numpy.cov() function - Python

The numpy.cov() function is used to calculate the covariance matrix of one or more numerical variables. Covariance shows how two variables change together. A positive value means both variables increase together, a negative value means one increases while the other decreases, and zero means there is no linear relationship.

This example showing how to calculate the covariance of a single 2D array where each row represents a variable.

import numpy as np
x = np.array([[1, 2, 3], [4, 5, 6]])
print(np.cov(x))

[[1. 1.]
 [1. 1.]]

Explanation:

• np.cov(x) calculates how the two variables change together
• Both variables increase at a similar rate, so the covariance is positive
• Diagonal values show how much each variable varies by itself (variance)

Syntax:

numpy.cov(m, y=None, rowvar=True, bias=False, ddof=None)

Parameters:

• m: Input data (1D or 2D array)
• y (Optional): second dataset
• rowvar: If True, rows are variables (default)
• bias: If True, normalizes using N instead of N-1
• ddof: Overrides normalization factor

Examples

Example 1: This example calculates the covariance matrix of a 2D array where each row is a variable and columns are observations.

import numpy as np
a = np.array([[0, 3, 4], [1, 2, 4], [3, 4, 5]])
print(np.cov(a))

[[4.33333333 2.83333333 2.        ]
 [2.83333333 2.33333333 1.5       ]
 [2.         1.5        1.        ]]

Explanation:

• np.cov(a) computes covariance between rows
• Positive values indicate the variables increase together
• Larger covariance means a stronger relationship between variables
• Variance values on the diagonal show how spread out each variable is

Example 2: This example finds covariance between two separate lists by stacking them as rows.

import numpy as np

x = [1.2, 2.1, 3.3, 4.5]
y = [2.5, 2.9, 3.7, 3.9]
data = np.stack((x, y))
print(np.cov(data))

[[2.0625     0.925     ]
 [0.925      0.43666667]]

Explanation:

• np.stack() combines both variables
• np.cov(data) calculates covariance between x and y
• As x increases, y also tends to increase this indicates a positive linear relationship between x and y

Example 3: This example shows how covariance changes when variables are arranged column-wise using rowvar=False.

import numpy as np

x = [1.2, 2.1, 3.3, 4.5]
y = [2.5, 2.9, 3.7, 3.9]
data = np.column_stack((x, y))
print(np.cov(data, rowvar=False))

[[2.0625     0.925     ]
 [0.925      0.43666667]]

Explanation: rowvar=False tells NumPy that each column is a variable