What is EDM?

Empirical Dynamic Modeling (EDM) is a nonparametric framework for analyzing time series data. Unlike traditional statistical approaches that assume a fixed model structure (e.g., ARIMA, VAR), EDM reconstructs the underlying dynamics directly from observed data using Takens’ embedding theorem.

Core Idea

A single observed time series can be used to reconstruct the multidimensional state space of the system that generated it. By embedding the time series into a higher-dimensional space using lagged copies of itself, EDM recovers the attractor — the geometric structure that governs the system’s dynamics.

The EDM Workflow

Embedding — Transform a scalar time series into a multidimensional state space using lagged coordinates
Prediction — Use nearest neighbors in the reconstructed state space to forecast future states
Inference — Distinguish deterministic chaos from stochasticity, estimate nonlinearity, and test causal relationships

Key Algorithms

Simplex Projection

Simplex projection is the simplest EDM prediction method. It finds the E+1 nearest neighbors (forming a simplex in E-dimensional space) around a query point and produces a weighted-average prediction. The weights are inversely proportional to distance.

Simplex projection is primarily used to determine the optimal embedding dimension E — the dimension at which prediction skill peaks indicates the true dimensionality of the attractor.

S-Map

Sequential Locally Weighted Global Linear Map (S-Map) extends simplex projection by fitting a local linear model at each prediction point. A locality parameter theta controls how strongly distance-weighted the regression is:

theta = 0: Global linear regression (equivalent to a standard linear model)
theta > 0: Increasingly local, nonlinear behavior

The degree to which prediction improves with increasing theta quantifies the nonlinearity of the system.

Convergent Cross Mapping (CCM)

CCM tests for causal relationships between variables in coupled dynamical systems. If variable X causally influences variable Y, then the attractor reconstructed from Y should contain information about X.

CCM measures this by checking whether predictions of X from the reconstructed state space of Y converge (improve) as more data is used. Convergence with increasing library size is the signature of causality.

When to Use EDM

EDM is particularly suited for:

Nonlinear dynamical systems where linear models fail
Short, noisy time series where parametric models overfit
Causal inference in coupled systems (e.g., ecology, neuroscience, climate)
State-dependent dynamics where relationships change over time