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设计实验中失拟检验的功效：样本量与重复观测分配指南AI 解读

Although there exists substantial literature on lack-of-fit tests for normal linear models, there is scant literature for determining the number or distribution of points to replicate when starting with an un-replicated base design like a factorial, response surface, or related constructions. In this article, we provide an easy-to-implement approach to determining power and sample size based on the standard lack-of-fit test. The method employs simply stated bounds on the…

包络 Huber 回归AI 解读

Huber regression (HR) is a popular flexible alternative to the least squares regression when the error follows a heavy-tailed distribution. We propose a new method called the enveloped Huber regression (EHR) by considering the envelope assumption that there exists some subspace of the predictors that has no association with the response, which is referred to as the immaterial part. More efficient estimation is achieved via the removal of the immaterial part. Different from…

超高维充分降维与同步变量选择的稀疏最小差异方法AI 解读

Sufficient dimension reduction (SDR) is known to be a powerful tool for achieving data reduction and data visualization in regression and classification problems. In this work, we study ultrahigh-dimensional SDR problems and propose solutions under a unified minimum discrepancy approach with regularization. When p grows exponentially with n, consistency results in both central subspace estimation and variable selection are established simultaneously for important SDR…

高维回归中的偏最小二乘预测AI 解读

We study the asymptotic behavior of predictions from partial least squares (PLS) regression as the sample size and number of predictors diverge in various alignments. We show that there is a range of regression scenarios where PLS predictions have the usual root-$n$ convergence rate, even when the sample size is substantially smaller than the number of predictors, and an even wider range where the rate is slower but may still produce practically useful results. We show also…

矩阵变量回归与包络模型AI 解读

Summary Modern technology often generates data with complex structures in which both response and explanatory variables are matrix valued. Existing methods in the literature can tackle matrix-valued predictors but are rather limited for matrix-valued responses. We study matrix variate regressions for such data, where the response Y on each experimental unit is a random matrix and the predictor X can be either a scalar, a vector or a matrix, treated as non-stochastic in terms…

充分降维中的诊断方法AI 解读

Sufficient dimension reduction in regression aims to reduce the predictor dimension by replacing the original predictors with some set of linear combinations of them without loss of information. Numerous dimension reduction methods have been developed based on this paradigm. However, little effort has been devoted to diagnostic studies within the context of dimension reduction. In this paper we introduce methods to check goodness-of-fit for a given dimension reduction…

包络方法与降秩回归AI 解读

We incorporate the nascent idea of envelopes (Cook et al., Statist. Sinica20, 927–1010) into reduced-rank regression by proposing a reduced-rank envelope model, which is a hybrid of reduced-rank and envelope regressions. The proposed model has total number of parameters no more than either of reduced-rank regression or envelope regression. The resulting estimator is at least as efficient as both existing estimators. The methodology of this paper can be adapted to other…

包络模型与方法的基础AI 解读

Envelopes were recently proposed by Cook, Li and Chiaromonte as a method for reducing estimative and predictive variations in multivariate linear regression. We extend their formulation, proposing a general definition of an envelope and a general framework for adapting envelope methods to any estimation procedure. We apply the new envelope methods to weighted least squares, generalized linear models and Cox regression. Simulations and illustrative data analysis show the…

充分降维中中心子空间的融合估计AI 解读

When studying the regression of a univariate variable Y on a vector x of predictors, most existing sufficient dimension-reduction (SDR) methods require the construction of slices of Y to estimate moments of the conditional distribution of X given Y. But there is no widely accepted method for choosing the number of slices, while a poorly chosen slicing scheme may produce miserable results. We propose a novel and easily implemented fusing method that can mitigate the problem…

信息密集型高维回归中的预测变量充分降维估计AI 解读

We study the asymptotic behavior of a class of methods for sufficient dimension reduction in high-dimension regressions, as the sample size and number of predictors grow in various alignments. It is demonstrated that these methods are consistent in a variety of settings, particularly in abundant regressions where most predictors contribute some information on the response, and oracle rates are possible. Simulation results are presented to support the theoretical conclusion.

坐标无关的稀疏充分降维与变量选择AI 解读

Sufficient dimension reduction (SDR) in regression, which reduces the dimension by replacing original predictors with a minimal set of their linear combinations without loss of information, is very helpful when the number of predictors is large. The standard SDR methods suffer because the estimated linear combinations usually consist of all original predictors, making it difficult to interpret. In this paper, we propose a unified method—coordinate-independent sparse…

基于似然的充分降维AI 解读

We obtain the maximum likelihood estimator of the central subspace under conditional normality of the predictors given the response. Analytically and in simulations we found that our new estimator can preform much better than sliced inverse regression, sliced average variance estimation and directional regression, and that it seems quite robust to deviations from normality.

