A Shrinkage Estimation of Central Subspace in Sufficient Dimension Reduction

Sliced regression is an effective dimension reduction method by replacing the original high-dimensional predictors with its appropriate low-dimensional projection. It is free from any probabilistic assumption and can exhaustively estimate the central subspace. In this article, we propose to incorporate shrinkage estimation into sliced regression so that variable selection can be achieved simultaneously with dimension reduction. The new method can improve the estimation accuracy and achieve better interpretability for the reduced variables. The efficacy of proposed method is shown through both simulation and real data analysis.     

大规模数据下基于充分降维的Leverage重要性抽样方法

随着信息技术的飞速发展,大规模数据在短时间内搜集并储存下来,为分析决策提供了巨大的信息量,也给统计建模带来了一定难度。对于样本容量大、变量个数少的数据,Leverage重要性抽样是一个简便可行的方法。本文发现,该方法中度量样本重要性的Leverage分数与因变量无关,而且在维度较大的情形下对样本没有区分程度,使得估计结果较差。为了同时考虑因变量和维度的影响,本文提出了基于充分降维的Leverage重要性抽样方法。该方法以不损失信息为前提,在充分降维的空间内重新计算Leverage分数,使得抽样更具有代表性。模拟数据分析显示,在样本容量较大的复杂数据中,相比于原始的Leverage重要性抽样方法,本文提出的方法可以降低估计的均方误差。三个实际数据也证实了该方法的可行性和有效性。     

Sparse Sufficient Dimension Reduction for Markov Blanket Discovery

In this article, we propose to use sparse sufficient dimension reduction as a novel method for Markov blanket discovery of a target variable, where we do not take any distributional assumption on the variables. By assuming sparsity on the basis of the central subspace, we developed a penalized loss function estimate on the high-dimensional covariance matrix. A coordinate descent algorithm based on an inverse regression is used to get the sparse basis of the central subspace. Finite sample behavior of the proposed method is explored by simulation study and real data examples.     

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.     

大数据下张量充分降维方法及其应用研究

在大数据时代,金融学、基因组学和图像处理等领域产生了大量的张量数据。Zhong等(2015)提出了张量充分降维方法,并给出了处理二阶张量的序列迭代算法。鉴于高阶张量在实际生活中的广泛应用,本文将Zhong等(2015)的算法推广到高阶,以三阶张量为例,提出了两种不同的算法:结构转换算法和结构保持算法。两种算法都能够在不同程度上保持张量原有结构信息,同时有效降低变量维度和计算复杂度,避免协方差矩阵奇异的问题。将两种算法应用于人像彩图的分类识别,以二维和三维点图等形式直观展现了算法分类结果。将本文的结构保持算法与K-means聚类方法、t-SNE非线性降维方法、多维主成分分析、多维判别分析和张量切片逆回归共五种方法进行对比,结果表明本文所提方法在分类精度方面有明显优势,因此在图像识别及相关应用领域具有广阔的发展前景。     

A Practical Procedure to Find Matching Priors for Frequentist Inference

We present a practical way to find matching priors via the use of saddlepoint approximations and obtain p-values of tests of an interest parameter in the presence of nuisance parameters. The advantages of our procedure are the flexibility in choosing different initial conditions so that one may adjust the performance of a test, and the less intensive computational efforts compared to a Markov Chain Monto Carlo method.     

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.     

Random sliced inverse regression

Sliced Inverse Regression (SIR; 1991) is a dimension reduction method for reducing the dimension of the predictors without losing regression information. The implementation of SIR requires inverting the covariance matrix of the predictors—which has hindered its use to analyze high-dimensional data where the number of predictors exceed the sample size. We propose random sliced inverse regression (rSIR) by applying SIR to many bootstrap samples, each using a subset of randomly selected candidate predictors. The final rSIR estimate is obtained by aggregating these estimates. A simple variable selection procedure is also proposed using these bootstrap estimates. The performance of the proposed estimates is studied via extensive simulation. Application to a dataset concerning myocardial perfusion diagnosis from cardiac Single Proton Emission Computed Tomography (SPECT) images is presented.     

An Approach for Estimating Causal Effects in Randomized Trials with Noncompliance

We developed methods for estimating the causal risk difference and causal risk ratio in randomized trials with noncompliance. The developed estimator is unbiased under the assumption that biases due to noncompliance are identical between both treatment arms. The biases are defined as the difference or ratio between the expectations of potential outcomes for a group that received the test treatment and that for the control group in each randomly assigned group. Although the instrumental variable estimator yields an unbiased estimate under a sharp null hypothesis but may yield a biased estimate under a non-null hypothesis, the bias of the developed estimator does not depend on whether this hypothesis holds. Then the estimate of the causal effect from the developed estimator may have a smaller bias than that from the instrumental variable estimator when the treatment effect exists. There is not yet a standard method for coping with noncompliance, and thus it is important to evaluate estimates under different assumptions. The developed estimator can serve this purpose. Its application to a field trial for coronary heart disease is provided.     

