Random Trajectory Models for Complex Phenomena

Abstract

Many models for complex phenomena use a model for strongly-interacting elements on a small scale to generate larger-scale simulations of some aspects of experimental realizations. These models may be agent-based (as in the case of discrete element method models for granular flow) or based on pattern-forming systems of PDEs (as in models for Raleigh-B\`enard convection patterns). Often these models are purely deterministic, producing a single simulation for each set of initial conditions. If observed realizations demonstrate between-realization variability for important aspects of the phenomenon, those aspects can be simulated by adding probabilistic components to the deterministic models to create random trajectory (RT) models. The RT model framework provides probabilistic models which can be fit to data and validated, together with a clear perspective on how difficult it can be to establish any kind of validity for a fitted model. It treats models as code with adjustable coefficients, rather than as systems of differential equations. It provides a simply stated necessary condition for these code models to be easily fit and verified, as well as an argument that this condition can almost never be checked. When the necessary condition cannot be checked, the RT model framework identifies the code models as black box models which may have the capacity for emulating the joint distributions of small collections of statistics observed on realizations, but which can only provide very weak evidence for any form of explanation for the emergence of any aspect of the phenomenon. The framework also provides a way to clearly understand why finding a useful code model and scientifically validating it may require many person-years of extra experimentation and statistical analysis undertaken after the first output-producing code model is constructed and contributed.