Forming a Co-variance matrix for a 2D numpy array

Question

I am trying to figure out a fully vectorised way to compute the co-variance matrix for a 2D numpy array for a given base kernel function. For example if the input is X = [[a,b],[c,d]] for a kernel function k(x_1,x_2) the covariance matrix will be

K=[[k(a,a),k(a,b),k(a,c),k(a,d)], [k(b,a),k(b,b),k(b,c),k(b,d)], [k(c,a),k(c,b),k(c,c),k(c,d)], [k(d,a),k(d,b),k(d,c),k(d,d)]].

how do I go about doing this? I am confused as to how to repeat the values and then apply the function and what might be the most efficient way of doing this.

Answer 1

You can use np.meshgrid to get two matrices with values for the first and second parameter to the k function.

In [8]: X = np.arange(4).reshape(2,2)    
In [9]: np.meshgrid(X, X)
Out[9]: 
[array([[0, 1, 2, 3],
        [0, 1, 2, 3],
        [0, 1, 2, 3],
        [0, 1, 2, 3]]), 
 array([[0, 0, 0, 0],
        [1, 1, 1, 1],
        [2, 2, 2, 2],
        [3, 3, 3, 3]])]

You can then just pass these matrices to the k function:

In [10]: k = lambda x1, x2: (x1-x2)**2

In [11]: X1, X2 = np.meshgrid(X, X)

In [12]: k(X1, X2)
Out[12]: 
array([[0, 1, 4, 9],
       [1, 0, 1, 4],
       [4, 1, 0, 1],
       [9, 4, 1, 0]])