Generalized additive models in R
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- Ζώνα Μαρής
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1 Generalized additive models in R Magne Aldrin, Norwegian Computing Center and the University of Oslo Sharp workshop, Copenhagen, October 2012
2 Generalized Linear Models - GLM y Distributed with mean µ and perhaps an additional parameter h(µ) = η η = β 0 + β 1 x 1 + β 2 x β p x p η: linear predictor h: link fuction 1
3 Generalized Additive Models - GAM η is additive, but each term can be non-linear η = β 0 + s 1 (x 1 ) + s 2 (x 2 ) + + s p (x p ) η: additive predictor s i ( ) is a smooth function, estimated from the data 2
4 Estimation of s Find s such that fit to data is as best as possible and s is as smooth as possible Can for instance be formulated as Minimize likelihood + i x λ is i (x)dx where s (x) = d 2 s(x)/d 2 x is the second derivative of s and λ s are constants that control the degree of smoothing 3
5 Effective number of parameters Very high λ i gives maximal smoothness a straight line described by 1 parameter (per explanatory variable) Smaller values of λ i gives more flexible functions effective number of parameters > 1 4
6 Choice of smoothness in the mgcv package Default is to estimate the smoothness of each s-function, controlled by λ i or a corresponding sp[i] by generalized cross-validation, minimizing - 2 log likelihood n/(n p) 2 where n = number of observations p = effective number of parameters This tends to give too unsmooth curves 5
7 Ex: Ozone data y = log(no 3 ) Gaussian > require(faraway) > data(ozone) > > require(mgcv) > gamobj<-gam(log(o3)~s(vh)+s(wind)+s(humidity)+s(temp)+s(ibh)+ s(dpg)+s(ibt)+s(vis)+s(doy), family=gaussian(link=identity),data=ozone) > summary(gamobj) Family: gaussian Link function: identity Formula: log(o3) ~ s(vh) + s(wind) + s(humidity) + s(temp) + s(ibh) + s(dpg) + s(ibt) + s(vis) + s(doy) Parametric coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <2e-16 *** 6
8 Approximate significance of smooth terms: edf Ref.df F p-value s(vh) ** s(wind) ** s(humidity) s(temp) ** s(ibh) *** s(dpg) e-11 *** s(ibt) s(vis) e-06 *** s(doy) < 2e-16 *** --- Signif. codes: 0 *** ** 0.01 * R-sq.(adj) = Deviance explained = 84% GCV score = Scale est. = n = 330 7
9 Plot results > pdf("gamozone.pdf") > par(mfrow=c(3,3)) > plot(gamobj) > dev.off() null device 1 8
10 s(vh,1) s(wind,1.02) vh wind humidity s(temp,3.8) s(ibh,2.77) s(humidity,2.41) temp ibh dpg s(ibt,1) s(vis,5.48) s(doy,4.61) s(dpg,3.29) ibt vis doy 9
11 Ex: Precipitation - binary logistic regression > Tryvann.dat<-read.table("/nr/user/aldrin/Sharp/data/Tryvann.dat") > > ### load the mgcv library > require(mgcv) > > gamobj<-gam(p01~p01.l1+p01.l2+p01.l3+ s(prep.l1)+s(prep.l2)+s(prep.l3), data=tryvann.dat,family=binomial(link=logit)) > summary(gamobj) Family: binomial Link function: logit Formula: P01 ~ P01.L1 + P01.L2 + P01.L3 + s(prep.l1) + s(prep.l2) + s(prep.l3) 10
