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First-Order Markov Process Estimation

A more useful inverse problem is the first-order markov process. Companies have a share of the marketplace where they do business. Generally, customers for a specific market space can move from company to company. The movement of customers can be visualized graphically as a flow diagram, as in Figure 12.17. The arrows represent movements of customers from one company to another.

Figure 12.17 Markov Transition Diagram

You can model the probability that a customer moves from one company to another using a first-order Markov model. Mathematically the model is:

where is a vector of market shares at time and is a matrix of unknown transition probabilities. The value represents the probability that a customer who is currently using company at time moves to company at time . The diagonal elements then represent the probability that a customer stays with the current company. The columns in sum to one.

Given market share information over time, you can estimate the transition probabilities . In order to estimate using traditional methods, you need at least observations. If you have fewer than transitions, you can use the ENTROPY procedure to estimate the probabilities.

Suppose you are studying the market share for four companies. If you want to estimate the transition probabilities for these four companies, you need a time series with four observations of the shares. Assume the current transition probability matrix is as follows:

The following SAS DATA step statements generate a series of market shares from this probability matrix. A transition is represented as the current period shares, y, and the previous period shares, x.

   data m;
               /* Known Transition matrix */
      array p[4,4] (0.7 .4 .0 .1
                    0.1 .5 .4 .0
                    0.0 .1 .6 .0
                    0.2 .0 .0 .9 ) ;
               /* Initial Market shares */
      array y[4] y1-y4 ( .4 .3 .2 .1 );
      array x[4] x1-x4;
      drop p1-p16 i;
      do i = 1 to 3;
          x[1] = y[1]; x[2] = y[2];
          x[3] = y[3]; x[4] = y[4];
          y[1] = p[1,1] * x1 + p[1,2] * x2 + p[1,3] * x3 + p[1,4] * x4;
          y[2] = p[2,1] * x1 + p[2,2] * x2 + p[2,3] * x3 + p[2,4] * x4;
          y[3] = p[3,1] * x1 + p[3,2] * x2 + p[3,3] * x3 + p[3,4] * x4;
          y[4] = p[4,1] * x1 + p[4,2] * x2 + p[4,3] * x3 + p[4,4] * x4;
          output;
      end;
   run;

The following SAS statements estimate the transition matrix by using only the first transition.

   proc entropy markov pure data=m(obs=1);
      model y1-y4 = x1-x4;
   run;

The MARKOV option implies NOINT for each model, that the sum of the parameters in each column is one, and chooses support points of and . This model can be expressed equivalently as

   proc entropy pure data=m(obs=1) ;
      priors y1.x1 0 1 y1.x2 0 1 y1.x3 0 1 y1.x4 0 1;
      priors y2.x1 0 1 y2.x2 0 1 y2.x3 0 1 y2.x4 0 1;
      priors y3.x1 0 1 y3.x2 0 1 y3.x3 0 1 y3.x4 0 1;
      priors y4.x1 0 1 y4.x2 0 1 y4.x3 0 1 y4.x4 0 1;
   
      model y1 = x1-x4 / noint;
      model y2 = x1-x4 / noint;
      model y3 = x1-x4 / noint;
      model y4 = x1-x4 / noint;
   
      restrict y1.x1 + y2.x1 + y3.x1 + y4.x1 = 1;
      restrict y1.x2 + y2.x2 + y3.x2 + y4.x2 = 1;
      restrict y1.x3 + y2.x3 + y3.x3 + y4.x3 = 1;
      restrict y1.x4 + y2.x4 + y3.x4 + y4.x4 = 1;
   run;

The transition matrix is given in Figure 12.18.

Figure 12.18 Estimate of by Using One Transition

Prior Distribution of Parameter T

The ENTROPY Procedure

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