Python | numpy.cov() function

Covariance provides the a measure of strength of correlation between two variable or more set of variables. The covariance matrix element C_ij is the covariance of xi and xj. The element Cii is the variance of xi. 

• If COV(xi, xj) = 0 then variables are uncorrelated
• If COV(xi, xj) > 0 then variables positively correlated
• If COV(xi, xj) > < 0 then variables negatively correlated

Syntax: numpy.cov(m, y=None, rowvar=True, bias=False, ddof=None, fweights=None, aweights=None)
Parameters:
m : [array_like] A 1D or 2D variables. variables are columns 
y : [array_like] It has the same form as that of m. 
rowvar : [bool, optional] If rowvar is True (default), then each row represents a variable, with observations in the columns. Otherwise, the relationship is transposed: 
bias : Default normalization is False. If bias is True it normalize the data points. 
ddof : If not None the default value implied by bias is overridden. Note that ddof=1 will return the unbiased estimate, even if both fweights and aweights are specified. 
fweights : fweight is 1-D array of integer frequency weights 
aweights : aweight is 1-D array of observation vector weights.
Returns: It returns ndarray covariance matrix 

Example #1: 

Python3

# Python code to demonstrate the
# use of numpy.cov
import numpy as np

x = np.array([[0, 3, 4], [1, 2, 4], [3, 4, 5]])

print("Shape of array:\n", np.shape(x))

print("Covariance matrix of x:\n", np.cov(x))

Output:

Shape of array:
 (3, 3)
Covariance matrix of x:
 [[ 4.33333333  2.83333333  2.        ]
 [ 2.83333333  2.33333333  1.5       ]
 [ 2.          1.5         1.        ]]

Example #2: 

Python3

# Python code to demonstrate the
# use of numpy.cov
import numpy as np

x = [1.23, 2.12, 3.34, 4.5]

y = [2.56, 2.89, 3.76, 3.95]

# find out covariance with respect  columns
cov_mat = np.stack((x, y), axis = 0)

print(np.cov(cov_mat))

Output:

[[ 2.03629167  0.9313    ]
 [ 0.9313      0.4498    ]]

Example #3: 

Python3

# Python code to demonstrate the
# use of numpy.cov
import numpy as np

x = [1.23, 2.12, 3.34, 4.5]

y = [2.56, 2.89, 3.76, 3.95]

# find out covariance with respect  rows
cov_mat = np.stack((x, y), axis = 1)

print("shape of matrix x and y:", np.shape(cov_mat))

print("shape of covariance matrix:", np.shape(np.cov(cov_mat)))

print(np.cov(cov_mat))

Output:

shape of matrix x and y: (4, 2)
shape of covariance matrix: (4, 4)
[[ 0.88445  0.51205  0.2793  -0.36575]
 [ 0.51205  0.29645  0.1617  -0.21175]
 [ 0.2793   0.1617   0.0882  -0.1155 ]
 [-0.36575 -0.21175 -0.1155   0.15125]]