Practical Tools and Guidelines for Exploring and Fitting Linear and Nonlinear Dynamical Systems Models

Abstract

A dynamical system is a system of variables that show some regularity in how they evolve over time. Change concepts described in most dynamical systems models are by no means novel to social and behavioral scientists, but most applications of dynamic modeling techniques in these disciplines are grounded on a narrow subset of—typically linear—theories of change. I provide practical guidelines, recommendations, and software code for exploring and fitting dynamical systems models with linear and nonlinear change functions in the context of four illustrative examples. Cautionary notes, challenges, and unresolved issues in utilizing these techniques are discussed.

Keywords:

• Differential equation
• functional data analysis
• dynamical systems
• GAM
• time-varying parameters

Notes

1 These solutions may equivalently be expressed in exact discrete time form (Harvey, 2001), which specifies the values of ${\mathit{y}}_i(t_{i,j})$ at discrete time point $t_{i,j}$ using the projected values of ${\mathit{y}}_i(t_{i,j-1})$ at a previous time point.

2 These time steps are distinct from the time intervals of the observed data—the former can be specified to be smaller than the latter to reduce the numerical errors that arise from such approximations.

3 In a standard leave-out-one cross-validation approach, the goal is to optimize fit by minimizing the sum of the squared discrepancies from predicting each observed data point using an approximation curve constructed using coefficients estimated using all but that specific data point. The GCV generalizes this kind of leave-one-out approaches by incorporating a weight function to accommodate scenarios with irregularly spaced data points and nonperiodic curves (Craven & Wahba, 1978).