Dimension reduction for the conditional kth moment in regression

The idea of dimension reduction without loss of information can be quite helpful for guiding the construction of summary plots in regression without requiring a prespecified model. Central subspaces are designed to capture all the information for the regression and to provide a population structure for dimension reduction. Here, we introduce the central k th-moment subspace to capture information from the mean, variance and so on up to the k th conditional moment of the regression. New methods are studied for estimating these subspaces. Connections with sliced inverse regression are established, and examples illustrating the theory are presented.     

Dimension Reduction in Regressions through Weighted Variance Estimation

Because sliced inverse regression (SIR) using the conditional mean of the inverse regression fails to recover the central subspace when the inverse regression mean degenerates, sliced average variance estimation (SAVE) using the conditional variance was proposed in the sufficient dimension reduction literature. However, the efficacy of SAVE depends heavily upon the number of slices. In the present article, we introduce a class of weighted variance estimation (WVE), which, similar to SAVE and simple contour regression (SCR), uses the conditional variance of the inverse regression to recover the central subspace. The strong consistency and the asymptotic normality of the kernel estimation of WVE are established under mild regularity conditions. Finite sample studies are carried out for comparison with existing methods and an application to a real data is presented for illustration.     

Using adaptively weighted large margin classifiers for robust sufficient dimension reduction

In this paper, we combine adaptively weighted large margin classifiers with Support Vector Machine (SVM)-based dimension reduction methods to create dimension reduction methods robust to the presence of extreme outliers. We discuss estimation and asymptotic properties of the algorithm. The good performance of the new algorithm is demonstrated through simulations and real data analysis.     

Dimension reduction summaries for balanced contrasts

We discuss the covariate dimension reduction properties of conditional density ratios in the estimation of balanced contrasts of expectations. Conditional density ratios, as well as related sufficient summaries, can be used to replace the covariates with a smaller number of variables. For example, for comparisons among k   populations the covariates can be replaced with k-1$k-1$ conditional density ratios. The dimension reduction properties of conditional density ratios are directly connected with sufficiency, the dimension reduction concepts considered in regression theory, and propensity theory. The theory presented here extends the ideas in propensity theory to situations in which propensities do not exist and develops an approach to dimension reduction outside of the potential outcomes or counterfactual framework. Under general conditions, we show that a principal components transformation of the estimated conditional density ratios can be used to investigate whether a sufficient summary of dimension lower than k-1$k-1$ exists and to identify such a lower dimensional summary.     

Inference on local average treatment effects for misclassified treatment

We develop point-identification for the local average treatment effect when the binary treatment contains a measurement error. The standard instrumental variable estimator is inconsistent for the parameter since the measurement error is nonclassical by construction. We correct the problem by identifying the distribution of the measurement error based on the use of an exogenous variable that can even be a binary covariate. The moment conditions derived from the identification lead to generalized method of moments estimation with asymptotically valid inferences. Monte Carlo simulations and an empirical illustration demonstrate the usefulness of the proposed procedure.     

Causal inference for recurrent events data with all-or-none compliance

ABSTRACT

In this article, causal inference in randomized studies with recurrent events data and all-or-none compliance is considered. We use the counting process to analyze the recurrent events data and propose a causal proportional intensity model. The maximum likelihood approach is adopted to estimate the parameters of the proposed causal model. To overcome the computational difficulties created by the mixture structure of the problem, we develop an expectation-maximization (EM) algorithm. The resulting estimators are shown to be consistent and asymptotically normal. We further estimate the complier average causal effect (CACE), which is defined as the difference of the average numbers of recurrence between treatment and control groups within the complier class. The corresponding inferential procedures are established. Some simulation studies are conducted to assess the finite sample performance of the proposed approach.     

Identification of Local Causal Effects with Missing Outcome Values and an Instrument for Non Response

Even in randomized experiments the identification of causal effects is often threatened by the presence of missing outcome values, with missingness possibly being non ignorable. We provide sufficient conditions under which the availability of a binary instrument for non response allows us to non parametrically point identify average causal effects in some latent subgroups of units, named Principal Strata, defined by their non response behavior in all possible combinations of treatment and instrument. Examples are provided as possible scenarios where our assumptions may be plausible.     

A Note on the Invariance Law for Surrogate Dimension Reduction

In this article, we give an alternative and much shorter proof for the invariance law of surrogate dimension reduction for measurement error regression that was established by Li and Yin (2007 Li , B. , Yin , X. ( 2007 ). On surrogate dimension reduction for measurement error regression: an invariance law . Ann. Statist. 35 : 2143 – 2172 . [Google Scholar]).