Calculating Tsallis Entropy for One-Dimensional Vectors

Tsallis entropy is a generalized information entropy measurement method commonly used in statistical mechanics and information theory. As an extension of Shannon entropy, Tsallis entropy introduces an adjustable parameter q (known as the entropy parameter) to characterize non-extensive properties of different systems.

To calculate Tsallis entropy for a one-dimensional vector, the first step involves probability distribution estimation of the vector. The standard implementation normalizes the vector so that the sum of all elements equals 1, thereby obtaining probability values for each element. The algorithm then computes Tsallis entropy using the mathematical formula that incorporates the q parameter, which users can adjust according to their requirements to obtain different entropy characteristics. The core function typically implements this calculation using vectorized operations for efficiency.

This functionality has been encapsulated into a header file function where users simply need to pass the one-dimensional vector and entropy parameter q to quickly compute the corresponding Tsallis entropy value. The implementation uses optimized numerical computation methods, making it easy to integrate into various data analysis or signal processing projects, particularly suitable for non-equilibrium statistics and complex system research applications. The function signature typically follows: double tsallis_entropy(const std::vector& input_vector, double q_param).

For different q values, Tsallis entropy exhibits distinct behaviors: when q approaches 1, Tsallis entropy reduces to classical Shannon entropy; when q is greater or less than 1, it can reflect varying degrees of non-extensivity in the system. This makes Tsallis entropy particularly advantageous for analyzing data with long-range correlations and non-exponential distribution characteristics. The implementation includes special case handling for q=1 to ensure numerical stability during the convergence to Shannon entropy.