Calculating Covariance with Python and Numpy

Question

I am trying to figure out how to calculate covariance with the Python Numpy function cov. When I pass it two one-dimentional arrays, I get back a 2x2 matrix of results. I don't know what to do with that. I'm not great at statistics, but I believe covariance in such a situation should be a single number. This is what I am looking for. I wrote my own:

def cov(a, b):

    if len(a) != len(b):
        return

    a_mean = np.mean(a)
    b_mean = np.mean(b)

    sum = 0

    for i in range(0, len(a)):
        sum += ((a[i] - a_mean) * (b[i] - b_mean))

    return sum/(len(a)-1)

That works, but I figure the Numpy version is much more efficient, if I could figure out how to use it.

Does anybody know how to make the Numpy cov function perform like the one I wrote?

Thanks,

Dave

Answer 1

When a and b are 1-dimensional sequences, numpy.cov(a,b)[0][1] is equivalent to your cov(a,b).

The 2x2 array returned by np.cov(a,b) has elements equal to

cov(a,a)  cov(a,b)

cov(a,b)  cov(b,b)

(where, again, cov is the function you defined above.)

Answer 2

Thanks to unutbu for the explanation. By default numpy.cov calculates the sample covariance. To obtain the population covariance you can specify normalisation by the total N samples like this:

Covariance = numpy.cov(a, b, bias=True)[0][1]
print(Covariance)

or like this:

Covariance = numpy.cov(a, b, ddof=0)[0][1]
print(Covariance)