Finding a Transition Matrix to Fit Given Data

  The transition matrix is used to find changes in the states of a stochastic system. The reverse problem, using the changes in the states to determine the transition matrix, is a little more difficult. The sum constraints on the probabilities simplifies the problem to one equation with two unknowns. To solve for the unknown components of P we need two data points represented by α and α' and their images β and β'.

We find that the transition matrix is dependent on three parameters corresponding to the initial point and its image and the relative change in the probabilities of the states.  A simple example helps to illustrate the process and clarify the definitions.

We see that the solution for the transition matrix gives the correct image, p^<1>, of p^<0> and the calculated relative change ρ' = ρ.