We introduce a novel sufficient dimension-reduction (SDR) method which is robust against outliers using α-distance covariance (dCov)in dimension-reduction problems. Under very mild conditions on the predictors, the central subspace is effectively estimated and model-free without estimating link function based on the projection on the Stiefel manifold. We establish the convergence property of the pro-posed estimation under some regularity conditions. We compare the performance of our method with existing SDR methods by simulation and real data analysis and show that our algorithm improves the computational efficiency and effectiveness.
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Inverse moment methods for sufficient forecasting using high-dimensional predictors
We consider forecasting a single time series using a large number of predictors in the presence of a possible nonlinear forecast function. Assuming that the predictors affect the response through the latent factors, we propose to first conduct factor analysis and then apply sufficient dimension reduction on the estimated factors to derive the reduced data for subsequent forecasting. Using directional regression and the inverse third-moment method in the stage of sufficient dimension reduction, the proposed methods can capture the nonmonotone effect of factors on the response. We also allow a diverging number of factors and only impose general regularity conditions on the distribution of factors, avoiding the undesired time reversibility of the factors by the latter. These make the proposed methods fundamentally more applicable than the sufficient forecasting method of Fan et al. (2017). The proposed methods are demonstrated both in simulation studies and an empirical study of forecasting monthly macroeconomic data from 1959 to 2016. Also, our theory contributes to the literature of sufficient dimension reduction, as it includes an invariance result, a path to perform sufficient dimension reduction under the high-dimensional setting without assuming sparsity, and the corresponding order-determination procedure.
more » « less- Award ID(s):
- 1953189
- PAR ID:
- 10544987
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Biometrika
- Volume:
- 109
- Issue:
- 2
- ISSN:
- 0006-3444
- Page Range / eLocation ID:
- 473 to 487
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Summary As high dimensional data become routinely available in applied sciences, sufficient dimension reduction has been widely employed and its research has received considerable attention. However, with the majority of sufficient dimension reduction methodology focusing on the dimension reduction step, complete analysis and inference after dimension reduction have yet to receive much attention. We couple the strategy of sufficient dimension reduction with a flexible semiparametric model. We concentrate on inference with respect to the primary variables of interest, and we employ sufficient dimension reduction to bring down the dimension of the regression effectively. Extensive simulations demonstrate the efficacy of the method proposed, and a real data analysis is presented for illustration.
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Summary A stepwise procedure, correlation pursuit (COP), is developed for variable selection under the sufficient dimension reduction framework, in which the response variable Y is influenced by the predictors X1,X2,…,Xp through an unknown function of a few linear combinations of them. Unlike linear stepwise regression, COP does not impose a special form of relationship (such as linear) between the response variable and the predictor variables. The COP procedure selects variables that attain the maximum correlation between the transformed response and the linear combination of the variables. Various asymptotic properties of the COP procedure are established and, in particular, its variable selection performance under a diverging number of predictors and sample size is investigated. The excellent empirical performance of the COP procedure in comparison with existing methods is demonstrated by both extensive simulation studies and a real example in functional genomics.
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Sufficient dimension reduction is popular for reducing data dimensionality without stringent model assumptions. However, most existing methods may work poorly for binary classification. For example, sliced inverse regression (Li, 1991) can estimate at most one direction if the response is binary. In this paper we propose principal weighted support vector machines, a unified framework for linear and nonlinear sufficient dimension reduction in binary classification. Its asymptotic properties are studied, and an efficient computing algorithm is proposed. Numerical examples demonstrate its performance in binary classification.more » « less
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Abstract Compositional data reside in a simplex and measure fractions or proportions of parts to a whole. Most existing regression methods for such data rely on log-ratio transformations that are inadequate or inappropriate in modeling high-dimensional data with excessive zeros and hierarchical structures. Moreover, such models usually lack a straightforward interpretation due to the interrelation between parts of a composition. We develop a novel relative-shift regression framework that directly uses proportions as predictors. The new framework provides a paradigm shift for regression analysis with compositional predictors and offers a superior interpretation of how shifting concentration between parts affects the response. New equi-sparsity and tree-guided regularization methods and an efficient smoothing proximal gradient algorithm are developed to facilitate feature aggregation and dimension reduction in regression. A unified finite-sample prediction error bound is derived for the proposed regularized estimators. We demonstrate the efficacy of the proposed methods in extensive simulation studies and a real gut microbiome study. Guided by the taxonomy of the microbiome data, the framework identifies important taxa at different taxonomic levels associated with the neurodevelopment of preterm infants.
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/biomet/asab037
