Bayes-informed mixture distribution for the EVD estimation and dynamic reliability analysis

In this paper, a Bayes-informed mixture distribution is proposed to capture the underlying implicit extreme value distribution of a nonlinear dynamic system subjected to stochastic seismic excitation. First, three possible candidate distributions considering the properties of the extreme value distribution are selected, i.e., Gumbel, inverse Gaussian and Gamma distributions. Then, the mixture of these possible candidate distributions is conducted from Bayesian perspective by using the proposed moment-generating function-guided likelihood function. The benchmark data for constructing the likelihood function is the function value of the moment-generating function related to the unknown extreme value distribution, which can be calculated by the Latinized partially stratified sampling technique. The predicted data can be calculated by the mixed moment-generating function of the mixture distribution. Finally, the mixture distribution's unknown parameters can be calibrated via the Bayesian inference and transitional Markov chain Monte Carlo so that the calibrated mixture distribution can be a representation of the implicit extreme value distribution. Regarding the dynamic reliability analysis, the first passage failure probability can be readily obtained from the mixture distribution. Three nonlinear dynamic systems subjected to the fully non-stationary seismic excitations, i.e., a single-degree-of-freedom system, a 3-D reinforced concrete frame structure and a practical highway bridge are investigated to validate the efficiency and accuracy of the proposed method for extreme value distribution estimation.