Probabilistic Performance-Pattern Decomposition (PPPD): Analysis framework and applications to stochastic mechanical systems

Numerous research efforts have been devoted to developing quantitative solutions to stochastic mechanical systems. In general, the problem is perceived as “solved” when a complete or partial probabilistic description on quantities of interest (QoIs) is determined. However, in the presence of complex system behavior, there is a critical need to go beyond computing probabilities. In fact, to gain a better understanding of the system, it is crucial to extract physical characterizations from the probabilistic structure of the QoIs, especially when the QoIs are computed in a data-driven fashion. Motivated by this perspective, the paper proposes a framework to obtain structuralized characterizations on behaviors of stochastic systems. The framework is named Probabilistic Performance-Pattern Decomposition (PPPD). PPPD analysis aims to decompose complex response behaviors, conditional to a prescribed performance state, into meaningful patterns in the space of system responses, and to investigate how the patterns are triggered in the space of basic random variables. To illustrate the application of PPPD, the paper studies three numerical examples: (1) an illustrative example with hypothetical stochastic processes input and output; (2) a stochastic Lorenz system with periodic as well as chaotic behaviors; and (3) a simplified shear-building model subjected to a stochastic ground motion excitation.