Question

I am stuck at a problem in matlab. How do I select a number ‘i’ with a probability ‘i’ in a matrix.
I want to do this in a matrix, where I select a number ‘i’ from each row, with probability ‘i’.

Any help is appreciated

a sample matrix( dont need to select zeros):

    w1= .47;
    w2= .023;
    m1= .06;
    m2= .037;
    x=rand(1,m1)<=m1;
    tolerance= 0.01;

    Transition=[m1 w1 0  w1 0  0   0;
        0  m1 w1 0  0  0   0;
        0  0  m1 w1 0  0   0;
        0  0  0  m2 w2 0   0;
        0  0  0  0  m2 w1  w1;
        0  0  0  0  0  m1  w1
        0  0  0  0  0  0   m1];

Answer 1

First, we could calculate a matrix of cumulative probabilities Tc in preparation for a strategy that finds the first column exceeding a random value between 0 and 1.

Tc = zeros(size(T));
Tc(:, 1) = T(:, 1);
for k = 2 : size(T, 2)
  Tc(:, k) = Tc(:, k - 1) + T(:, k);
end

To now draw numbers from each row, we first draw p = rand(size(T, 2), 1) and then find the column for which p falls into a bucket of cumulative probability:

for k = 1 : size(T, 1)
  col = find(T(k, :) > p(k), 1, 'first');
  if isempty(col)
    fprintf('nothing drawn for row %d\n', k);
  else
    fprintf('row %d, col %d, p = %f\n', k, col, T(k, col));
  end
end

This works because order does not matter. For example, for some p = 0.2, some arbitrary cumulative distribution could increase from 0.65 to 0.85. The probability for rand to yield a value in this interval is indeed 0.2. If this is the last non-zero entry of T, rand will return a value with 0.15 probability for which no number is drawn. In another example, if all entries in T are zero, the cumulative distribution will never exceed what can be drawn by rand. For a last example, if there are two entries at 0.5, each, the cumulative distribution will exceed what’s drawn by rand half and half at either column.