On the use of the maximum entropy method for reliability evaluation involving stochastic process modeling

From a probabilistic perspective, a stochastic process (or random field) is completely specified by its joint probability distribution, which can be further decomposed into marginal distributions and a multivariate copula function. This paper proposes a non-parametric approach based on the maximum entropy theory to build the two parts of the distribution of stochastic processes. The specification of a stochastic process is interpreted as moment constraints and classified into two categories. The first category describes the uncertainty of the random quantity at an arbitrary point. The second category defines the dependence between the random quantities at any two points. The marginal distribution and copula function are then developed by maximizing the entropy under two classes of constraints, which are formulated as two optimization problems. The proposed method is applied to two reliability problems: a beam reliability evaluation considering time-dependent corrosion, and a tunnel reliability evaluation considering soil spatial variability. The first example illustrates the method when only partial information (marginal distribution and autocorrelation function) is available, while the second example shows the development of a random field model based on data. The comparison of the non-parametric approach with typical parametric models indicates the flexibility of the proposed method in capturing the variation of random quantities in time or space.