Question

Remember the formula for covariance

\[\text{Cov}(X, Y) = \frac{\sum_{i=1}^n(X_i - \bar{X})(Y_i - \bar{Y})}{n-1}\]

where \(\text{Cov}(X, X)\) is the sample variance of \(X\).

def cov(x, y):
    """Returns coaraiance of vectors x and y)."""
    xbar = x.mean()
    ybar = y.mean()
    return np.sum((x - xbar)*(y - ybar))/(len(x) - 1)

X = np.random.random(10)
Y = np.random.random(10)

np.array([[cov(X, X), cov(X, Y)], [cov(Y, X), cov(Y,Y)]])

array([[ 0.0727, -0.0092],
       [-0.0092,  0.0798]])

# This can of course be calculated using numpy's built in cov() function
np.cov(X, Y)

array([[ 0.0727, -0.0092],
       [-0.0092,  0.0798]])

# Extension to more vairables is done in a pair-wise way
Z = np.random.random(10)
np.cov([X, Y, Z])

array([[ 0.0727, -0.0092,  0.0383],
       [-0.0092,  0.0798, -0.0299],
       [ 0.0383, -0.0299,  0.0541]])