3.4 Shrinkage Methods Prostate Cancer Data Example (Continued) Ridge Regression

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1 3.3.4 Prstate Cancer Data Example (Cntinued) 3.4 Shrinkage Methds 61 Table 3.3 shws the cefficients frm a number f different selectin and shrinkage methds. They are best-subset selectin using an all-subsets search, ridge regressin, the lass, principal cmpnents regressin and partial least squares. Each methd has a cmplexity parameter, and this was chsen t minimize an estimate f predictin errr based n tenfld crss-validatin; full details are given in Sectin Briefly, crss-validatin wrks by dividing the training data randmly int ten equal parts. The learning methd is fit fr a range f values f the cmplexity parameter t nine-tenths f the data, and the predictin errr is cmputed n the remaining ne-tenth. This is dne in turn fr each ne-tenth f the data, and the ten predictin errr estimates are averaged. Frm this we btain an estimated predictin errr curve as a functin f the cmplexity parameter. Nte that we have already divided these data int a training set f size 67 and a test set f size 30. Crss-validatin is applied t the training set, since selecting the shrinkage parameter is part f the training prcess. The test set is there t judge the perfrmance f the selected mdel. The estimated predictin errr curves are shwn in Figure 3.7. Many f the curves are very flat ver large ranges near their minimum. Included are estimated standard errr bands fr each estimated errr rate, based n the ten errr estimates cmputed by crss-validatin. We have used the ne-standard-errr rule we pick the mst parsimnius mdel within ne standard errr f the minimum (Sectin 7.10, page 244). Such a rule acknwledges the fact that the tradeff curve is estimated with errr, and hence takes a cnservative apprach. Best-subset selectin chse t use the tw predictrs lcvl and lweight. The last tw lines f the table give the average predictin errr (and its estimated standard errr) ver the test set. 3.4 Shrinkage Methds By retaining a subset f the predictrs and discarding the rest, subset selectin prduces a mdel that is interpretable and has pssibly lwer predictin errr than the full mdel. Hwever, because it is a discrete prcess variables are either retained r discarded it ften exhibits high variance, and s desn t reduce the predictin errr f the full mdel. Shrinkage methds are mre cntinuus, and dn t suffer as much frm high variability Ridge Regressin Ridge regressin shrinks the regressin cefficients by impsing a penalty n their size. The ridge cefficients minimize a penalized residual sum f

2 62 3. Linear Methds fr Regressin All Subsets Ridge Regressin Subset Size Lass Degrees f Freedm Principal Cmpnents Regressin Shrinkage Factr s Number f Directins Partial Least Squares Number f Directins FIGURE 3.7. Estimated predictin errr curves and their standard errrs fr the varius selectin and shrinkage methds. Each curve is pltted as a functin f the crrespnding cmplexity parameter fr that methd. The hrizntal axis has been chsen s that the mdel cmplexity increases as we mve frmleftt right. The estimates f predictin errr and their standard errrs were btained by tenfld crss-validatin; full details are given in Sectin The least cmplex mdel within ne standard errr f the best is chsen, indicated by thepurple vertical brken lines.

3 3.4 Shrinkage Methds 63 TABLE 3.3. Estimated cefficients and test errr results, fr different subset and shrinkage methds applied t the prstate data. The blank entries crrespnd t variables mitted. Term LS Best Subset Ridge Lass PCR PLS Intercept lcavl lweight age lbph svi lcp gleasn pgg Test Errr Std Errr squares, ˆβ ridge =argmin β { N ( yi β 0 i=1 p ) 2 p x ij β j + λ β 2 j }. (3.41) Here λ 0 is a cmplexity parameter that cntrls the amunt f shrinkage: the larger the value f λ, the greater the amunt f shrinkage. The cefficients are shrunk tward zer (and each ther). The idea f penalizing by the sum-f-squares f the parameters is als used in neural netwrks, where it is knwn as weight decay (Chapter 11). An equivalent way t write the ridge prblem is ˆβ ridge =argmin β subject t N ( y i β 0 i=1 p βj 2 t, p ) 2, x ij β j (3.42) which makes explicit the size cnstraint n the parameters. There is a net-ne crrespndence between the parameters λ in (3.41) and t in (3.42). When there are many crrelated variables in a linear regressin mdel, their cefficients can becme prly determined and exhibit high variance. A wildly large psitive cefficient n ne variable can be canceled by a similarly large negative cefficient n its crrelated cusin. By impsing a size cnstraint n the cefficients, as in (3.42), this prblem is alleviated. The ridge slutins are nt equivariant under scaling f the inputs, and s ne nrmally standardizes the inputs befre slving (3.41). In additin,

