Results 1  10
of
55
On directional regression for dimension reduction
 J. Amer. Statist. Ass
, 2007
"... By slicing the region of the response (Li, 1991, SIR) and applying local kernel regression (Xia et al., 2002, MAVE) to each slice, a new dimension reduction method is proposed. Compared with the traditional inverse regression methods, e.g. sliced inverse regression (Li, 1991), the new method is fre ..."
Abstract

Cited by 38 (3 self)
 Add to MetaCart
(Show Context)
By slicing the region of the response (Li, 1991, SIR) and applying local kernel regression (Xia et al., 2002, MAVE) to each slice, a new dimension reduction method is proposed. Compared with the traditional inverse regression methods, e.g. sliced inverse regression (Li, 1991), the new method is free of the linearity condition (Li, 1991) and enjoys much improved estimation accuracy. Compared with the direct estimation methods (e.g., MAVE), the new method is much more robust against extreme values and can capture the entire central subspace (Cook, 1998b, CS) exhaustively. To determine the CS dimension, a consistent crossvalidation (CV) criterion is developed. Extensive numerical studies including one real example confirm our theoretical findings. KEY WORDS: crossvalidation; earnings forecast; minimum average variance estimation; sliced inverse regression; sufficient dimension reduction
Sufficient Dimension Reduction via Squaredloss Mutual Information Estimation
"... The goal of sufficient dimension reduction in supervised learning is to find the lowdimensional subspace of input features that is ‘sufficient ’ for predicting output values. In this paper, we propose a novel sufficient dimension reduction method using a squaredloss variant of mutual information as ..."
Abstract

Cited by 34 (29 self)
 Add to MetaCart
(Show Context)
The goal of sufficient dimension reduction in supervised learning is to find the lowdimensional subspace of input features that is ‘sufficient ’ for predicting output values. In this paper, we propose a novel sufficient dimension reduction method using a squaredloss variant of mutual information as a dependency measure. We utilize an analytic approximator of squaredloss mutual information based on density ratio estimation, which is shown to possess suitable convergence properties. We then develop a natural gradient algorithm for sufficient subspace search. Numerical experiments show that the proposed method compares favorably with existing dimension reduction approaches. 1
Sufficient dimension reduction and prediction in regression
 Philosophical Transactions of the Royal Society A
"... Dimension reduction for regression is a prominent issue today because technological advances now allow scientists to routinely formulate regressions in which the number of predictors is considerably larger than in the past. While several methods have been proposed to deal with such regressions, pri ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
(Show Context)
Dimension reduction for regression is a prominent issue today because technological advances now allow scientists to routinely formulate regressions in which the number of predictors is considerably larger than in the past. While several methods have been proposed to deal with such regressions, principal components still seem to be the most widely used across the applied sciences. We give a broad overview of ideas underlying a particular class of methods for dimension reduction that includes principal components, along with an introduction to the corresponding methodology. New methods are proposed for prediction in regressions with many predictors.
Dimension reduction for nonelliptically distributed predictors
 Ann. Statist
, 2009
"... Sufficient dimension reduction methods often require stringent conditions on the joint distribution of the predictor, or, when such conditions are not satisfied, rely on marginal transformation or reweighting to fulfill them approximately. For example, a typical dimension reduction method would requ ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
Sufficient dimension reduction methods often require stringent conditions on the joint distribution of the predictor, or, when such conditions are not satisfied, rely on marginal transformation or reweighting to fulfill them approximately. For example, a typical dimension reduction method would require the predictor to have elliptical or even multivariate normal distribution. In this paper, we reformulate the commonly used dimension reduction methods, via the notion of “central solution space, ” so as to circumvent the requirements of such strong assumptions, while at the same time preserve the desirable properties of the classical methods, such as √ nconsistency and asymptotic normality. Imposing elliptical distributions or even stronger assumptions on predictors is often considered as the necessary tradeoff for overcoming the “curse of dimensionality, ” but the development of this paper shows that this need not be the case. The new methods will be compared with existing methods by simulation and applied to a data set.
Fourier Methods for Estimating The Central Subspace and The Central Mean Subspace in Regression
"... In high dimensional regression, it is important to estimate the central and central mean subspaces, to which the projections of the predictors preserve sufficient information about the response and the mean response, respectively. Using the Fourier transform, we have derived the candidate matrices w ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
In high dimensional regression, it is important to estimate the central and central mean subspaces, to which the projections of the predictors preserve sufficient information about the response and the mean response, respectively. Using the Fourier transform, we have derived the candidate matrices whose column spaces recover the central and central mean subspaces exhaustively. Under the normality assumption of the predictors, explicit estimates of the central and central mean subspaces are derived. Bootstrap procedures are used for determining dimensionality and choosing tuning parameters. Simulation results and an application to a real data are reported. Our methods demonstrate competitive performance compared to SIR, SAVE and other existing methods. The approach proposed in the paper provides a novel view on sufficient dimension reduction and may lead to more powerful tools in the future.
A new algorithm for estimating the effective dimensionreduction subspace
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2008
"... The statistical problem of estimating the effective dimensionreduction (EDR) subspace in the multiindex regression model with deterministic design and additive noise is considered. A new procedure for recovering the directions of the EDR subspace is proposed. Many methods for estimating the EDR su ..."
Abstract

