Exploring Mutual Information Using MATLAB: A Visual Guide
As an essential statistic of information theory, mutual information plays a crucial role in many fields, including signal processing, image recognition, data compression, and biology. Essentially, mutual information measures the statistical dependence or correlation between two variables, offering critical insights into the relationships between events or signals. In this article, we’ll explore mutual information using MATLAB, an intuitive and powerful computing environment widely used across many domains.
What is Mutual Information?
Let’s start with the basic definition of mutual information. Given two random variables X and Y, mutual information is defined as the reduction in the uncertainty of one variable when the other variable is known. In other words, mutual information measures the amount of information one variable tells you about the other. This is calculated using the entropy function, which takes the probability distribution of each variable as input:
I(X;Y) = H(X) – H(X|Y) = H(Y) – H(Y|X)
Here, H(X) represents the entropy of X, and H(X|Y) is the conditional entropy of X given Y. The same holds for Y.
The result of the mutual information can be expressed in bits, indicating the reduction in entropy or uncertainty. The higher the mutual information, the stronger the dependence between X and Y. Conversely, when mutual information is zero, the two variables are independent.
How to Calculate Mutual Information Using MATLAB
MATLAB provides several built-in functions for calculating mutual information. One of the most common ones is ‘mutualinfo’, which takes two vectors of the same length as input and returns the mutual information in bits. Let’s take a simple example where we have two variables X and Y, both with n samples:
X = randn(1,n); % Generate n Gaussian samples
Y = sin(X) + randn(1,n); % Add some noise to X and take the sine function
Now, let’s calculate the mutual information between X and Y:
mi = mutualinfo(X,Y); % Calculate mutual information
The resulting value of ‘mi’ indicates the strength of the dependence between X and Y. Note that MATLAB also offers other functions for calculating mutual information, such as ‘entropy’, ‘condentropy’, and ‘jointentropy’, which can be used to create more complex measures of dependence.
Visualizing Mutual Information using MATLAB
While the numeric values of mutual information can be informative, visualizing the relationship between variables can lead to a deeper understanding of the correlation. Here, MATLAB provides excellent tools for creating 2D and 3D scatter plots, histograms, and other graphical representations. Let’s illustrate this using the same example as before, where we have two variables X and Y:
scatter(X,Y) % Create a scatter plot of X vs Y
xlabel(‘X’) % Add X-axis label
ylabel(‘Y’) % Add Y-axis label
title(‘Mutual Information’) % Add title to the plot
The resulting plot shows how X and Y are related and highlights the mutual information between them. We can also use MATLAB to create 2D histograms of joint probability distributions:
hist3([X’,Y’],’EdgeColor’,’none’) % Create a 2D histogram
xlabel(‘X’) % Add X-axis label
ylabel(‘Y’) % Add Y-axis label
title(‘Joint Probability Distribution’) % Add title to the plot
This plot provides an intuitive view of the joint probability distribution of X and Y, which can be useful for identifying patterns and for selecting appropriate models.
Conclusion
Exploring mutual information using MATLAB offers a wide range of opportunities for better understanding the correlation between variables and their dependence. In this article, we discussed the basic definition of mutual information, how to calculate it using MATLAB, and how to visualize the correlation using scatter plots and histograms. Understanding the concepts and tools of mutual information is a fundamental step towards becoming proficient in manipulating data and building effective models in many fields of science and engineering.
