ポアソン回帰モデル

事後シミュレーション比較のプログラム

#include <oxstd.h>
#include <oxprob.h>
#import <maximize>
// mx: independent variable, vy: dependent variable
// We need to use them in the following subroutine. So set them static.
static decl mx,vm,vy;
fLK(const dX, const adFunc, const avScore, const amHess){
// fLK will calculate the value of log posterior density as adFunc[0]
decl vb;vb=dX;
adFunc[0]=-vb'*vb/(2*1.0)+sumc((vy.*mx))*vb-sumc(vm.*exp(mx*vb));
// 1st derivative vector
if(avScore){
avScore[0]=-vb/(1.0)+sumc(vy.*mx)'-sumc(vm.*exp(mx*vb).*mx)';}
// 2nd derivative matrix, Hessian matrix
if(amHess){
amHess[0]=-1/1.0*unit(rows(vb))-mx'*(vm.*exp(mx*vb).*mx);}
//
return 1;
}
//
main(){
decl cburn,cm,cn,cnobs,crepeat;
decl car_accept,car_count,cmh_accept,cmh_count;
decl dh_n,dh_o,dlhat,dl_n,dl_o,dfrac;
decl vb,vb1,vb_o,vb_n,veig,vbhat,vl1hat;
decl file,mdata,mbeta_s,mB1,mB1root,ml2hat;
// Read Data File
// X1-X8, m_i, y
cnobs=34;file = fopen("ship.txt");fscan(file,"%#m",cnobs,10,&mdata);fclose(file);
// dependent varible: yesvm
vy=mdata[][9]; // y
//independent variables: Constant X1-X8
mx=ones(cnobs,1)~mdata[][0:7];
cm=columns(mx);vm=mdata[][8];
// When m_i is very large, the sample seems to have high autocorrelation.
// To accelerate the convergence, we set mean of m_i =1 as follows:
vm=vm/meanc(vm); // m
//
// Suppose that the prior for beta is set as follows: vb ~ N(0,I).
vbhat=zeros(cm,1);MaxBFGS(fLK,&vbhat,&dlhat,0,1);
fLK(vbhat,&dlhat,&vl1hat,&ml2hat);
// vbhat : posterior mode 
// dlhat : log posterior evaluated at mode (constant term ignored)
// vl1hat: 1st derivative vector of log posterior evaluated at mode
// ml2hat: 2nd derivative matrix of log posterior evaluated at mode
//         Hessian matrix
mB1=-invert(ml2hat);
// check whether mB1 is positive definite (just in case)
// veig: eigenvalues of mB1. should be > 0 for positive definite matrix
eigen(mB1,&veig);
if(min(veig)<=0)
{println("mB1 is not positive definite! Something may be wrong with the program.");}
mB1root=choleski(mB1);vb1=mB1*(vl1hat+invert(mB1)*vbhat);
//
// Burn-in period: crepeat, Number of samples: cburn
// Since convergence seems slow, we take crepeat=200,000
//
 crepeat=200000;cburn=20000;
//
mbeta_s=zeros(crepeat,cm);
car_accept=car_count=cmh_accept=cmh_count=0;
vb=vbhat;
for(cn=-cburn;cn<crepeat;++cn){
// Calculate mode for different vy 
vbhat=zeros(cm,1);
MaxNewton(fLK,&vbhat,&dlhat,0,0);
fLK(vbhat,&dlhat,&vl1hat,&ml2hat);
mB1=-invert(ml2hat);
 eigen(mB1,&veig);
 if(min(veig)<=0)
 {println("mB1 is not positive definite! Something may be wrong with the program.");}
mB1root=choleski(mB1);vb1=mB1*(vl1hat+invert(mB1)*vbhat);
if(fmod(cn,100)==0){println("cn=",cn);}
//
// A-R Step
  vb_o=vb;
  fLK(vb_o,&dl_o,0,0);
  dh_o=dlhat+vl1hat'*(vb_o-vbhat)+0.5*(vb_o-vbhat)'*ml2hat*(vb_o-vbhat);
  // generate new beta
do{
  vb_n=vb1+mB1root*rann(cm,1);
  fLK(vb_n,&dl_n,0,0);
  dh_n=dlhat+vl1hat'*(vb_n-vbhat)+0.5*(vb_n-vbhat)'*ml2hat*(vb_n-vbhat);
  car_count+=1;
  }while(ranu(1,1)>=exp(dl_n-dh_n));
  car_accept+=1;
// M-H Step
  dfrac=exp(dl_n+min(dl_o|dh_o)-dl_o-min(dl_n|dh_n));
  if(ranu(1,1)<=dfrac){vb=vb_n;cmh_accept+=1;}
  cmh_count+=1;
  // Add the following lines for posterior sumlation comparison
  // Generate y|m,lambda ~ Poi (m*lambda) where lambda=exp(x*beta)
   decl vmlambda,i;
   vmlambda=vm.*exp(mx*vb);vy=zeros(cnobs,1);
   for(i=0;i<cnobs;++i){vy[i]=ranpoisson(1,1,vmlambda[i]);}
// 
  if(cn>=0){mbeta_s[cn][]=vb';}
}
format(200);
file = fopen("beta_psc.txt","w");fprint(file,"%15.10f",mbeta_s);fclose(file);
// check whether the argorithm was efficient enough.
println("AR acceptance=",car_accept/car_count);
println("MH acceptance=",cmh_accept/cmh_count);
}