Lecture 2: Prelude to the big shrink
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1 Lecture 2: Prelude to the bg shrnk
2 Last tme A slght detour wth vsualzaton tools (hey, t was the frst day... why not start out wth somethng pretty to look at?) Then, we consdered a smple 120a-style regresson problem that seemed to requre somethng other than a lnear ft The concept of an orthogonal polynomal was ntroduced and some very prelmnary computatonal smplfcatons of the usual OLS estmates were hnted at
3 Today We frst consder orthogonal polynomals n more detal We wll then revst the OLS crteron and spend a lttle tme examnng how to compute regresson coeffcents We derve a smple expressons for our frend the resdual sum of squares (RSS) n ths context and reexamne subset selecton We end by consderng the bas-varance tradeoff and get our frst glmpse at so-called shrnkage estmates
4 Fsh dmensons (II) Last tme we had a look at a data set collected to examne mercury content n fsh We quckly boled down a farly nnocuous problem nvolvng the length and weght of fsh To model these data, we consdered elaboratons of the smple lnear model; polynomals seemed lke a natural way to go weght length
5 A smple polynomal ft > ft = lm(weght~poly(length,degree=2),data=fsh) > summary(ft) Call: lm(formula = weght ~ poly(length, degree = 2), data = fsh) Resduals: Mn 1Q Medan 3Q Max Coeffcents: Estmate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** poly(length, degree = 2) < 2e-16 *** poly(length, degree = 2) e-14 *** --- Sgnf. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 Resdual standard error: on 95 degrees of freedom Multple R-Squared: , Adjusted R-squared: F-statstc: on 2 and 95 DF, p-value: < 2.2e-16
6 A smple polynomal ft > newdata = data.frame(length=seq(25,60,len=100)) > y = predct(ft,newdata=newdata) > lnes(newdata$len,y) weght length Not a bad ft, but what s R dong under the hood?
7 Orthogonal desgns The R command poly creates the desgn matrx assocated wth the space of orthogonal polynomals n fsh length Ths s just one example of an desgn matrx that has orthogonal (n ths case, orthonormal) columns Many facts about ordnary least squares become smpler n ths context
8 plot of X[,1] plot of X[,2] Orthogonal polynomals n R > X = poly(fsh$len,degree=5) > class(x) [1] "poly" "matrx" # X s a just a fancy matrx > round(t(x)%*%x, dgts=2) plot of X[,3] plot of X[,5] plot of X[,4]
9 plot of X[,1] plot of X[,2] Orthogonal polynomals n R By examnng the cross terms n the expresson t(x)%*%x we see that all the off-dagonal elements are zero so > sum(x[,1]*x[,2]) [1] 0 and that the dagonal elements are 1 or > sum(x[,1]^2) [1] 1 plot of X[,3] plot of X[,5] plot of X[,4]
10 Regresson revsted Recall the basc assumptons for the normal lnear model We have a response y varables x 1, x 2,..., x p and (possble) predctor We hypothesze a lnear model for the response y = β 1x 1 + β 2x 2 + β px p + ɛ where ɛ varance has a normal dstrbuton wth mean 0 and σ 2
11 Regresson revsted Then, we collect data... We have a n observatons y 1,..., y n and each response s assocated wth the predctors x 1, x 2,..., x p y Then, accordng to our lnear model y = β 1x 1 + β 2x 2 + β px p + ɛ ɛ and we assume the are ndependent, each havng a normal dstrbuton wth mean 0 and varance σ 2
12 Regresson revsted To estmate the coeffcents β1, β2,..., βp we turn to OLS, ordnary least squares (ths s also maxmum lkelhood under our normal lnear model) We want to choose β 1, β 2,..., β p to mnmze the OLS crteron n [y β 1 x 1 β 2 x 2 β p x p ] 2 =1 You recall that ths s just the sum of squared errors that we ncur f we predct y wth β 1 x 1 + β 2 x β p x p
13 Regresson revsted Whle we would never do ths n even the smplest regresson context, let s see what happens f we expand the OLS crteron Ths looks bad, but t s just algebra! [y β 1 x 1 β p x p ] 2 = y 2 + β1 2 x βp 2 x 2 p 2β 1 y x 1 2β p y x p 2β 1 β 2 x 1 x 2 2β (p 1) β p x (p 1) x p
14 Regresson revsted Whle we would never do ths n even the smplest regresson context, let s see what happens f we expand the OLS crteron Squared terms Cross terms wth y Remanng cross terms Ths looks bad, but t s just algebra! [y β 1 x 1 β p x p ] 2 = y 2 + β1 2 x βp 2 x 2 p 2β 1 y x 1 2β p y x p 2β 1 β 2 x 1 x 2 2β (p 1) β p x (p 1) x p
15 Regresson revsted For orthogonal polynomals, we recall that cross terms not nvolvng the response are zero The expresson n R > sum(x[,1]*x[,2]) x 1 x 2 = 0 [1] 0 and what s more, the squared terms not nvolvng the response are one The expresson n R > sum(x[,1]^2) [1] 1 The equvalent math expresson The equvalent math expresson x 2 1 = 1
16 Regresson revsted Squared terms Cross terms wth y And so thngs smplfy... Remanng cross terms [y β 1 x 1 β p x p ] 2 = y 2 + β1 2 x βp 2 x 2 p 2β 1 y x 1 2β p y x p 2β 1 β 2 x 1 x 2 2β (p 1) β p x (p 1) x p
17 Regresson revsted And so thngs smplfy... Squared terms Cross terms wth y These terms are zero [y β 1 x 1 β p x p ] 2 = y 2 + β1 2 x βp 2 x 2 p 2β 1 y x 1 2β p y x p 2β 1 β 2 x 1 x 2 2β (p 1) β p x (p 1) x p
18 Regresson revsted And so thngs smplfy... [y β 1 x 1 β p x p ] 2 = 1 1 y 2 + β1 2 x βp 2 x Many of these are 1 2 p Cross terms wth y 2β 1 y x 1 2β p y x p These terms are zero 2β 1 β 2 x 1 x 2 2β (p 1) β p x (p 1) x p
19 Regresson revsted Collectng terms, we arrve at the followng expresson [y β 1 x 1 β p x p ] 2 = y 2 + β βp 2 2β 1 y x 1 2β p y x p Mnmzng the sum of squared errors s now trval!
