Foundations of MDE

MDE leverages the Generalized Embedding Theorem employing Empirical Dynamic Modeling to evaluate dynamical manifolds. The generalized embedding theorem is a generalization of Taken's theorem from time-delays to multivariate observations.

Time Series as Observations of a Dynamic System

Dynamic systems are both state and time dependent. Here one can see the direct connection between state and time dependence.

Takens Theorem

Takens theorem is a remarkable mathematical result that allows one to reconstruct a representation of the system dynamics (state-space manifold) from a single (univariate, 1-D) timeseries observed from the system. The embedding results in generation of a higher-dimensional representation of the system.

MDE Generalized Embedding

MDE leverages the generalized embedding theorem to discover optimally informative multidimensional manifolds for a target observable of a multivariate complex system. Specifically, given a target observable, scan all other observables to find the best 1-D predictor of the target, ensuring the predictor has causal inference with the target. With this 1-D vector scan all remaining observables to find the 2-D embedding with best predictability and causal inference. This greedy algorithm is iterated up to the point that no further prediction skill improvement can be produced.