Question

I have two matricies, X and Y, with each column representing multiple realizations of a random variable;

X = [x_11  x_21  .... x_n1
     x_12  x_22  .... x_n2
      .     .    ....  .
      .     .    ....  .
     x_1m  x_2m  .... x_nm]

And where Y is a function of X: Y = f(X)

Y = [y_11  y_21  .... y_n1
     y_12  y_22  .... y_n2
      .     .    ....  .
      .     .    ....  .
     y_1m  y_2m  .... y_nm]

I want to find the covariance matrix between the variables x_n and y_n;

E{(X - E{Y}) * (Y - E{Y})^H}

Where ()^H denotes the Hermitian Transpose of the vector

In matlab, when I run cov(X,Y) on the matricies, (each 1000 trials of 20 variables) I only get a 2×2 matrix back, which leads me to believe that it is treating each matrix as a single “variable” somehow. If I concatenate the two matricies and call cov on the result:

cov( [X Y] )

I get a 40×40 matrix, with the result of cov( X ) in the top left, the result of cov( Y ) in the bottom right, and the matrix I want in the top right and bottom left, but is there a way to calculate this without having to resort to this?

Thanks

Answer 1

cov(X,Y) is equivalent to cov([x(:) y(:)]). But [x(:) y(:)] is 20000 by 2 for you, and cov() treats rows as observations and columns as dimensions, so you get a 2 by 2 covariance matrix.

I would just implement it myself by the definition:

bsxfun(@minus,x,mean(x))'*bsxfun(@minus,y,mean(y))/(size(x,1)-1)

If you have an older version of matlab that doesn’t support bsxfun(), just use repmat().