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Since the early 1900s, numerous research efforts have been devoted todeveloping quantitative solutions to stochastic mechanical systems. In general,the problem is perceived as solved when a complete or partial probabilisticdescription on the quantity of interest (QoI) is determined. However, in thepresence of complex system behavior, there is a critical need to go beyond mereprobabilistic descriptions. In fact, to gain a full understanding of thesystem, it is crucial to extract physical characterizations from theprobabilistic structure of the QoI, especially when the QoI solution isobtained in a data-driven fashion. Motivated by this perspective, the paperproposes a framework to obtain structuralized characterizations on behaviors ofstochastic systems. The framework is named Probabilistic Performance-PatternDecomposition (PPPD). PPPD analysis aims to decompose complex responsebehaviors, conditional to a prescribed performance state, into meaningfulpatterns in the space of system responses, and to investigate how the patternsare triggered in the space of basic random variables. To illustrate theapplication of PPPD, the paper studies three numerical examples: 1) anillustrative example with hypothetical stochastic processes input and output;2) a stochastic Lorenz system with periodic as well as chaotic behaviors; and3) a simplified shear-building model subjected to a stochastic ground motionexcitation.

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