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Shrinkage Regression Shrinkage regression Rolf Sundberg Volume 4, pp 1994–1998 in Encyclopedia of Environmetrics (ISBN 0471 899976) Edited by Abdel H. El-Shaarawi and Walter W. Piegorsch John Wiley & Sons, Ltd, Chichester, 2002 Shrinkage regression unrealistically large values. Under exact collinearity, bOLS is not even uniquely defined. In these situations it can pay substantially to use shrinkage methods that Shrinkage regression refers to shrinkage methods trade bias for variance. Not only can they give more of estimation or prediction in regression situations, realistic estimates of ˇ, but they are motivated even useful when there is multicollinearity among the stronger for construction of a predictor of y from x. regressors. With a term borrowed from approximation A simple and as we will see fundamental shrink- theory, these methods are also called regulariza- age construction is ridge regression (RR), according tion methods. Such situations occur frequently in to which the OLS estimator is replaced by environmetric studies, when many chemical, biologi- 1 cal or other explanatory variables are measured, as bRR D Sxx C υI sxy 4 when Branco et al. [3] wanted to model toxicity of n D 19 industrial wastewater samples by pH and for a suitable ridge constant υ>0 (to be specified). seven metal concentrations, and found it necessary to Addition of whatever υ>0 along the diagonal of use dimensionality-reducing techniques. As another Sxx clearly makes this factor less close to singular. example, to be discussed further below, Andersson To see more precisely how RR shrinks, consider et al. [1] constructed a quantitative structure–acti- an orthogonal transformation that replaces the com- vity relationship (QSAR) to model polychlorinated ponents of x by the principal components (PCs), biphenyl (PCB) effects on cell-to-cell communication ti D ci x,iD 1,...,p. Here the coefficient vectors inhibition in the liver, having a sample of size n D 27 ci are the eigenvectors of Sxx, with eigenvalues i, of PCB congeners that represented a set of 154 differ- say. Correspondingly, Sxx is replaced by the diago- ent PCBs, and using 52 physicochemical properties nal matrix of these eigenvalues i, and RR adds υ to of the PCBs as explanatory variables. them all. When OLS divides the ith component of the Suppose we have n independent observations vector sty by i, RR divides by i C υ. Hence, RR of x,yD x1,...,xp,y from a standard multiple shrinks bOLS much in principal directions with small regression model , but only little in other principal directions. Variable selection might appear to shrink, by 0 Y D ˛ C ˇ x C ε1 restricting attention to a lower-dimensional subspace U r < p of the model space of the xs. Variable 2 r varε D (see Linear models). The ordinary least selection improves precision when it omits vari- squares (OLS) estimator can be written ables with negligible influence, but if it is used to eliminate multicollinearities, then it does not neces- b D S1s 2 OLS xx xy sarily shrink and will make interpretation difficult. More sophisticated ways of restricting attention to where Sxx is the sum of squares and products matrix lower-dimensional subspaces Ur of regressors are of the centered x variables and sxy is the vector of their sums of products with y. The OLS estimator is represented by methods such as principal compo- the best fitting and minimum variance linear unbiased nents regression (PCR) and partial least squares (or estimator (best linear unbiased estimator, BLUE), projection to latent structures; PLS) regression. Both with variance methods construct new regressors, linear in the old regressors, and y is regressed (by least squares, LS) 2 1 on a relatively small number of these PCs or PLS varbOLS D Sxx 3 factors. This number r, as well as the ridge constant That OLS yields the best fit does not say, however, υ in RR, is typically chosen by a cross-validation that bOLS is best in a wider sense, or even a good criterion. As an example, Andersson et al. [1] used choice. We will discuss here alternatives to be used PLS regression with two PLS factors to replace the when the x variables are (near) multicollinear, that 52 original variables in the regression. This model is when there are linear combinations among them explained R2 D 84% of the variability, and in a cross- 2 that show little variation. The matrix Sxx is then near validation predicted 80% (necessarily less than R , singular, so varbOLS will have very large elements. but quite close in this case). They could also interpret Correspondingly, the components of bOLS may show their model