WDAVIS@UNCVM1.BITNET (Walter Davis) (02/01/90)
I apologize for the cross-posting.
A colleague and I have some mobility data from which we've derived a
transition matrix P, where p gives the probability of moving from
ij
state i to state j in one generation ( sum (p ) = 1 ).
j ij
We are interested in finding the mean first passage time from
each state to each other state, assuming a Markov process.
According to my relatively ancient book on Finite Markov
Chains (Kemeny and Snell, 1960), to find these mean first passage
times, we need to find the 'limiting matrix' A where
P to the nth power converges to A as N goes to infinity.
Each row of A is the 'limiting vector' alpha where alpha equals
{a1,a2,...ak} where k is the number of states. This vector has the
property that:
alpha*P = alpha.
This leads to K+1 equations with K unknowns:
1= a1 + a2 + ... + ak
a1=p11*a1 + p21*a2 + ... + pk1*ak
a2=p12*a1 + p22*a2 + ... + pk2*ak
.
.
.
ak=p1k*a1 + p2k*a2 + ... + pkk*ak
where all pij are known.
What I plan to do is to simply raise P to some very high
power (say 100) and use that for A. But if there's a relatively
easy way to solve the above equations, I'd prefer that.
If there were only a few categories, I could do this by hand. But there
are a lot of categories. I'm hoping for one of two things:
1) it's been too long since I've done this and there's a simple solution
to this that I'm not seeing.
2) someone has SAS code written to do this
thanks in advance,
Walter Davis <WDAVIS@UNCVM1>
Dept. of Sociology
UNC - Chapel Hill