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Empirical Dynamic Modelling (EDM, aka Emiprical Dynamics) is an analytical approach to studying nonlinear systems (such as ecosystems, financial systems, or human physiology) based on reconstructing the underlying attractor manifold of the system from observational time series.

Introduction

Visual representation of trajectories on the Lorenz attractor

In the natural sciences dynamic systems are often described as a series of differential equations or difference equations, such as the three differential equations that define the classic Lorenz attractor. The equations completely determine how the system changes in time based on current conditions and capture the interactions between different variables. However, in more abstract mathematics a dynamical system is more fundamentally defined as a flow of trajectories on a multidimensional manifold.

When equations are already known for a system, the dynamic attractor can be obtained by integrating the equations through time. However, the dynamic attractor can also be recovered from time-series observations of the system without having to know the correct dynamic equations . Thus, while many analytical approaches seek to understand systems by studying or reconstructing underlying equations, empirical dynamic modeling centers on reconstructing the attractor manifold directly from data.

Attractor Reconstruction

The central step in empirical dynamic modeling is reconstructing the attractor manifold and trajectories from time series. The process is grounded in the embedding theorems of Whitney and Takens, as well as their extensions.

Uses

Forecasting

Points that are nearby on the attractor manifold follow similar trajectories forward in time. This means that the behavior at one point on the manifold (e.g. that corresponds to the current conditions in an ecosystem) can be predicted based on the behavior of the system at other points in time when the system was in a similar state . There are several implementations of this idea of nearest-neighbor forecasting including simplex projection and S-maps . Empircal dynamic modeling has been successfully applied to forecasting in many cases, including:

• diatom population dynamics
• disease outbreaks
• sunspot activity
• neurological flight control in fruit flies
• population dynamics of Albacore tuna

Note that if the system is chaotic, nearby trajectories on the attractor manifold eventually diverge, so forecasts in these cases are limited to short term behavior.

Classifying Dynamics

Forecasting with EDM can be used as a practical criterion for distinguishing random noise from chaos and distinguishing nonlinear dynamics from linear dynamics .

Causality

Main article: convergent cross mapping

Empirical dynamic modeling has given rise to convergent cross mapping (CCM), which is a practical method for determining causal relationships in nonlinear systems .