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# Weather in the land of OZ -- never rains 2 days in a row

So, last class, we introduced a simple model for weather in the land of Oz.

In Oz, the weather switches among 3 states every day... Sunny, Rainy, or Snowy. And given the state of the weather today, we know how to predict the state of the weather tomorrow.

Prob( x -> y ) Sunny today Rainy today Snowy today
Sunny tomorrow 1/2 1/2 1/4
Rainy tomorrow 1/4
0
1/4
Snowy tomorrow 1/4 1/2 1/2

Show how learn composition can be rewritten as $$p(t+1) = A p(t)$$ where

$A := \frac{1}{4} \begin{bmatrix} 2 & 2 & 1 \\ 1 & 0 & 1\\ 1 & 2 & 2\end{bmatrix}$

We can simulate this process with a very simple python algorithm

[Show code]
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  #!/usr/bin/env python2 from scipy import rand s = 'Sunny' r = 'Rainy' w = 'Snowy' transitions = {s: (s,s,r,w), r:(s,s,w,w), w:(s,r,w,w)} states = transitions.keys() def rndStep(state): return transitions[state][int(rand()*4)] def rndPath(t, x, tmax): # x is the initial state # t is the initial time # tmax is the time to stop at while t < tmax: yield t, x t, x = t + 1, rndStep(x) def main(): for t, x in rndPath(0,s,10): print t, x main()

One example of the weather we might observer is then ...

0 Sunny
1 Sunny
2 Sunny
3 Sunny
4 Rainy
5 Snowy
6 Snowy
7 Rainy
8 Snowy
9 Sunny

This is not a very general method (though it is fast). Better methods like aliasing exist.

## Path integrals

If it is sunny today, what is the chance that it will slow in 3 days? (we don't care about the weather for the next 2 days)

Do an example calculation of weather prediction out the long way.

$p(t)^T A^t p(0)$

Since the columns some to $$1$$, there must be an eigenvalue equal to $$1$$.
(has simple eigenvalues and steady-state) The characteristic polynomial $$z^{3} - z^{2} -{z}/{16} + {1}/{16} = 0$$ factors to $\left(z - 1\right) \left(4 z - 1\right) \left(4 z + 1\right) = 0$
Calculate equilibrium distribution using reduced row-eschelon form. ($$I-A$$ will be singular! But we also want $$p_1 + p_2 + p _ 2 = 1$$.)
$p(\infty) = \frac{1}{5} \begin{bmatrix}2\\1\\2\end{bmatrix}$ So, after forgetting the initial condition, we expect it to be sunny $$2/5$$th of the time, snowy $$2/5$$th of the time, and rainy only $$1/5$$th of the time.