[R] Limit distribution of continuous-time Markov process

From: <gschultz_at_scriptpro.com>
Date: Thu, 05 Jun 2008 08:41:37 -0500

I have (below) an attempt at an R script to find the limit distribution of
a continuous-time Markov process, using the formulae outlined at http://www.uwm.edu/~ziyu/ctc.pdf, page 5.

First, 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.

Second, 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.

---------------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 for this example is a matrix with all values = 1/3.)

Grant D. Schultz



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