HMM Procedure

Let , , denote a discrete time-series random variable whose sample space is a set of nonnegative integers.

The variable is said to follow a Poisson HMM distribution, with the parameter , if it follows a Poisson distribution in each state . The conditional probability distribution is denoted as , where is a positive integer.

The variable is the so-called state and follows the first-order Markov chain; that is,

where denotes the conditional probability. The range of is a finite set, . The transition probability from state i to state j is expressed as

The matrix is called the transition probability matrix (TPM). The last element in the model is the initial state probability vector (ISPV), , of the first state, :

Because the state variable is unobservable and follows a Markov chain, and because is from a Poisson distribution that depends on a different state, the model is called the Poisson hidden Markov model (Poisson HMM).

Consider a univariate Poisson HMM model that has two regimes, and .

The initial state probability vector and transition probability matrix are as follows:

%let pi1 = 0.7;
%let a11 = 0.9;
%let a22 = 0.8;
%let lambda1 = 5;
%let lambda2 = 10;

%macro dgpPsHMM(table,T,nSections,seed);
data &table;
   call streaminit(&seed.);
   secLen = ceil(&T./&nSections.);
   do t = 1 to &T.;
      sec = floor((t-1)/secLen)+1;
      if(t=1) then do;

   /* initial probability distribution */
      p = &pi1.;
   end;
   else do;
   /* transition probability matrix */
   if(lags=1) then p = &a11.;
   else p = 1-&a22.;
   end;
   u = rand("Uniform");
   if(u<=p) then s=1;
   else s = 2;
   if(s=1) then do;
   /* at state 1 */
      y = rand("Poisson",&lambda1.);
   end;
   else do;
   /* at state 2 */
      y = rand("Poisson",&lambda2.);
   end;
   output;
   lags = s;
   end;
run;
proc sort data=&table.; by sec t; run;
%mend;

%let T = 1000 ;
%let nSections = 2;
%let seed = 1234;
%dgpPsHMM(hmmpsdt,&T.,&nSections.,&seed.);

data mylib.hmmpsdt;
   set hmmpsdt;
run;

The following code estimates a bistate Poisson HMM:

proc hmm data=mylib.hmmpsdt;
   id time=t;
   model y / type=Poisson nstate=2;
run;

The number of observations and model information are shown in Figure 63.

The estimates of the ISPV are shown in Figure 64.

The estimates of the TPM are shown in Figure 65.

The estimates of the lambda are shown in Figure 66.

The estimates of all parameters are shown in Figure 67, which displays columns for parameter name, estimated value, standard error, t value, and p-value.