4 64 3. Linear Methds fr Regressin ntice that the intercept β 0 has been left ut f the penalty term. Penalizatin f the intercept wuld make the prcedure depend n the rigin chsen fr Y ; that is, adding a cnstant c t each f the targets y i wuld nt simply result in a shift f the predictins by the same amunt c. It can be shwn (Exercise 3.5) that the slutin t (3.41) can be separated int tw parts, after reparametrizatin using centered inputs: each x ij gets replaced by x ij x j. We estimate β 0 by ȳ = 1 N N 1 y i. The remaining cefficients get estimated by a ridge regressin withut intercept, using the centered x ij. Hencefrth we assume that this centering has been dne, s that the input matrix X has p (rather than p + 1) clumns. Writing the criterin in (3.41) in matrix frm, RSS(λ) =(y Xβ) T (y Xβ)+λβ T β, (3.43) the ridge regressin slutins are easily seen t be ˆβ ridge =(X T X + λi) 1 X T y, (3.44) where I is the p p identity matrix. Ntice that with the chice f quadratic penalty β T β, the ridge regressin slutin is again a linear functin f y. The slutin adds a psitive cnstant t the diagnal f X T X befre inversin. This makes the prblem nnsingular, even if X T X is nt f full rank, and was the main mtivatin fr ridge regressin when it was first intrduced in statistics (Herl and Kennard, 1970). Traditinal descriptins f ridge regressin start with definitin (3.44). We chse t mtivate it via (3.41) and (3.42), as these prvide insight int hw it wrks. Figure 3.8 shws the ridge cefficient estimates fr the prstate cancer example, pltted as functins f df(λ), the effective degrees f freedm implied by the penalty λ (defined in (3.50) n page 68). In the case f rthnrmal inputs, the ridge estimates are just a scaled versin f the least squares estimates, that is, ˆβ ridge = ˆβ/(1 + λ). Ridge regressin can als be derived as the mean r mde f a psterir distributin, with a suitably chsen prir distributin. In detail, suppse y i N(β 0 + x T i β,σ2 ), and the parameters β j are each distributed as N(0,τ 2 ), independently f ne anther. Then the (negative) lg-psterir density f β, withτ 2 and σ 2 assumed knwn, is equal t the expressin in curly braces in (3.41), with λ = σ 2 /τ 2 (Exercise 3.6). Thus the ridge estimate is the mde f the psterir distributin; since the distributin is Gaussian, it is als the psterir mean. The singular value decmpsitin (SVD) f the centered input matrix X gives us sme additinal insight int the nature f ridge regressin. This decmpsitin is extremely useful in the analysis f many statistical methds. The SVD f the N p matrix X has the frm X = UDV T. (3.45)

5 3.4 Shrinkage Methds 65 Cefficients lcavl lweight age lbph svi lcp gleasn pgg45 df(λ) FIGURE 3.8. Prfiles f ridge cefficients fr the prstate cancer example, as the tuning parameter λ is varied. Cefficients are pltted versus df(λ), the effective degrees f freedm. A vertical line is drawn at df = 5.0, thevaluechsenby crss-validatin.

6 66 3. Linear Methds fr Regressin Here U and V are N p and p p rthgnal matrices, with the clumns f U spanning the clumn space f X, and the clumns f V spanning the rw space. D is a p p diagnal matrix, with diagnal entries d 1 d 2 d p 0 called the singular values f X. If ne r mre values d j =0, X is singular. Using the singular value decmpsitin we can write the least squares fitted vectr as X ˆβ ls = X(X T X) 1 X T y = UU T y, (3.46) after sme simplificatin. Nte that U T y are the crdinates f y with respect t the rthnrmal basis U. Nte als the similarity with (3.33); Q and U are generally different rthgnal bases fr the clumn space f X (Exercise 3.8). Nw the ridge slutins are X ˆβ ridge = X(X T X + λi) 1 X T y = UD(D 2 + λi) 1 DU T y = p d 2 j u j d 2 j + λut j y, (3.47) where the u j are the clumns f U. Nte that since λ 0, we have d 2 j /(d2 j + λ) 1. Like linear regressin, ridge regressin cmputes the crdinates f y with respect t the rthnrmal basis U. It then shrinks these crdinates by the factrs d 2 j /(d2 j + λ). This means that a greater amunt f shrinkage is applied t the crdinates f basis vectrs with smaller d 2 j. What des a small value f d 2 j mean? The SVD f the centered matrix X is anther way f expressing the principal cmpnents f the variables in X. The sample cvariance matrix is given by S = X T X/N, and frm (3.45) we have X T X = VD 2 V T, (3.48) which is the eigen decmpsitin f X T X (and f S, up t a factr N). The eigenvectrs v j (clumns f V) are als called the principal cmpnents (r Karhunen Leve) directins f X. The first principal cmpnent directin v 1 has the prperty that z 1 = Xv 1 has the largest sample variance amngst all nrmalized linear cmbinatins f the clumns f X. This sample variance is easily seen t be Var(z 1 )=Var(Xv 1 )= d2 1 N, (3.49) and in fact z 1 = Xv 1 = u 1 d 1. The derived variable z 1 is called the first principal cmpnent f X, and hence u 1 is the nrmalized first principal

