Envelope methods

Abstract

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

Envelopes: A new chapter in partial least squares regression

“…An introduction to envelopes is available from the monograph, 36 which contains many illustrations of how envelopes can be used to improve the application of regression, and a brief overview from. 37…”

“…Each of these and others demonstrate a potential for envelope methodology to achieve reduction in estimative and predictive variation beyond that attained by standard methods, sometimes by amounts equivalent to increasing the sample size many times over. An introduction to envelopes is available from the monograph, 36 which contains many illustrations of how envelopes can be used to improve the application of regression, and a brief overview from 37 …”

Ultrasound Methods for the Assessment of Liver Steatosis: A Critical Appraisal

“…According to Cook, the use of envelope statistics is a relatively new method, a form of targeted dimension reduction, with the goal of reducing the estimative and predictive variation relative to the standard multivariate statistical methods, sometimes by increasing the sample size many times over [ 45 ].…”