Charlier series approximation for nonhomogenious Poisson processes

초록

This study investigates the Charlier series approximation for modeling nonhomogeneous Poisson processes. It focuses on mixtures of Poisson distributions and Markov-Modulated Poisson processes to address complex temporal data patterns, such as hospital admission rates. The Charlier series approximation is constructed by expanding probability mass functions using Charlier orthogonal polynomials, which allow for adjustments to reflect higher-order moments like skewness and kurtosis. These polynomials are combined with a Poisson weight function to create flexible approximations tailored to the variability in event rates. Two artificial examples demonstrate the method’s effectiveness in capturing dynamic event behaviors. A real-world application to hospital admission data further highlights its practical utility. Performance is assessed using Kullback-Leibler divergence, quantifying the improvement over simple Poisson models. The results show that the Charlier series provides enhanced data fitting and deeper insights into complex probabilistic structures.

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