Chapter 7. Transformation

Transcription

1 Chapter 7 Trasformatio

2 7.. Trasformatio Is liear regressio appropriate?

3 7.. Trasformatio The assumptio of liear relatioship does ot alwas hold We ca trasform The predictor The respose Both to achieve the liear relatioship

4 Power trasformatio Power trasformatio U U Wat a liear relatioship BraiWt BodWt e λ= a - b 0 i.e. log U c 0.33 d 0.5 Which λ will ou choose?

5 Practical suggestios Log rule: log trasform is useful whe Observatios are positive Rage of variable is huge i.e. the biggest observatios is a much bigger tha the smallest Rage rule: No trasformatio is useful if Rage of variable is too small

6 Iterpretatio λ > 0 BraiWt BodWt BraiWt Artificial usuall has o phsical meaig λ = 0 : log trasformatio BodWt e Correspodig to a phsical model allometric model e log BraiWt log BodWt BraiWt BodWt e Multiplicative error

7 Improvig Power trasformatio Power trasformatio Scaled power trasformatio Advatage U lim 0 U s log log Cotiuous fuctio of λ : Preserve the directio of associatio True model : E egative assocatio b/w ad Power trasform: E / positive assocatio b/w ad 0 0 Scaled power trasform: E s egative assocatio b/w ad s

8 Procedures to look for trasformatio Method : Draw ma fitted curves i. e. plot x ˆ for various x where ˆ ˆ ˆ 0 x 0... Method : Draw ma scatter plots vs vs / vs log Method 3: plot λ agaist RSS of fittig agaist ψ λ the fid the λ that miimizes RSS. Or choose λ i the set --/0

9 Example =Height of tree =diameter of tree M: Draw ma curves M: The best scatterplots M3: Miimize RSS: RSSλ=0=3. RSSλ==44.5 RSSλ=-=54.8. Coclusio: Height = β o + β logdiameter + e

10 Methods for multiple regressio Three approaches Iverse fitted value plot ˆ Plot agaist Fid trasformatio for that matches the above patter Box Cox trasformatio A modificatio of scaled power trasformatio but applied to. Modified power trasform for each predictor

11 Iverse fitted value plot. Fit a liear regressio betwee ad get the fitted value ˆ ˆ. Plot ˆ -axis agaist x-axis 3. Fix a λ fit ˆ agaist s ad obtai ˆ ˆ ˆ 0 s 4. Draw the fitted curve ˆ o the graph see if it matches the patter i. ˆ ˆ ˆ ˆ ˆ Match 0 s 5. Repeat 3-4 to search for the best λ sa λ* * ad s areliearl related Regress * agaist s

12 Example of Iverse fitted value Read data highwa.data=read.table"c:/highwa.txt"header=t #Or libraralr3; highwa.data=highwa Step : Multiple regressio fit=lmrate~logadt+logtrks+shld+logledata=highwa.data Step : Plot fitted values agaist.hat=fit$fitted.values =highwa.data$rate plot.hat ablielm.hat~ Step 3+4: Regressio: Fitted value agaist trasformed ad plot the Newl fitted values Psi.0=log fit=lm.hat~psi.0 poitsfit$fitted.valuescol= Trial : Step 3+4: Psi.mius=-/- fit=lm.hat~psi.mius poitsfit$fitted.valuescol=3 More R techiques: Sort to draw the lie. order.=order ordered.=[order.] ordered.fit=fit$fitted.values[order.] ordered.fit=fit$fitted.values[order.] liesordered.ordered.fittpe="l"col= liesordered.ordered.fittpe="l"col=3 I this case λ=0 seems to be the best.

13 Box-Cox trasformatio. Modified power famil. Advatage: Uit of is the same as for all λ 3. Model Assumptio: 4. How to choose λ? Fix a λ fit model * for ad obtai RSSλ Tr various λ ad fid the oe which miimizes RSSλ * ' x x E M 0 if log... 0 if S M M

14 Example of Box-Cox trasformatio E M x ' x Modified power famil * M... log highwa.data=read.table"c:/highwa.txt"header=t =highwa.data$rate =legth gm=prod^{/}... if 0 if 0 Choose log or λ=- 0.5 #A lambda=- Trasform.A=-gm^*/- fit.a=lmtrasform.a~logadt+logtrks+shld+logledata=highwa.data Rss.A=sumfit.A$residuals^ #G lambda= Trasform.G=//gm*^- fit.g=lmtrasform.g~logadt+logtrks+shld+logledata=highwa.data Rss.G=sumfit.G$residuals^ plotc--/0/3/crss.arss.brss.crss.drss.erss.frss.gtpe="l"

15 Example of Box-Cox trasformatio # Read data highwa.data=read.table"c:/highwa.txt"header=t =highwa.data$rate =legth gm=prod^{/} #A lambda=- Trasform.A=-gm^*/- fit.a=lmtrasform.a~logadt+logtrks+shld+logle data=highwa.data Rss.A=sumfit.A$residuals^ #B lambda=-/ Trasform.B=-*gm^3/*^-/- fit.b=lmtrasform.b~logadt+logtrks+shld+logle data=highwa.data Rss.B=sumfit.B$residuals^ #C lambda=0 Trasform.C=gm*log fit.c=lmtrasform.c~logadt+logtrks+shld+logl edata=highwa.data Rss.C=sumfit.C$residuals^ #D lambda=/3 Trasform.D=3*gm^/3*^/3- fit.d=lmtrasform.d~logadt+logtrks+shld+logledata= highwa.data Rss.D=sumfit.D$residuals^ #E lambda=/ Trasform.E=*gm^/*sqrt- fit.e=lmtrasform.e~logadt+logtrks+shld+logledata= highwa.data Rss.E=sumfit.E$residuals^ #F lambda= Trasform.F= fit.f=lmtrasform.f~logadt+logtrks+shld+logledata=h ighwa.data Rss.F=sumfit.F$residuals^ #G lambda= Trasform.G=//gm*^- fit.g=lmtrasform.g~logadt+logtrks+shld+logledata= highwa.data Rss.G=sumfit.G$residuals^ plotc-- /0/3/cRss.ARss.BRss.CRss.DRss.ERs s.frss.gtpe="l"

16 Modified power trasformatio for all predictors Modified power famil Trasform predictors so that each pair of variables i the scatterplot matrix has a liear relatioship p p M M M p 0 if log... 0 if S M

17 Modified power trasformatio for all predictors Trasformatio with modified power famil... p M M... M p p Not a eas task. Ol use it if other methods do ot work well

18 Trasformatio of o-positive variables Problem of o-positive variables e.g. λ= S x S x we ca t distiguish betwee x ad x. logx is udefied if x<0. Solutios U Fid a sufficietl large ad trasform U to eo-johso trasformatio S U U 0 J U S U U 0 x

19 Fial Remarks No eed to trasform factors e.g. x F 0 F 0 group group we look at β to see the mea differet betwee the groups. Trasformig the dumm does t help. There is o correct wa of trasformatio oce ou come up with trasformatio... p p 0 which looks roughl liear i the scatterplot matrix the it is ok to fit p p p