Covariance with np.cov() in NumPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of np.cov() with two 1D arrays

What is the output of the following code snippet?

NumPy

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

A

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

B

[[0.66666667 0.66666667]
 [0.66666667 0.66666667]]

C

[[1.5 1.5]
 [1.5 1.5]]

D

[[2. 2.]
 [2. 2.]]

💡 Hint

Recall that np.cov() by default calculates sample covariance with normalization by (N-1).

The covariance matrix shows the covariance between x and y. Since both arrays increase linearly with the same step, covariance is 0.66666667. The sample covariance divides by (N-1) = 2, so the variance and covariance values are 0.66666667.

❓ data_output

intermediate

Shape of covariance matrix from 2D array input

Given the code below, what is the shape of the covariance matrix produced by np.cov()?

NumPy

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

A (2, 2)

B (3, 3)

C (6, 6)

D (1, 1)

💡 Hint

np.cov() treats rows as variables and columns as observations by default.

The input has 2 rows (variables) and 3 columns (observations). The covariance matrix shape is (number of variables, number of variables), so (2, 2).

🔧 Debug

advanced

Identify the error in covariance calculation

What error will this code raise, if any?

NumPy

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

A No error, outputs a covariance matrix

B TypeError: unsupported operand type(s) for -: 'list' and 'list'

C ValueError: all the input arrays must have the same length

D IndexError: index out of range

💡 Hint

Check if input arrays to np.cov() have the same length.

np.cov() requires input arrays to have the same number of observations. Here, x has length 3 and y length 2, causing a ValueError.

🧠 Conceptual

advanced

Effect of rowvar parameter in np.cov()

Given a 2D array with shape (3, 4), what does setting rowvar=False do in np.cov()?

A Treats columns as variables and rows as observations

B Treats rows as variables and columns as observations

C Raises an error because rowvar must be True

D Returns a covariance matrix with shape (4, 4) but with zeros

💡 Hint

rowvar=False changes the interpretation of rows and columns.

By default, rowvar=True means each row is a variable. Setting rowvar=False means each column is a variable, so covariance is computed between columns.

🚀 Application

expert

Calculate covariance matrix for multiple variables

You have a dataset with 3 variables and 5 observations: Variable A: [2, 4, 6, 8, 10] Variable B: [1, 3, 5, 7, 9] Variable C: [5, 7, 9, 11, 13] Using np.cov(), what is the covariance between Variable A and Variable C?

NumPy

import numpy as np
A = np.array([2, 4, 6, 8, 10])
B = np.array([1, 3, 5, 7, 9])
C = np.array([5, 7, 9, 11, 13])
data = np.array([A, B, C])
cov_matrix = np.cov(data)
print(cov_matrix[0, 2])

A 9.0

B 8.0

C 12.5

D 10.0

💡 Hint

Calculate covariance manually or trust np.cov() output for these linear sequences.

Variables A and C increase linearly with same step size. Covariance is calculated as sum((A-meanA)*(C-meanC))/(N-1). This equals 10.0.