# [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.

Thanks!

--j

--

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

```