"Kevin" <[EMAIL PROTECTED]> wrote: > I am kind of thinking how to calculate the invariant distribution > for the large transition matrix of a Markov Chain.
OK, I'm with you so far. > The general Lumpability is not applicable here. Hmm, I don't understand the importance of this statement. I'll plow forward all the same. > Actually, each column in the transition matrix is just like a > Binomial distribution with probability P(i), where i is the > index of the column. Can you give me some suggestion on how > to calculate the invariant distribution efficiently? As you know the invariant distribution for a discrete Markov chain is the eigenvector associated with eigenvalue 1. There is a package for large-scale eigenvalue problems called ARPACK that you might use. With ARPACK you define a function to compute the scalar (dot) product of two columns and this function is called instead of computing the scalar product in the usual way. For a matrix with special structure (e.g., most column elements are zero) it can be much faster. I don't remember for sure, but I think ARPACK can avoid storing the matrix (since it has the scalar product function). Also, iirc ARPACK can find the eigenvectors associated with the largest so-many eigenvalues. You can find ARPACK on the web. It is in Fortran. You didn't say how big a problem you have, but keep in mind that 1000 by 1000 isn't "large" anymore; such a problem can be solved by standard techniques, as implemented, for example, by Octave. > I am more interested in the mean and the variance > of the states, any simpler ways to calculate them? Could be, I don't know. Maybe someone else has an idea here. For what it's worth, Robert Dodier -- Life isn't all beer and skittles, but beer and skittles, or something better of the same sort, must form a good part of every Englishman's education. -- Thomas Hughes (1822-96) . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
