Empirical dynamic modeling for beginners

Abstract

Natural systems are often complex and dynamic (i.e. nonlinear), making them difficult to understand using linear statistical approaches. Linear approaches are fundamentally based on correlation. Thus, they are illposed for dynamical systems, where correlation can occur without causation, and causation may also occur in the absence of correlation. ''Mirage correlation'' (i.e., the sign and magnitude of the correlation change with time) is a hallmark of nonlinear systems that results from state dependency. State dependency means that the relationships among interacting variables change with different states of the system. In recent decades, nonlinear methods that acknowledge state dependence have been developed. These nonlinear statistical methods are rooted in state space reconstruction, i.e. lagged coordinate embedding of time series data. These methods do not assume any set of equations governing the system but recover the dynamics from time series data, thus called empirical dynamic modeling (EDM). EDM bears a variety of utilities to investigating dynamical systems. Here, we provide a step-by-step tutorial for EDM applications with rEDM, a free software package written in the R language. Using model examples, we aim to guide users through several basic applications of EDM, including (1) determining the complexity (dimensionality) of a system, (2) distinguishing nonlinear dynamical systems from linear stochastic systems, and quantifying the nonlinearity (i.e. state dependence), (3) determining causal variables, (4) forecasting, (5) tracking the strength and sign of interaction, and (6) exploring the scenario of external perturbation. These methods and applications can be used to provide a mechanistic understanding of dynamical systems.

Climate change and interspecific interactions drive species alternations between anchovy and sardine in the western North Pacific: Detection of causality by convergent cross mapping

“…Suppose that we do not know the true causal relationship and suspect X and Y as the cause and effect, respectively ( X and Y are the “candidate cause” and “candidate effect,” respectively). Since the candidate effect time series must be stationary (Chang, Ushio, & Hsieh, ), it should be first‐differenced or logarithmic‐differenced before the analysis, as necessary.…”

Rapid evolution leads to differential population dynamics and top‐down control in resurrectedDaphniapopulations

“…Differences in dynamics often arise from differences in interactions between the state variables of the system (Chang, Ushio, & Hsieh, ; May, ; Mougi & Kondoh, ). In the following paragraphs, we discuss the differences in interactions between phytoplankton, Daphnia juveniles and Daphnia adults among populations and treatments that might explain the differences in top‐down control of algae by the different Daphnia subpopulations.…”

Does past evolutionary history under different mating regimes influence the demographic dynamics of interspecific competition?

“…To examine the strength of interspecific interactions during the competition experiment, we analyzed the time series of census data using EDM, which is based on the state space reconstruction, for example, from a single time series with lagged coordinate embedding: x t = { x ( t ), x ( t − τ ), x ( t − 2 τ ),…, x ( t − ( E − 1) τ )}, where x ( t ) is the value of variable x at time t , τ is the embedding lag and E is the embedding dimension. EDM is an analysis method for deterministic, nonlinear systems (Chang, Ushio, & Hsieh, ). For the applicability of EDM to C. chinensis ‐ C. maculatus competitive dynamics, see Kawatsu and Kishi ().…”

“…is the value of variable x at time t, τ is the embedding lag and E is the embedding dimension. EDM is an analysis method for deterministic, nonlinear systems (Chang, Ushio, & Hsieh, 2017). For the applicability of EDM to C. chinensis-C. maculatus competitive dynamics, see Kawatsu and Kishi (2018).…”