Creating a matrix of arbitrary size where rows sum to 1?

Question

My task is to create a program that simulates a discrete time Markov Chain, for an arbitrary number of events. However, right now the part I'm struggling with is creating the right stochastic matrix that will represent the probabilities. A right stochastic matrix is a matrix that has row entries that sum to 1. And for a given size, I kind of know how to write the matrix that does that, however, the problem is that I don't know how to do that for an arbitrary size.

Any help is appreciated.

(Note that this isn't a homework problem, it's only for extra credit in my Math class and the professor doesn't mind the use of outside sources.)

Answer 1

Using @MBo's idea:

In [16]: matrix = np.random.rand(3,3)

In [17]: matrix/matrix.sum(axis=1)[:,None]
Out[17]:
array([[ 0.25429337,  0.22502947,  0.52067716],
       [ 0.17744651,  0.42358254,  0.39897096],
       [ 0.36179247,  0.28707039,  0.35113714]])

In [18]:

Answer 2

Generate NxN matrix with random values.
For every row:
Find sum of row S

S[j] = Sum(0..N-1){A[j, i]}

Then subtract (S-1)/N from every value in this row

A[j, i] = A[j, i] - (S[j] - 1) / N

If you need only non-negative values, generate non-negative randoms, and divide every value in row by sum of this row

A[j, i] = A[j, i] / S[j]

Answer 3

Here is some code:

import random

precision = 1000000

def f(n) :
    matrix = []
    for l in range(n) :
        lineLst = []
        sum = 0
        crtPrec = precision
        for i in range(n-1) :
            val = random.randrange(crtPrec)
            sum += val
            lineLst.append(float(val)/precision)
            crtPrec -= val
        lineLst.append(float(precision - sum)/precision)
        matrix.append(lineLst)
    return matrix


matrix = f(5)
print matrix

I assumed the random numbers have to be positive, the sum of numbers on a raw has to be 1. I used a precision give in variable 'precision', if this is 1000 it means that the random numbers will have 3 digits after the comma. In y example 6 digits are used, you may use more.

Output:

[[0.086015, 0.596464, 0.161664, 0.03386, 0.121997], 
[0.540478, 0.040961, 0.374275, 0.003793, 0.040493], 
[0.046263, 0.249761, 0.460089, 0.006739, 0.237148], 
[0.594743, 0.125554, 0.142809, 0.056124, 0.08077], 
[0.746161, 0.151382, 0.068062, 0.005772, 0.028623]]