

In [1]:

    
import numpy as np

Stochastic Matrices

These guys are matrices that predict a probability distribution iteratively. You present a "current" distribution, then you multiply by the "transition matrix" and it tells you the "next" distribution.



In [55]:

    
#
# define a transition matrix
#
T = np.array([[0.1, 0.2, 0.3],
             [0.5,0.3,0.6],
             [0.4,0.5,0.1]])
T









    Out[55]:





array([[0.1, 0.2, 0.3],
       [0.5, 0.3, 0.6],
       [0.4, 0.5, 0.1]])



In [56]:

    
#
# Start out with a random prob distribution vector
#

x0 = np.array([np.random.rand(3)]).T
x0 = x0/x0.sum()
x0









    Out[56]:





array([[0.65827967],
       [0.29726598],
       [0.04445434]])



In [57]:

    
x1 = T.dot(x0)
x1









    Out[57]:





array([[0.13861747],
       [0.44499224],
       [0.4163903 ]])



In [58]:

    
x2 = T.dot(x1)
x2









    Out[58]:





array([[0.22777728],
       [0.45264058],
       [0.31958214]])



In [59]:

    
x3 = T.dot(x2)
x3









    Out[59]:





array([[0.20918049],
       [0.4414301 ],
       [0.34938942]])



In [60]:

    
xn = x3
for i in range(100):
    xn = T.dot(xn)
    
xn









    Out[60]:





array([[0.21290323],
       [0.44516129],
       [0.34193548]])



In [61]:

    
vals, vecs = np.linalg.eig(T)
vals









    Out[61]:





array([ 1.       , -0.1381966, -0.3618034])



In [62]:

    
vecs









    Out[62]:





array([[ 0.35463375,  0.80901699, -0.30901699],
       [ 0.74150694, -0.5       , -0.5       ],
       [ 0.5695633 , -0.30901699,  0.80901699]])



In [63]:

    
vecs.sum(axis=0)









    Out[63]:





array([ 1.66570400e+00, -8.32667268e-16,  0.00000000e+00])



In [64]:

    
vecs/vecs.sum(axis=0)[0]









    Out[64]:





array([[ 0.21290323,  0.48569073, -0.18551735],
       [ 0.44516129, -0.30017338, -0.30017338],
       [ 0.34193548, -0.18551735,  0.48569073]])



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