In this article we will discuss how to compute a variance-covariance matrix using Python.

Table of Contents

• Introduction
• Variance-covariance matrix explained
• Create a sample DataFrame
• Compute variance-covariance matrix
• Conclusion

Introduction

A variance-covariance matrix is a square matrix (has the same number of rows and columns) that gives the covariance between each pair of elements available in the data.

Covariance measures the extent to which to variables move in the same direction.

In the variance-covariance matrix, variances of variables appear on the diagonal and covariances of variables are all other elements of the matrix.

To continue following this tutorial we will need the following Python library: pandas.

If you don’t have them installed, please open “Command Prompt” (on Windows) and install them using the following code:

pip install pandas

Variance-Covariance Matrix Explained

A covariance matrix is:

• Symmetric
  The square matrix is equal to its transpose: \( A = A^T \).
• Positive semi-definite
• With main diagonal containing the variances (covariances of variables on themselves)

$$
cov_{x,y,z} = \left[ \begin{array}{ccc}
cov_{x,x} & cov_{x,y} & cov_{x,z} \\
cov_{y,x} & cov_{y,y} & cov_{y,z} \\
cov_{z,x} & cov_{z,y} & cov_{z,z}
\end{array} \right]
= \left[ \begin{array}{ccc}
\sigma^2_{x} & \sigma_{xy} & \sigma_{xz} \\
\sigma_{yx} & \sigma^2_{y} & \sigma_{yz} \\
\sigma_{zx} & \sigma_{zy} & \sigma^2_{z}
\end{array} \right]
$$

where each covariance can be computed by the following formula (replacing x, y, z values):

$$
cov_{x,y} = E[(X – E[X])(Y – E[Y])] = \frac{\sum(x_i – \bar{x})(y_i – \bar{y})}{N-1}
$$

Create a sample DataFrame

Let’s create a sample Pandas DataFrame with three variables: Age, Experience, Salary with a few observations for each:

import pandas as pd

df = pd.DataFrame(
    {'Age': [25, 32, 37],
    'Experience': [2, 6, 9],
    'Salary': [2000, 3000, 3500]}
    )

print(df)

And we get:

   Age  Experience  Salary
0   25           2    2000
1   32           6    3000
2   37           9    3500

Compute variance-covariance matrix using Python

Using the .cov() method of the Pandas DataFrame we are are able to compute the variance-covariance matrix using Python:

cov_matrix = df.cov()

print(cov_matrix)

And we get:

                    Age   Experience         Salary
Age           36.333333    21.166667    4583.333333
Experience    21.166667    12.333333    2666.666667
Salary      4583.333333  2666.666667  583333.333333

Conclusion

In this article we discussed how to compute a variance-covariance matrix using Python.

Feel free to leave comments below if you have any questions or have suggestions for some edits and check out more of my Statistics articles.

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