ポアソン回帰モデル

プログラム

#include <oxstd.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/2000+sumc((vy.*mx))*vb-sumc(vm.*exp(mx*vb));
// use the following instead for the improper prior
// adFunc[0]=sumc((vy.*mx))*vb-sumc(vm.*exp(mx*vb));
// 1st derivative vector
if(avScore){
avScore[0]=-vb/1000+sumc(vy.*mx)'-sumc(vm.*exp(mx*vb).*mx)';}
// use the following for the improper prior
// avScore[0]=sumc(vy.*mx)'-sumc(vm.*exp(mx*vb).*mx)';}
// 2nd derivative matrix, Hessian matrix
if(amHess){
amHess[0]=-1/1000*unit(rows(vb))-mx'*(vm.*exp(mx*vb).*mx);}
// use the following for the improper prior
//amHess[0]=-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]; // m
// Suppose that the prior for beta is set as follows: vb ~ N(0,1000I).
// To use improper prior, edit the subroutine fLK as mentioned above.  
vbhat=zeros(cm,1);MaxBFGS(fLK,&vbhat,&dlhat,0,1);
fLK(vbhat,&dlhat,&vl1hat,&ml2hat);
// If improper prior is used, print the following lines to get
// maximum likelihood estimates
//  println("Maximum Likelihood Estimate=",vbhat');
//  println("Standard Deviation of M.L.E.=",sqrt(diagonal(invert(-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=20,000.
crepeat=20000;cburn=2000;
mbeta_s=zeros(crepeat,cm);
car_accept=car_count=cmh_accept=cmh_count=0;
vb=vbhat;
for(cn=-cburn;cn<crepeat;++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;
  if(cn>=0){mbeta_s[cn][]=vb';}
}
format(200);
file = fopen("beta.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);
}