Partial envelopes for efficient estimation in multivariate linear regression

Su, Zhihua; Cook, R. Dennis

doi:10.1093/biomet/asq063

Cited by 59 publications

(65 citation statements)

References 6 publications

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“…We therefore refer to the RRR model as fully supervised. The SupSVD model (2) is also connected with the envelope model that was recently proposed by Cook et al [8] and further developed in [32,7,11,10]. The envelope model is a parsimonious model for multivariate regression that is based on the assumption that variation in the response can be divided into two parts: a material part that is related to the predictor, and ) -an immaterial part that is unrelated to the predictor.…”

Section: Connections With Existing Modelsmentioning

confidence: 99%

Supervised singular value decomposition and its asymptotic properties

Yang

Nobel

et al. 2016

Journal of Multivariate Analysis

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a b s t r a c tA supervised singular value decomposition (SupSVD) model has been developed for supervised dimension reduction where the low rank structure of the data of interest is potentially driven by additional variables measured on the same set of samples. The SupSVD model can make use of the information in the additional variables to accurately extract underlying structures that are more interpretable. The model is general and includes the principal component analysis model and the reduced rank regression model as two extreme cases. The model is formulated in a hierarchical fashion using latent variables, and a modified expectation-maximization algorithm for parameter estimation is developed, which is computationally efficient. The asymptotic properties for the estimated parameters are derived. We use comprehensive simulations and a real data example to illustrate the advantages of the SupSVD model.

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Section: Connections With Existing Modelsmentioning

confidence: 99%

Supervised singular value decomposition and its asymptotic properties

Yang

Nobel

et al. 2016

Journal of Multivariate Analysis

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“…() introduced the novel idea of an envelope for parsimonious parameterizations of multivariate statistical problems. An expository introduction to the underlying structure of an envelope was given by Su and Cook (). Concentrating on reduction in the y ‐dimension and assuming normal errors and non‐stochastic predictors x , Cook et al .…”

Section: Envelopesmentioning

confidence: 99%

“…Whereas PLS is an integral part of the chemometrics culture, envelope methodology is new and not yet widely recognized in statistics. As shown in past studies (Cook et al ., ; Su and Cook, , ) envelope methodology has the potential to achieve substantial gains in efficiency in a variety of multivariate problems and thus also has the potential to become wonted methodology. The gains in efficiency that are afforded by envelopes are achieved by a targeted form of dimension reduction that can effectively separate information that is material for the goals at hand from that which is immaterial.…”

Section: Introductionmentioning

confidence: 99%

Envelopes and Partial Least Squares Regression

Cook

Helland

2013

Journal of the Royal Statistical Society Series B: Statistical Methodology

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118

152

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Summary We build connections between envelopes, a recently proposed context for efficient estimation in multivariate statistics, and multivariate partial least squares (PLS) regression. In particular, we establish an envelope as the nucleus of both univariate and multivariate PLS, which opens the door to pursuing the same goals as PLS but using different envelope estimators. It is argued that a likelihood‐based envelope estimator is less sensitive to the number of PLS components that are selected and that it outperforms PLS in prediction and estimation.

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“…The partial envelope model (Su and Cook 2011) is an extension of the envelope model in which some predictors are of special interest. Suppose X can be partitioned into X 1 ∈ R p 1 and X 2 ∈ R p 2 , p 1 + p 2 = p, and the regression coefficient becomes β = (β T 1 , β T 2 ) T correspondingly.…”

Section: Discussionmentioning

confidence: 99%

Simultaneous Envelopes for Multivariate Linear Regression

Cook

Zhang

2015

Technometrics

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We introduce envelopes for simultaneously reducing the predictors and the responses in multivariate linear regression, so the regression then depends only on estimated linear combinations of X and Y. We use a likelihood-based objective function for estimating envelopes and then propose algorithms for estimation of a simultaneous envelope as well as for basic Grassmann manifold optimization. The asymptotic properties of the resulting estimator are studied under normality and extended to general distributions. We also investigate likelihood ratio tests and information criteria for determining the simultaneous envelope dimensions. Simulation studies and real data examples show substantial gain over the classical methods, like partial least squares, canonical correlation analysis, and reduced-rank regression. This article has supplementary material available online.

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Partial envelopes for efficient estimation in multivariate linear regression

Cited by 59 publications

References 6 publications

Supervised singular value decomposition and its asymptotic properties

Supervised singular value decomposition and its asymptotic properties

Envelopes and Partial Least Squares Regression

Simultaneous Envelopes for Multivariate Linear Regression

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