7 3.4 Shrinkage Methds Largest Principal Cmpnent Smallest Principal Cmpnent X 1 X2 FIGURE 3.9. Principal cmpnents f sme input data pints. The largest principal cmpnent is the directin that maximizes the variance f the prjected data, and the smallest principal cmpnent minimizes that variance. Ridge regressin prjects y nt these cmpnents, and then shrinks the cefficients f the lw variance cmpnents mre than the high-variance cmpnents. cmpnent. Subsequent principal cmpnents z j have maximum variance d 2 j /N, subject t being rthgnal t the earlier nes. Cnversely the last principal cmpnent has minimum variance. Hence the small singular values d j crrespnd t directins in the clumn space f X having small variance, and ridge regressin shrinks these directins the mst. Figure 3.9 illustrates the principal cmpnents f sme data pints in tw dimensins. If we cnsider fitting a linear surface ver this dmain (the Y -axis is sticking ut f the page), the cnfiguratin f the data allw us t determine its gradient mre accurately in the lng directin than the shrt. Ridge regressin prtects against the ptentially high variance f gradients estimated in the shrt directins. The implicit assumptin is that the respnse will tend t vary mst in the directins f high variance f the inputs. This is ften a reasnable assumptin, since predictrs are ften chsen fr study because they vary with the respnse variable, but need nt hld in general.

8 68 3. Linear Methds fr Regressin In Figure 3.7 we have pltted the estimated predictin errr versus the quantity df(λ) = tr[x(x T X + λi) 1 X T ], = tr(h λ ) = p d 2 j + λ. (3.50) d 2 j This mntne decreasing functin f λ is the effective degrees f freedm f the ridge regressin fit. Usually in a linear-regressin fit with p variables, the degrees-f-freedm f the fit is p, the number f free parameters. The idea is that althugh all p cefficients in a ridge fit will be nn-zer, they are fit in a restricted fashin cntrlled by λ. Nte that df(λ) =p when λ = 0 (n regularizatin) and df(λ) 0 as λ. Of curse there is always an additinal ne degree f freedm fr the intercept, which was remved apriri. This definitin is mtivated in mre detail in Sectin and Sectins In Figure 3.7 the minimum ccurs at df(λ) = 5.0. Table 3.3 shws that ridge regressin reduces the test errr f the full least squares estimates by a small amunt The Lass The lass is a shrinkage methd like ridge, with subtle but imprtant differences. The lass estimate is defined by ˆβ lass = argmin β N ( y i β 0 i=1 subject t p ) 2 x ij β j p β j t. (3.51) Just as in ridge regressin, we can re-parametrize the cnstant β 0 by standardizing the predictrs; the slutin fr ˆβ 0 is ȳ, and thereafter we fit a mdel withut an intercept (Exercise 3.5). In the signal prcessing literature, the lass is als knwn as basis pursuit (Chen et al., 1998). We can als write the lass prblem in the equivalent Lagrangian frm { 1 ˆβ lass =argmin β 2 N ( yi β 0 i=1 p ) 2 p } x ij β j + λ β j. (3.52) Ntice the similarity t the ridge regressin prblem (3.42) r (3.41): the L 2 ridge penalty p 1 β2 j is replaced by the L 1 lass penalty p 1 β j. This latter cnstraint makes the slutins nnlinear in the y i, and there is n clsed frm expressin as in ridge regressin. Cmputing the lass slutin

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