Empirically Detecting Causality

Phenomenological models mathematically describe relationships among empirically observed phenomena without attempting to explain underlying mechanisms. Within the context of NLTS, phenomenological modeling goes beyond phase space reconstruction to extract equations governing real-world system dynamics from a single or multiple observed time series. Phenomenological models provide several benefits. They can be used to characterize the dynamics of variable interactions; for example, whether an incremental increase in one variable drives a marginal increase/decrease in the growth rate of another, and whether these dynamic interactions follow systematic patterns over time. They provide an analytical framework for data driven science still searching for credible theoretical explanation. They set a descriptive standard for how the real world operates so that theory is not misdirected in explaining fanciful behavior. The success of phenomenological modeling depends critically on selection of governing parameters. Model dimensionality, and the time delays used to synthesize dynamic variables, are guided by statistical tests run for phase space reconstruction. Other regression and numerical integration parameters can be set on a trial and error basis within ranges providing numerical stability and successful reproduction of empirically-detected dynamics. We illustrate phenomenological modeling with solutions of the Lorenz model so that we can recognize the dynamics that need to be reproduced.