12 Parametric coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) e-15 *** P01.L < 2e-16 *** P01.L e-04 *** P01.L e-06 *** --- Signif. codes: 0 *** ** 0.01 *
13 Approximate significance of smooth terms: edf Ref.df Chi.sq p-value s(prep.l1) e-07 *** s(prep.l2) s(prep.l3) Signif. codes: 0 *** ** 0.01 * R-sq.(adj) = Deviance explained = 9.81% UBRE score = Scale est. = 1 n = 3420 > 12
14 Plot results > pdf("gamlogistic.pdf") > par(mfrow=c(2,2)) > plot(gamobj) > dev.off() null device 1 13
15 s(prep.l1,2.53) Prep.L Prep.L2 s(prep.l3,1.11) s(prep.l2,1.01) Prep.L3 14
16 Ex: Simulated data y Poisson > n<-200 > x1<-rnorm(n) # N(0,1) > x2<-runif(n,-10,10) # Uniform(-10,10) > x3<-rnorm(n) > > eta< *x1-0.2*x2^2 + 0*x3 > mu<-exp(eta) > > y<-rpois(n=n,lambda=mu) > > data.obj<-data.frame(y=y,x1=x1,x2=x2,x3=x3) > > gamobj<-gam(y~s(x1)+s(x2)+s(x3),family=poisson(link=log), data=data.obj) 15
17 Results > summary(gamobj) Family: poisson Link function: log Formula: y ~ s(x1) + s(x2) + s(x3) Parametric coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) e-13 *** --- Signif. codes: 0 *** ** 0.01 *
18 Approximate significance of smooth terms: edf Ref.df Chi.sq p-value s(x1) <2e-16 *** s(x2) <2e-16 *** s(x3) Signif. codes: 0 *** ** 0.01 * R-sq.(adj) = Deviance explained = 97.4% UBRE score = Scale est. = 1 n =
19 Plot results > pdf("gampoisson.pdf") > par(mfrow=c(2,2)) > plot(gamobj) > dev.off() null device 1 18
20 s(x1,1.51) x x2 s(x3,1) s(x2,4.36) x3 19
21 Option: Terms can be forced to be linear > ### The terms can be forced to be linear > gamobj<-gam(log(o3)~vh+wind+s(humidity)+s(temp)+s(ibh)+ s(dpg)+ibt+s(vis)+s(doy), family=gaussian(link=identity),data=ozone) > summary(gamobj) Family: gaussian Link function: identity Formula: log(o3) ~ vh + wind + s(humidity) + s(temp) + s(ibh) + s(dpg) + ibt + s(vis) + s(doy) Parametric coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) * vh ** wind ** ibt
22 Approximate significance of smooth terms: edf Ref.df F p-value s(humidity) s(temp) ** s(ibh) *** s(dpg) e-11 *** s(vis) e-06 *** s(doy) < 2e-16 *** --- Signif. codes: 0 *** ** 0.01 * R-sq.(adj) = Deviance explained = 83.9% GCV score = Scale est. = n =
23 Controlling the smoothness by the sp option > gamobj<-gam(log(o3)~s(vh)+s(wind)+s(humidity)+s(temp)+s(ibh)+ s(dpg)+s(ibt)+s(vis)+s(doy),sp=rep(0.5,9), family=gaussian(link=identity),data=ozone) > summary(gamobj) Family: gaussian Link function: identity Formula: log(o3) ~ s(vh) + s(wind) + s(humidity) + s(temp) + s(ibh) + s(dpg) + s(ibt) + s(vis) + s(doy) Parametric coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <2e-16 *** --- Signif. codes: 0 *** ** 0.01 *
24 Approximate significance of smooth terms: edf Ref.df F p-value s(vh) s(wind) s(humidity) s(temp) e-07 *** s(ibh) * s(dpg) e-06 *** s(ibt) s(vis) *** s(doy) e-06 *** --- Signif. codes: 0 *** ** 0.01 * R-sq.(adj) = Deviance explained = 78.6% GCV score = Scale est. = n =