利用片内协方差改进回归中心子空间估计AI 解读

Popular methods for estimating the central subspace in regression require slicing a continuous response. However, slicing can result in loss of information and in some cases that loss can be substantial. We use intraslice covariances to construct improved inference methods for the central subspace. These methods are optimal within a class of quadratic inference functions and permit chi-squared tests of conditional independence hypotheses involving the predictors. Our…

异质子总体回归中的充分降维AI 解读

Summary Sliced inverse regression is one of the widely used dimension reduction methods. Chiaromonte and co-workers extended this method to regressions with qualitative predictors and developed a method, partial sliced inverse regression, under the assumption that the covariance matrices of the continuous predictors are constant across the levels of the qualitative predictor. We extend partial sliced inverse regression by removing the restrictive homogeneous covariance…

单指标回归中的方向估计AI 解读

We propose a general dimension-reduction method that combines the ideas of likelihood, correlation, inverse regression and information theory. We do not require that the dependence be confined to particular conditional moments, nor do we place restrictions on the predictors or on the regression that are necessary for methods like ordinary least squares and sliced-inverse regression. Although we focus on single-index regressions, the underlying idea is applicable more…

通过逆回归实现充分降维AI 解读

A family of dimension-reduction methods, the inverse regression (IR) family, is developed by minimizing a quadratic objective function. An optimal member of this family, the inverse regression estimator (IRE), is proposed, along with inference methods and a computational algorithm. The IRE has at least three desirable properties: (1) Its estimated basis of the central dimension reduction subspace is asymptotically efficient, (2) its test statistic for dimension has an…

不依赖模型形式的变量选择AI 解读

Summary The importance of variable selection in regression has grown in recent years as computing power has encouraged the modelling of data sets of ever-increasing size. Data mining applications in finance, marketing and bioinformatics are obvious examples. A limitation of nearly all existing variable selection methods is the need to specify the correct model before selection. When the number of predictors is large, model formulation and validation can be difficult or even…

收缩切片逆回归札记AI 解读

We employ Lasso shrinkage within the context of sufficient dimension reduction to obtain a shrinkage sliced inverse regression estimator, which provides easier interpretations and better prediction accuracy without assuming a parametric model. The shrinkage sliced inverse regression approach can be employed for both single-index and multiple-index models. Simulation studies suggest that the new estimator performs well when its tuning parameter is selected by either the…

迭代 Hessian 变换的维数确定AI 解读

The central mean subspace (CMS) and iterative Hessian transformation (IHT) have been introduced recently for dimension reduction when the conditional mean is of interest. Suppose that X is a vector-valued predictor and Y is a scalar response. The basic problem is to find a lower-dimensional predictor ηTX such that E(Y|X)=E(Y|ηTX). The CMS defines the inferential object for this problem and IHT provides an estimating procedure. Compared with other methods, IHT requires fewer…

充分降维中的预测变量贡献检验AI 解读

We develop tests of the hypothesis of no effect for selected predictors in regression, without assuming a model for the conditional distribution of the response given the predictors. Predictor effects need not be limited to the mean function and smoothing is not required. The general approach is based on sufficient dimension reduction, the idea being to replace the predictor vector with a lower-dimensional version without loss of information on the regression. Methodology…

含类别预测变量回归中的条件均值降维AI 解读

Consider the regression of a response Y on a vector of quantitative predictors $\X$ and a categorical predictor W. In this article we describe a first method for reducing the dimension of $\X$ without loss of information on the conditional mean $\mathrm{E}(Y|\X,W)$ and without requiring a prespecified parametric model. The method, which allows for, but does not require, parametric versions of the subpopulation mean functions $\mathrm{E}(Y|\X,W=w)$, includes a procedure for…

多元回归降秩的无预设模型检验AI 解读

We propose a test of dimension in multivariate regression. This test is in the spirit of tests on the rank of the coefficient matrix in a multivariate linear model, but it does not require a prespecified model. The test may be particularly useful at the outset of an analysis before a multivariate model is posited, because it can lead to low-dimensional summary plots that are inferred to contain all of the sample information on the multivariate mean function.

回归中条件第 k 阶矩的降维AI 解读

Summary 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 kth-moment subspace to capture information from the mean, variance and so on up to the kth conditional moment of…

切片逆回归的扩展AI 解读

Sliced inverse regression (SIR) and an associated chi-squared test for dimension have been introduced as a method for reducing the dimension of regression problems whose predictor variables are normal. In this article the assumptions on the predictor distribution, under which the chi-squared test was proved to apply, are relaxed, and the result is extended. A general weighted chi-squared test that does not require normal regressors for the dimension of a regression is given.…