Phase Space Reconstruction from Time Series Data: Where History Meets Theory

In ‘dissipative’ dynamical systems, variables evolve asymptotically toward low‐dimensional ‘attractors’ that define their dynamical properties. Unfortunately, real‐world dynamical systems are generally too complex for us to directly observe these attractors. Fortunately, there is a method—‘phase space reconstruction’—that can be used to indirectly detect attractors in real‐world dynamical systems using time series data on a single variable (Broomhead and King, 1985; Schaffer and Kott, 1985; Kott et al, 1988; Williams,1997). Armed with this knowledge, we can formulate more accurate and informative models of real‐world dynamical systems. We begin by introducing the concept of phase space attractors within the context of a dynamic ISLM model. We next demonstrate how phase space reconstruction faithfully reproduces one of the model’s attractors. Finally, we discuss how phase space reconstruction fits into a more general ‘diagnostic’ modeling approach that relies on historical data to guide and test the deterministic formulation of theoretical dynamical models. As an example of diagnostic modeling, we test how closely the attractor generated by the dynamic ISLM model visually approximates the attractor reconstructed from time series data on real‐world interest rates.