[R] Stochastic (transition) matrices: how to determine distributions and variance?

Jonathan Greenberg greenberg at ucdavis.edu
Sun Aug 30 09:10:52 CEST 2009

(apologies for the cross-posting, and for this being a more general 
stats question rather than a specific-to-R one.  I assure you I will be 
doing the actual analysis in R :)

I am trying to determine the distribution and variance for a classic 
stochastic (transition) matrix problem such that:

let x(t) be an initial state vector consisting of counts of classes A, B 
and C:
x(t) = [A(t),B(t),C(t)]
T is the stochastic (transition) matrix for these classes consisting of 
the transition probabilities between each combination of A,B and C:

       pAA pBA pCA
T=   pAB pBB pCB
       pAC pBC pCC

By doing matrix multiplication of Tx(t) we can determine the *mean* 
counts of these classes at t+1 such that:
x mean (t+1) = Tx(t) = [A mean (t+1),B mean (t+1),C mean (t+1)]

What I want to know is what is a) what is the *distribution* of 
A(t+1),B(t+1) and C(t+1), and what is the variance around these mean 
values?  Since pXY are stochastic probabilities, it seems that the 
distribution and variance should be calculable.




Jonathan A. Greenberg, PhD
Postdoctoral Scholar
Center for Spatial Technologies and Remote Sensing (CSTARS)
University of California, Davis
One Shields Avenue
The Barn, Room 250N
Davis, CA 95616
Cell: 415-794-5043
AIM: jgrn307, MSN: jgrn307 at hotmail.com, Gchat: jgrn307

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