within the context of their chemometric 2 Shrinkage regression application, as referring basically to the position pat- The most important role of CR is perhaps not as a tern of the chlorine atoms. practical regression method, however, but as a theo- In the standard form of PCR, the new regres- retical device embracing several methods. For D 0 sors ti D ci x are the first PCs of the (centered) x we get back OLS, and the algorithm will stop after from a PCs analysis. This means that the cis are the one step r D 1.For D 1 the criterion function eigenvectors of Sxx that explain as much variation (5) is proportional to the covariance squared and will as possible in x. The space Ur is then spanned by yield PLS, and as !1, Vt will dominate and the first r PCs. In this way directions in x with little we get PCR. A much more general yet simple result, variation are shrunk to 0. The connection with vari- which relates to RR and provides an even wider ation in y is only indirect, however. PLS regression understanding, has been given in [2], as follows. is more efficient in this respect, because it constructs Suppose we are given a criterion that is a func- 2 a first regressor t1 D c1 x to maximize covariance tion of R t, y and Vt, nondecreasing in each of between t1 and y (among coefficient vectors c1 of these arguments. The latter demand is quite natural, fixed length), and then successively t2, t3,etc.soas because we want both of these quantities to have high to maximize covariance with the y residuals after LS values. The regressor maximizing any such criterion regression of y on the previous set of regressors. As under the constraint jcjD1 is then proportional to an algorithm for construction of the PLS regression the RR estimator (4) for some υ, possibly negative or subspaces Ur of the x space, spanned by t1,...tr, infinite. Conversely, for each υ ½ 0orυ<max in this is one of several possible. For a long time PLS (4) there is a criterion for which this is the solution. was only algorithmically defined, until Helland [10] The first factor PLS is obtained as υ !š1,and gave a more explicit characterization by showing that υ !max corresponds to the first factor PCR. The Ur for PLS is spanned by the sequence of vectors sxy, first factor CR with varying ½ 0 usually (but not r1 Sxxsxy,...,Sxx sxy. This representation is of more always) requires the whole union of υ ½ 0andυ< theoretical than practical interest, however, because max. The use of c D bRR from (4) to form a sin- there are numerically better algorithms. gle regressor in LS regression is called least squares A step towards a general view on the choice ridge regression (LSRR) in [2], and differs from RR of shrinkage regression was taken by Stone and by a scalar factor that will be near 1 when υ is small. Brooks [13], who introduced continuum regression Hence RR is closely related to the other construc- (CR), with the criterion function tions. Furthermore, experience shows that typically RR and LSRR for some best small υ yield a similar 2 R t, yVt 5 regression as (the best r)PLSand(bestr) PCR. If different constructions yield quite similar esti- where Rt, y is the sample correlation between t D mators/predictors, what shrinkage method is then c x and y, Vt is the sample variance of t,and ½ 0 preferable? Here is some general advice. is a parameter to be specified. This function of t (or c) should be maximized under the constraint jcjD1. In ž RR and LSRR have the advantage that they only previous terms involve a single factor. 2 ž LSRR only shrinks to compensate for multi- 2 c sxy collinearity, whereas RR always shrinks. R t, y / 6 c Sxxc ž RR has a Bayesian interpretation as posterior Vt / c Sxxc 7 mean for a suitable prior for ˇ (see Bayesian methods and modeling). As in PLS, when a first c D c1, t D t1, has been ž LSRR (like PLS and PCR) yields residuals found, the criterion is applied again on the residuals orthogonal to the fitted relation, unlike RR. from LS regression of y on t1 to find the next c, c D ž PLS and PCR have interpretations in terms of c2, etc. The new c will automatically be orthogonal latent factors, such that for PCR essentially all to the previous ones. In order to institute a regression x variability is in Ur for suitable r, whereas method of this algorithm, the number r of regressors for PLS all x variability that influences y is in and the parameter are chosen so as to optimize the corresponding Ur for a typically somewhat some cross-validation measure. smaller r than for PCR. Shrinkage regression 3 ž The latent factor structure in PCR and effects on shrinkage regression.
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