PubH 7440 Spring 2010 Midterm 2 April

Transcription

1 ubh 7440 Sprg 00 Mderm Aprl roblem a: Because \hea^_ s a lear combao of ormal radom arables wll also be ormal. Thus he mea ad arace compleel characerze he dsrbuo. We also use ha he Z ad \hea^{-}_ are depede. N N N α ( σ α Z N( α α σ α N( 0 σ ( α ( α α α σ N( 0 σ ( α ( α σ σ ( α COV ( X Y ( σ Where X N ( α σ ad Y N( 0 σ ( α Noe : EX ad VAR(X do o deped up Y ad ce ersa. Hece her coarace 0. roblem b: Whle s rue ha Alder s prodes beer MCMC coergece some cases he model preseed here does o prese a auocorrelao problems wh ordar Gbbs samplg. Therefore he oerrelaxao of Adler s s compesag for a problem ha does o exs creag a ew problem of large *egae* lag- auocorrelao (ad he large pose coug hs alerag maer. I he fgures below hs effec s que clear. Frs he full codoals: Full Baes Model:

2 ( ( ( ( ( ( ( N c N f ~ ( ( ad (where of Codoal Full ~ ( ad (where of Codoal Full τ σ π π τ σ

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4 roblem c: Addg he addoal modelg ucera of prors o \au^ ad \sgma^ shows esseall he same resul. Large alerag-sg auocorrelao Adlers ersus er good resuls for Gbbs. Noce ha we cao use Adler s o he arace parameer updaes sce her full codoals are o Gaussa. So hese use Gbbs seps eher mehod. ( ( ( ( ( ( ( ( ( ( ( ( 0 0 / 0 / 0 / / 5 0 / 0 0 / 0 ~ ( : of Codoal Full 5 ~ ( : of Codoal Full before. as ad of Codoal Full τ σ IG IG f

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6 roblem b: The frs problem we oce s hgh auocorrelao ad cross correlao \bea_0 ad \bea_. The eases fx as earl all lear models s o ceer he screeg coarae. Dog do fxes he coergece problem. bea alpha chas :3 sample: bea chas :3 sample: bea0 ode mea sd.50% meda 97.50% sar sample alpha bea bea The esmaed mea effec of \alpha he proporo of spaal arabl amog he clusers s 0.7 (95% Credble Ieral: We do o fd a sgfca effec of he proporo of screeg a cou upo lae deeco raes ((\bea_.00; 95% Credble Ieral: roblem c: As \alpha dcaes here s a lo of spaal arabl compared o o-spaal arabl. Ths ca be see hrough he predced Sadardzed Moral Raos (\ha{smr} ad spaal resduals for each cou. The lower hrd of Mesoa has hgher moral rae ha he mddle hrd swach of Mesoa. No surprsgl he spaal resduals are clusered smlarl o he \ha{smr}s. The \ph s represe he o-spaal arabl ad usurprsgl show o spaal paer. Examg he erals esmaes for ddual \ph s we oce ha he earl all oerlap zero dcag ha we mgh wa o remoe he heerogee effec from he model. \ha{\hea}: darker blue (lower ercle lgher blue (mddle ercle red (upper ercle

7 \ha{\ph}: darker blue (lower ercle; lgher blue (mddle ercle; red (upper ercle

8 \ha{smr}: dark blue (lower ercle; lgh blue (mddle ercle; red (upper ercle roblem e: Icorporag sure measureme error dd o chage he affec of \bea_. We aler he preous model b supposg ha he raes x_ T_ follow a N(T_ 00 dsrbuo. The rue screeg raes T_ hae a N(50 00 pror dsrbuo for smplc. Such a pror s somewha ague bu ceered a approxmael he oerall screeg rae. Fg hs model we do o fd a sgfca effec of he proporo of screeg a cou upo lae deeco raes ((\bea_.00; 95% Credble Ieral: Icorporag he addoal ucera abou he screeg coarae dd o ad he eral for \bea_ much bu f we had o fxed \mu_0 \dela ad \lambda we would hae see a more subsaal crease he \bea_ eral. bea chas :3 sample: ode mea sd.50% meda 97.50% sar sample alpha

9 bea bea roblem f: I s smple o replace a fxed precso \dela for he obsered screeg rae x_ T_ wh \dela r_ a precso mulpled b he sure sample sze so ha larger coues hae greaer precso. The we ca use he same herarchcal srucure eher fxg \dela or placg a gamma pror o.

10 # roblem s; ; c( ; legh( ers 000 hea marx(rowerscol0 mu marx(rowerscol hea[] rep(00 mu[] rorm( mea(hea[] sqr(/ #alpha 0 alpha # Regular Gbbs Sampler # Oer-relaxed Gbbs Sampler for ( :ers { for (j :0 { codmu ([j]* mu[-]*s/(s codsd sqr((s*/(s hea[j] codmu alpha * (hea[-j] - codmu codsd * sqr(-alpha^ * rorm( } meahea mea(hea[] codmu meahea codsd sqr(/ mu[] codmu alpha * (hea[-j] - codmu codsd * sqr(- alpha^ * rorm( } x( plo(seq(:ersmupe'l' plo(seq(:ershea[]pe'l' par(mfrowc(34omac(3 for ( :0 acf(hea[] mapase('hea ' acf(mu ma'mu' le(pase('alpha 'alphaouert # roblem c ers 000 hea marx(rowerscol0 mu marx(rowerscol s marx(rowerscol marx(rowerscol hea[] rep(00 mu[] 0 s[] /rgamma(shape5*/- scale /(sum(^ * [] /rgamma(shape/ scale /((*mu[]^ alpha 0 #alpha # Regular Gbbs Sampler # Oer-relaxed Gbbs Sampler for ( :ers {

11 for (j :0 { lasmu ([j]*[-] mu[-]*s[-]/(s[-][-] lassd sqr((s[-]*[-]/(s[-][-] hea[j] lasmu alpha*(hea[-j] - lasmu lassd * sqr(-alpha^ * rorm( } meahea mea(hea[] * s[] /rgamma(shape5*/ - scale /(sum((-hea[]^ [] /rgamma(shape/ scale /(sum((hea[]-mu[- ]^ mu[] mea(hea[] alpha*(mu[-] - mea(hea[] sqr([]/ * sqr(-alpha^ * rorm( } x( par(mfrowc(34omac(3 for ( :0 acf(hea[]mapase('hea ' acf(muma'mu' le(pase('alpha 'alphaouert

12 model { for ( : sumnumnegh { weghs[] <- } for ( : regos { scree[] ~ dorm(t[] 500 T[] ~ dorm( O[] ~ dpos(mu[] log(mu[] <- log(e[] bea0 bea*(scree[] - mea(scree[] ph[] hea[] hea[] ~ dorm(0.0au.h x[] <- hea[] ph[] SMRha[] <- 00 * mu[] / E[] SMRraw[] <- 00* O[] / E[] } ph[:regos] ~ car.ormal(adj[] weghs[] um[] au.c bea0 ~ dorm(0.0.0e-5 # ague pror o grad ercep bea ~ dorm(0.0.0e-5 # ague pror o coarae effec au.h ~ dgamma(.0e-3.0e-3 au.c ~ dgamma(.0e-.0e- sd.h <- sd(hea[] # margal SD of heerogee effecs sd.c <- sd(ph[] # margal SD of spaal effecs alpha <- sd.c / (sd.h sd.c } #Is ls(au.h au.c bea0 0 bea 0 heac( ph c(