Simultaneous Envelopes for Multivariate Linear Regression

Cook, R. Dennis; Zhang, Xin

doi:10.1080/00401706.2013.872700

Cited by 52 publications

(49 citation statements)

References 22 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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“…They also showed that asymptotically the envelope estimatorβ is always better than the standard maximum likelihood estimator b. In other work, simultaneous envelopes (Cook & Zhang, 2015b) allow for simultaneously reducing the response and random predictor vectors.…”

Section: Model For Predictor Envelopesmentioning

confidence: 96%

Envelope methods

Cook

2019

WIREs Computational Stats

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An envelope is a relatively new construct for decreasing estimative and predictive variation relative to standard methods in multivariate statistics, sometimes by amounts equivalent to increasing the sample size many times over. Essentially a form of targeted dimension reduction that is descendent from sufficient dimension reduction, an envelope inherits its underlying philosophy from Fisher's notion of sufficient statistics. The initial development of envelope methods took place largely in the context of the multivariate linear model, resulting in response envelopes for response reduction, predictor envelopes for predictor reduction, simultaneous envelopes for response and predictor reduction and partial envelopes for specialized considerations, each demonstrating a potential for substantial reduction in estimative variation. These advances demonstrated that there are close connections between envelopes and some standard multivariate methods like partial least squares regression and canonical correlations. Subsequently, envelope methodology has been adapted and extended to diverse areas, including envelopes for regressions with matrix and tensor‐valued responses, envelopes for spatial statistics, quantile envelopes for quantile regression, Bayesian response envelopes. Sparse versions of response and predictor envelopes have also been developed. More generally, there is also envelope methodology for reducing the variation in any asymptotically normal vector‐valued estimator. These advances have opened a new chapter in multivariate statistics, allowing variation that is material to the goals of an analysis to be separated effectively from immaterial variation that serves only to confound estimation and prediction. This article is categorized under: Statistical Models > Multivariate Models Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Dimension Reduction

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“…Recent developments of envelopes under other settings may be found in, for example, Su and Cook (), Cook et al . () and Cook and Zhang (). In particular, Cook and Zhang () studied the relationship between envelopes and canonical correlation analysis.…”

Section: Review Of Envelopesmentioning

confidence: 98%

“…() and Cook and Zhang (). In particular, Cook and Zhang () studied the relationship between envelopes and canonical correlation analysis. They concluded that, although there is a weak population connection, canonical correlation analysis is not well suited as a basis for envelope methodology.…”

Section: Review Of Envelopesmentioning

confidence: 99%

Matrix Variate Regressions and Envelope Models

Ding

Cook

2017

Journal of the Royal Statistical Society Series B: Statistical Methodology

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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 of the conditional distribution Y|X. We propose models for matrix variate regressions and then develop envelope extensions of these models. Under the envelope framework, redundant variation can be eliminated in estimation and the number of parameters can be notably reduced when the matrix variate dimension is large, possibly resulting in significant gains in efficiency. The methods proposed are applicable to high dimensional settings.

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Simultaneous Envelopes for Multivariate Linear Regression

Cited by 52 publications

References 22 publications

Supervised singular value decomposition and its asymptotic properties

Supervised singular value decomposition and its asymptotic properties

Envelope methods

Matrix Variate Regressions and Envelope Models

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