PIPs and POPs: The reduction of complex dynamical systems using principal interaction and oscillation patterns

A general method is described for constructing simple dynamical models to approximate complex dynamical systems with many degrees of freedom. The technique can be applied to interpret sets of observed time series or numerical simulations with high‐resolution models, or to relate observation and simulations. The method is based on a projection of the complete system on to a smaller number of “principal interaction patterns” (PIPs). The coefficients of the PIP expansion are assumed to be governed by a dynamic model containing a small number of adjustable parameters. The optimization of the dynamical model, which in the general case can be both nonlinear and time‐dependent, is carried out simultaneously with the construction of the optimal set of interaction patterns. In the linear case the PIPs reduce to the eigenoscilations of a first‐order linear vector process with stochastic forcing (principal oscillation patterns, or POPs). POPs are linearly related to the “principal prediction patterns” used in linear forecasting applications. The POP analysis can also be applied as a diagnostic tool to compress the extensive information contained in the high‐dimensional cross‐spectral covariance matrix representing the complete second‐moment structure of the system.