I am attempting to write an R script to find the limit distribution of
a continuous-time Markov process, using the formulae outlined
athttp://www..uwm.edu/~ziyu/ctc.pdf, page 5.
1. Is there a better exposition of a practical algorithm for doing
this? I have not found an R package that does this specifically, nor
anything on the web.
2. The script below will give the right answer, _if_ I "normalize"
the rate matrix, so that the average rate is near 1.0, and _if_ I
tweak the multiplier below (see **), and then watch for the Answer to
converge to a matrix where the rows to sum to 1.0. (This multiplier
is "t" in the PDF whose URL is above.) Is there a known way to get
this to converge?
Thank you very much.
---------------R script:--------------
# The rate matrix:
Q <- matrix(c(-1, 1, 0, 1, -2, 1, 0, 1, -1), ncol=3, byrow=T);
M <- eigen(Q)$vectors # diagonalizer matrix
D <- ginv(eigen(Q)$vectors) %*% Q %*% eigen(Q)$vectors # Diagonalized
form
Sum <- matrix(c(rep(0, 9)), ncol=3, byrow=T);
for (i in 0:60)
{ # Naive, Taylor series summation:
Temp <- D;
diag(Temp) <- (4 * diag(D)) ^ i; # ** =4
Sum <- Sum + Temp / factorial(i);}
Answer <- M %*% Sum %*% ginv(M);
Answer;
# Right answer is the matrix with all values = 1/3.
