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

TL;DR

This paper introduces PPPD, a framework for decomposing complex stochastic system responses into meaningful patterns, providing deeper physical insights beyond traditional probabilistic descriptions, with applications to various stochastic mechanical systems.

Contribution

The paper proposes the novel PPPD framework to extract physical characterizations from probabilistic data of stochastic systems, enhancing understanding of complex behaviors.

Findings

Decomposes stochastic responses into meaningful patterns.

Applied PPPD to chaotic and stochastic models.

Provides physical insights beyond probabilistic descriptions.

Abstract

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.