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Stochastic Matrices

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Date: 03/17/98 at 09:46:14
From: Wayland
Subject: eigenvectors

Hi:

eigenvectors. In stochastic hydrology there is an equation that
is used:

p(t+1) = p(t)T

where p(t+1) and p(t) are nx1 probability row vectors at two different
time periods t and t+1, and T is a called a transitional probability
matrix (nxn) whose rows sum to one. As t approaches infinity, p(t+1)
approaches p(t). According to Eigenvector theory, the eigenvalue of
this equation should then be one and solving for T should yield the
identity matrix. It is true that the equation is satisfied for the
identity matrix, but for other T matrices it converges to a unique
vector that depends on the matrix, but not on p(0). I have tried to
solve for this vector but when I put in the conditions that the
probabilities sum to one, the problem is overspecified.

Thanks

Wayland
```

```
Date: 03/17/98 at 15:36:07
From: Doctor Anthony
Subject: Re: eigenvectors

I have worked through an example and hope it answers your queries on
stochastic matrices.

Below is an example of the distribution of trade between depots.

The service specialist of Metrobug Heating and Air Conditioning
Company make their calls with a fleet of radio-dispatched service
trucks. Established procedure is that when a call requesting service
is received by the office, the dispatcher sends the next available
truck to respond to the call. One of the service specialists keeps
records of his service calls, and his data are summarize in the table
below. Formulate this situation. Find the transition diagram and the
one-step transition matrix.

---------------------------------------------------------------------
District of         District of                 Percent of
Current Call        Next Call                   Calls
---------------------------------------------------------------------
East               East                       50
Central                    40
West                       10
---------------------------------------------------------------------
Central            East                       10
Central                    60
West                       30
---------------------------------------------------------------------
West               East                       30
Central                    60
West                       10
---------------------------------------------------------------------

It is more usual to write the transition matrix with columns adding to
1 rather than rows in the way you have shown it. This allows you to
operate on a vector representing the present state as a product M.x
where x is the vector of present position. Example below.

East   Central   West

East [.5       .1      .3][1]    [.5]
Central |.4       .6      .6||0| =  |.4|
West [.1       .3      .1][0]    [.1]

If we take the stationary vector, that is we find the vector that does
not change any more, we have:

M[a]    [a]
|b| =  |b|
[c]    [c]

we get the vector  [ 9/38]
|21/38]
[ 8/38]

This vector is found by solving the three equations below for a, b, c.

a(.5-1)  +  .1b     +  .3c    = 0
.4a      +  b(.6-1) +  .6c    = 0
.1a      +  .3b     + c(.1-1) = 0

we get  a = 9/38,   b = 21/38,   c = 8/38

This is in fact a quick way to find M^(infinity) as you will see
below. This will always work for a stochastic matrix but is obviously
not a general method for powers of matrices.

So, over the long term, East gets 9/38 of the trade, Central gets

If you raised M to an infinite power you will find that it settles
down to the matrix shown below, and ANY starting vector will end up
with the vector shown on the right hand side.

[9/38   9/38    9/38][1/2]    [ 9/38]
|21/38  21/38  21/38||1/4| =  |21/38]
[8/38    8/38   8/38][1/4]    [ 8/38]

This shows that after an infinite application of the matrix every
starting vector will end up with the same distribution. The columns of
M^(infinity) are all the same and are given by the vector:

[a]
|b|
[c]

-Doctor Anthony,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
College Probability
High School Probability

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