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Regression methods for high dimensional multicollinear data
Authors:Lorna S Aucott  Paul H Garthwaite  James Currall
Institution:1. Department of Mathematical Sciences , University of Aberdeen , AB24 3UE, Aberdeen, United Kingdom;2. Computing Service , Glasgow University , G12 8QQ, Glasgow, United Kingdom
Abstract:To compare their performance on high dimensional data, several regression methods are applied to data sets in which the number of exploratory variables greatly exceeds the sample sizes. The methods are stepwise regression, principal components regression, two forms of latent root regression, partial least squares, and a new method developed here. The data are four sample sets for which near infrared reflectance spectra have been determined and the regression methods use the spectra to estimate the concentration of various chemical constituents, the latter having been determined by standard chemical analysis. Thirty-two regression equations are estimated using each method and their performances are evaluated using validation data sets. Although it is the most widely used, stepwise regression was decidedly poorer than the other methods considered. Differences between the latter were small with partial least squares performing slightly better than other methods under all criteria examined, albeit not by a statistically significant amount.
Keywords:Biased regression  data reduction  high-dimensional data  latent root regression  near infrared spectra  partial least squares  principal components regression  stepwise regression
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