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Analyzing the Population Changes

The state data for 1980 in the table of regional population distribution above could be entered in Maple using the commands:

S[80] := vector( [.2167,.2599, .1906, .3327 ] );
You can enter the data S[90] for 1990 similarly.

Let T be the 4 x 4 matrix

\left( \begin{array}{cccc}
.8166 & .085 & .01 & .01 \\
.01 ...
...01 & .97 & .01 \\
.0744 & .01 & .01 & .97
\end{array} \right)

This problem will start out by testing the appropriateness of T as a possible transition matrix describing regional population shifts over a decade.

By multiplying T on the left by the row vector [1,1,1,1], show that the sums of the columns of T are close to 1. (Explain why multiplying by this row vector is calculating the column sum.)
Show that T approximately accounts for the change from 1980 to 1990, i.e. $T \circ S[80]$ is approximately S[90].
Compute $T^{50} \circ S[80]$, $T^{100} \circ S[80]$, and $T^{200} \circ S[80]$, and compare these results. What does the comparison suggest ?
Use the row operations package to approximately solve the system $T \circ w = w$. Explain why a state vector w satisfying this equation would represent an unchanging (steady-state ) population distribution for this model.
Maple Comments:
Compare your steady state answer to the result of $T^{100} \circ S[80]$ above. In converting your solution to one the sum of whose components is 1, you may find it helpful to use the Maple commands
v_sum := add(v[i],i=1..4);
if v is a vector or
v_sum := add(v[i,1],i=1..4);
if v is a 4 x 1 matrix to add up the 4 components of v and
mult := u -> u/v_sum;
The latter define a helper function multiplying any number by 1/v_sum, and then apply that function to each entry of v.

The transition matrix T is just one Markov chain model consistent with the data. You might find it interesting to think about other possibilities.

The fact that 3 and 4 agree can be shown to hold in general for Markov matrices as long as some power has all its entries strictly positive. But it's not obvious ....

next up previous
Next: About this document ... Up: Due in Recitation on Previous: Markov Chain Models