20 Regresson revsted For example, we can dfferentate ths expresson wth respect to and solve to fnd β 1 β 1 = y x 1
21 Regresson revsted In general, for our orthogonal polynomals from R, we can compute the regresson coeffcents wth smple sums The expresson n R > sum(fsh$weght*x[,1]) [1] The equvalent math expresson β 1 = y x 1 > sum(fsh$weght*x[,2]) [1] > sum(fsh$weght*x[,5]) [1] β 2 =. β p = y x 2 y x p
22 Results from the ft n R > ft = lm(weght~poly(length,degree=5),data=fsh) > summary(ft) Call: lm(formula = weght ~ poly(length, degree = 5), data = fsh) Resduals: Mn 1Q Medan 3Q Max Coeffcents: Estmate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** poly(length, degree = 5) < 2e-16 *** poly(length, degree = 5) e-14 *** poly(length, degree = 5) poly(length, degree = 5) poly(length, degree = 5) Sgnf. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 Resdual standard error: 210 on 92 degrees of freedom Multple R-Squared: , Adjusted R-squared: F-statstc: on 5 and 92 DF, p-value: < 2.2e-16
23 Regresson revsted From these expressons, we can also easly derve the dstrbuton of the regresson estmates under the normal lnear model We have assumed that y = β 1x 1 + β 2x 2 + β px p + ɛ Substtutng nto the expresson for β 1 we get β 1 = y x 1 = (β 1x β px p + ɛ )x 1 Usng orthogonalty we fnd β 1 = β 1 + ɛ x 1
24 Regresson revsted Ths last expresson tells us everythng ɛ Snce the are assumed normal, our estmates are also normally dstrbuted Each ɛ has mean 0 and varance σ 2 so that β 1 = β 1 + ɛ x 1 has mean β 1 and varance var( β 1 ) = var(ɛ )x 2 1 = σ 2 x 2 1 = σ 2
25 Regresson revsted Contnung the smplfcaton parade, let s look at the resdual sum of squares, the mnmum value of the OLS crteron [ y β 1 x 1 β p x p ] 2 = y 2 + β β p 2 2 β 1 y x 1 2 β p y x p = y 2 + β β 2 p 2 β 2 1 β 2 p = y 2 p β 2 k k=1
26 Model selecton Let s put some of ths to work: How do we decde whch varables to nclude n a model? In 120a you learned about several model selecton crtera; AIC, BIC and Mallows Cp In regresson problems, AIC s gven by RSS + 2pσ 2 The goal s to fnd models that make these crtera small
27 Model selecton Let s put some of ths to work: How do we decde whch varables to nclude n a model? Usng our expresson for the resdual sum of squares, the AIC crteron becomes n y 2 p β 2 k + 2pσ 2 = n y 2 + p (2σ 2 β k) 2 =1 k=1 =1 k=1 That means, we fnd a mnmum by keepng all the varables for whch the regresson coeffcents satsfy β k σ > 2
28 Model selecton Have we seen ths before? If we know the error varance σ 2, then ths s just a hypothess test (a z-test rather than a t-test) β k σ > 2 The cutoff value of 1.41 corresponds to a sgnfcance level of 0.16 (Why?)
29 Results from the ft n R > ft = lm(weght~poly(length,degree=5),data=fsh) > summary(ft) Call: lm(formula = weght ~ poly(length, degree = 5), data = fsh) Resduals: Mn 1Q Medan 3Q Max Coeffcents: Estmate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** poly(length, degree = 5) < 2e-16 *** poly(length, degree = 5) e-14 *** poly(length, degree = 5) poly(length, degree = 5) poly(length, degree = 5) Sgnf. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 Resdual standard error: 210 on 92 degrees of freedom Multple R-Squared: , Adjusted R-squared: F-statstc: on 5 and 92 DF, p-value: < 2.2e-16
30 Model selecton Therefore, when usng AIC or a smlar knd of crteron, we are n fact dong smple hypothess tests to decde whether to keep or kll coeffcents When the regressors do not have ths nce structure, we had to resort to stepwse methods to approxmately mnmze AIC AIC coeffcent estmate!4! !4! In the orthogonal case, t s easy; at the left s another way to thnk about the AIC crteron beta!hat
31 Model selecton BIC dffers from AIC because t has a slghtly heaver penalty on model sze RSS + pσ 2 log n Ths translates to a keep or kll rule of the form BIC coeffcent estmate!4! β k σ > log n!4! beta!hat For our fsh data, log(98) = 4.6
32 Next tme In ths context, we see model selecton as applyng a knd of thresholdng operaton on the OLS estmates If they reach a certan sgnfcance level, we keep them, otherwse we drop the assocated varables from our model Next tme, we wll see stuatons n whch we not only drop terms, but also possbly shrnk the remanng coeffcents
33 Next tme To motvate ths, we wll frst consder the overall error n our estmated coeffcents p ( β k 2 βk) 2 k=1 By consderng the expected value of ths quantty, we wll derve the so-called bas-varance tradeoff But that joy wll wat for next lecture!
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