univariate time series data. We propose a nonparametric dimension reduction technique for both mean and variance functions of time series. This method does not require any model specification and instead we seek directions in both the mean and variance functions such that the conditional distribution of the current observation given the vector of past observations is the same as that of the current observation given a few linear combinations of the past observations without loss of inferential information. The directions of the mean and variance functions are estimated by maximizing the Kullback–Leibler distance function. The consistency of the proposed estimators is established. A computational procedure is introduced to detect lags of the conditional mean and variance functions in practice. Numerical examples and simulation studies are performed to illustrate and evaluate the performance of the proposed estimators.