Maximum entropy distribution with fractional moments for reliability analysis

An important task in reliability engineering is the approximation of the probability distribution function from limited sample data, both in terms of modeling of input random variables and response quantities from numerical simulations. Good estimation of the distribution tails is crucial, but challenging. In principle, the maximum entropy density with prescribed moments produces the least biased distribution, but it suffers from several drawbacks. Recently, maximum entropy with fractional moments has become popular as the fractions can be optimized, allowing a reduced number of moments to accurately characterize the distribution. However, the optimization problem is difficult to solve efficiently because the objective function is non-convex and non-continuous. This paper outlines a new method based on maximum entropy and fractional moments. Genetic algorithm is found to be a robust method for finding the global optimal solution for the fractional orders. The number of moments, which controls the bias-variance trade-off, is also optimized using the Akaike information criterion to avoid an overcomplex model. A new location variable is introduced, thus allowing negative data to be modeled. The proposed method is tested on several examples, including theoretical distributions, a truss system, and wind turbine loads. To investigate the bias and variance errors of the exceedance probabilities predicted by the proposed method, for each application, many datasets collected from the same population are analyzed. The speedup over Monte Carlo simulation is calculated from the variance error. The proposed method is shown to have very low bias errors and high speedups for the examples examined.