Is there a statistical difference between generating many random vectors vs a single random matrix
matlab, matrix, random, statistics
Solution
Just to add to the answer of @natan (+1), run the following code:
%# Store the seed
Rng1 = rng;
%# Get a matrix of random numbers
X = rand(3, 3);
%# Restore the seed
rng(Rng1);
%# Get a matrix of random numbers one vector at a time
Y = nan(3, 3);
for n = 1:3
Y(:, n) = rand(3, 1);
end
%# Test for differences
if any(any(X - Y ~= 0)); disp('Error'); end;
You'll note that there is no difference between `X` and `Y`. That is, there is no difference between building a matrix in one step, and building a matrix from a sequence of vectors.
However, there is a difference between my code and yours. Note I am populating the matrix by columns, not rows, since when `rand` is used to construct a matrix in one step, it populates by column. By the way, I'm not sure if you realize, but as a general rule you should always try and perform vector operations on the columns of matrices, not the rows. I explained why in a response to a question on SO the other day; see here for more...
Regarding the question of independence/dependence, one needs to be careful with the language one uses. The sequence of numbers generated by `rand` are perfectly dependent. For the vast majority of statistical tests, they will appear to be independent - nonetheless, in theory, one could construct a statistical test that would demonstrate the dependency between a sequence of numbers generated by `rand`.
Final thought, if you have a copy of Greene's "Econometric Analysis", he gives a neat discussion of random number generation in section 17.2.
Problem
Is there a statistical difference between generating a series of paths for a montecarlo simulation using the following two methods (note that by path I mean a vector of 350 points, normally distributed): A) ``` for path = 1:300000 Zn(path, :) = randn(1, 350); end ``` or the far more efficient B) ``` Zn = randn(300000, 350); ``` I just want to be sure there is no funny added correlation or dependence between the rows in method B that isn't present in method A. Like maybe method B distributes normally over 2 dimensions where A is over 1 dimension, so maybe that makes the two statistically different? If there is a difference then I need to know the same for uniform distributions (i.e. rand instead of randn)