25 Plot results > pdf("gamsmoothozone.pdf") > par(mfrow=c(3,3)) > plot(gamobj) > dev.off() null device 1 24
26 s(vh,1.63) s(wind,2.56) vh wind humidity s(temp,1.85) s(ibh,1.47) s(humidity,1.56) temp ibh dpg s(ibt,1.71) s(vis,2.04) s(doy,1.41) s(dpg,1.96) ibt vis doy 25
27 AIC or BIC to choose degree of smoothness > gamobj<-gam(log(o3)~s(vh)+s(wind)+s(humidity)+s(temp)+s(ibh)+ s(dpg)+s(ibt)+s(vis)+s(doy), family=gaussian(link=identity),data=ozone) > sp<-gamobj$sp > > tuning.scale<-c(1e-5,1e-4,1e-3,1e-2,1e-1,1e0,1e1,1e2,1e3,1e4,1e5) > ###tuning.scale<-c( ,0.0001,0.001,0.01,0.1,1,10,100,1000,10000 > scale.exponent<-log10(tuning.scale) > n.tuning<-length(tuning.scale) > edf<-rep(na,n.tuning) > min2ll<-rep(na,n.tuning) > aic<-rep(na,n.tuning) > bic<-rep(na,n.tuning) 26
28 > for (i in 1:n.tuning) { + gamobj<-gam(log(o3)~s(vh)+s(wind)+s(humidity)+s(temp)+s(ibh)+ s(dpg)+s(ibt)+s(vis)+s(doy), family=gaussian(link=identity),data=ozone, sp=tuning.scale[i]*sp) + min2ll[i]<--2*loglik(gamobj) + edf[i]<-sum(gamobj$edf)+1 + aic[i]<-aic(gamobj) + bic[i]<-bic(gamobj) + } 27
29 Plot results > pdf("aicbicozone.pdf") > par(mfrow=c(2,2)) > plot(scale.exponent,min2ll,type="b",main="-2 log likelihood") > plot(scale.exponent,edf,ylim=c(0,70),type="b",main="effective number > plot(scale.exponent,aic,type="b",main="aic") > plot(scale.exponent,bic,type="b",main="bic") > dev.off() null device 1 28
30 2 log likelihood effective number of parameters min2ll edf scale.exponent scale.exponent AIC BIC aic bic scale.exponent scale.exponent 29
31 Scaling corresponding to minimum BIC > min.bic<-1e100 > opt.tuning.scale<-null > for (i in 1:n.tuning) { + if (bic[i]<min.bic) { + min.bic<-bic[i] + opt.tuning.scale<-tuning.scale[i] + } + } > opt.sp<-opt.tuning.scale*sp 30
32 Estimate final model with optimal value of sp > gamobj<-gam(log(o3)~s(vh)+s(wind)+s(humidity)+s(temp)+s(ibh)+ s(dpg)+s(ibt)+s(vis)+s(doy), family=gaussian(link=identity),data=ozone,sp=opt.sp) > summary(gamobj) Family: gaussian Link function: identity Formula: log(o3) ~ s(vh) + s(wind) + s(humidity) + s(temp) + s(ibh) + s(dpg) + s(ibt) + s(vis) + s(doy) Parametric coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <2e-16 *** --- Signif. codes: 0 *** ** 0.01 *
33 Approximate significance of smooth terms: edf Ref.df F p-value s(vh) s(wind) * s(humidity) s(temp) e-05 *** s(ibh) ** s(dpg) e-07 *** s(ibt) s(vis) e-05 *** s(doy) < 2e-16 *** --- Signif. codes: 0 *** ** 0.01 * R-sq.(adj) = Deviance explained = 81% GCV score = Scale est. = n =
34 Plot results > pdf("gamoptsmoothozone.pdf") > par(mfrow=c(3,3)) > plot(gamobj) > dev.off() null device 1 33
35 s(vh,1) s(wind,1) vh wind humidity s(temp,2.1) s(ibh,1.66) s(humidity,1.37) temp ibh dpg s(ibt,1) s(vis,3.17) s(doy,2.46) s(dpg,1.73) ibt vis doy 34
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