Calculating Information Entropy and Mutual Information Between Discrete Time Series Variables

To compute the information entropy and mutual information between two discrete time series variables, follow this implementation workflow: 1. Preprocess continuous variables by discretizing them using binning methods (e.g., equal-width or equal-frequency binning) if the input time series are continuous. In code, this can be implemented using numpy.histogram or pandas.cut functions. 2. Calculate the probability distribution for each variable by counting frequency occurrences. This involves creating frequency tables using collections.Counter in Python or the tabulate function in MATLAB, then normalizing counts to obtain probabilities. 3. Compute the joint probability distribution by counting co-occurrence frequencies of value pairs. Implement this using 2D histogram functions (numpy.histogram2d) or contingency table creation, ensuring proper handling of missing data pairs. 4. Calculate the information entropy for each variable using the formula H(X) = -Σ p(x)log₂p(x). Code implementation involves summing over all probabilities multiplied by their base-2 logarithms, with special handling for zero probabilities. 5. Compute the joint entropy using H(X,Y) = -ΣΣ p(x,y)log₂p(x,y). This requires nested iteration over all possible value combinations from the joint distribution matrix. 6. Derive mutual information using I(X;Y) = H(X) + H(Y) - H(X,Y). This symmetric measure quantifies the shared information between variables and can be implemented as a simple arithmetic operation using previously calculated entropy values. The algorithm provides quantitative measures for analyzing dependency structures between time series variables, with typical applications in feature selection and correlation analysis. Code implementations should include validation checks for probability distributions summing to 1 and handling of logarithmic calculations for zero probabilities.