Judging model reduction of complex systems

Model reduction is a common goal in the study of complex systems, consisting of many components with a complex interaction structure. The quality of such reduction, however, may not be reflected correctly in the stepwise prediction error in the model since it ignores the global geometry of the dynamics. Here we introduce a general two-step framework, consisting of dimensionality reduction of the time series followed by modeling of the resulting time series, and propose the use of the shadowing distance to measure the quality of the second step. Using coupled oscillator networks as a prototypical example, we demonstrate that our approach can outperform those based on stepwise error and suggest that it sheds light on the problem of identifying and modeling low-dimensional dynamics in large-scale complex systems.