  Since the early 1900s, 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 the quantity of interest (QoI) is determined. However, in the
presence of complex system behavior, there is a critical need to go beyond mere
probabilistic descriptions. In fact, to gain a full understanding of the
system, it is crucial to extract physical characterizations from the
probabilistic structure of the QoI, especially when the QoI solution is
obtained 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.