Cited by 12 (7 self)
 Add to MetaCart
The statistical problem of estimating the effective dimensionreduction (EDR) subspace in the multiindex regression model with deterministic design and additive noise is considered. A new procedure for recovering the directions of the EDR subspace is proposed. Many methods for estimating the EDR subspace perform principal component analysis on a family of vectors, say ˆβ1,..., ˆ βL, nearly lying in the EDR subspace. This is in particular the case for the structureadaptive approach proposed by Hristache et al. (2001a). In the present work, we propose to estimate the projector onto the EDR subspace by the solution to the optimization problem minimize max ˆβ ℓ=1,...,L ⊤ ℓ (I − A) ˆ βℓ subject to A ∈ Am∗, where Am ∗ is the set of all symmetric matrices with eigenvalues in [0,1] and trace less than or equal to m ∗ , with m ∗ being the true structural dimension. Under mild assumptions, √ nconsistency of the proposed procedure is proved (up to a logarithmic factor) in the case when the structural dimension is not larger than 4. Moreover, the stochastic error of the estimator of the projector onto the EDR subspace is shown to depend on L logarithmically. This enables us to use a large number of vectors ˆβℓ for estimating the EDR subspace. The empirical behavior of the algorithm is studied through numerical simulations.
2009a Likelihoodbased sufficient dimension reduction
 J. Amer. Statist. Ass
"... 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 directio ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
(Show Context)
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.
Asymptotics for sliced average variance estimation
, 2005
"... In this paper, we systematically study the consistency of sliced average variance estimation (SAVE). The findings reveal that when the response is continuous, the asymptotic behavior of SAVE is rather different from that of sliced inverse regression (SIR). SIR can achieve n consistency even when eac ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
(Show Context)
In this paper, we systematically study the consistency of sliced average variance estimation (SAVE). The findings reveal that when the response is continuous, the asymptotic behavior of SAVE is rather different from that of sliced inverse regression (SIR). SIR can achieve n consistency even when each slice contains only two data points. However, SAVE cannot be √ n consistent and it even turns out to be not consistent when each slice contains a fixed number of data points that do not depend on n, where n is the sample size. These results theoretically confirm the notion that SAVE is more sensitive to the number of slices than SIR. Taking this into account, a bias correction is recommended in order to allow SAVE to be √ n consistent. In contrast, when the response is discrete and takes finite values, √ n consistency can be achieved. Therefore, an approximation through discretization, which is commonly used in practice, is studied. A simulation study is carried out for the purposes of illustration. 1. Introduction. Dimension