A unified probability expression on inverse Gaussian distribution in a hierarchical form and its application to the state estimation for actual energy stochastic systems

In the theoretical framework of the analysis of stochastic systems, statistical methodologies for random fluctuations have been proposed based on the inverse Gaussian distribution function. By considering the flexibility of the inverse Gaussian distribution function, the orthogonally expanded expression of the probability function is first derived using Schmit's orthogonalization technique, whose expansion coefficients hierarchically reflect the lower and higher order statistics of phenomena. Next, by adopting the Bayesian viewpoint, a computer-aided algorithm for estimating the unknown state of energy stochastic systems under random measurement noise is established. Finally, the effectiveness of the theoretical results has been experimentally confirmed