Manifold Dimensional Expansion (MDE)

Manifold dimensional expansion is a causal discovery and dimensionality reduction technique designed to identify low dimensional maximally predictive observables of a multivariate dynamical system. In contrast to many machine learning and statistical representations the discovered variables are not latent, abstract representations of complex system components, but observables.

The algorithm is based on a greedy implementation of the generalized Takens embedding theorem. However, instead of using time delays for dimensionality expansion, observables that improve forecast skill of a target variable are added until no further improvement can be achieved. The default predictor is the Empirical Dynamic Modeling (EDM) simplex function providing a fully nonlinear predictor.

Causal inference is performed with Convergent Cross Mapping (CCM) ensuring the added observable is part of the same dynamical system as the target time series.

Output is a DataFrame of observation vectors defining a low-dimensional manifold with optimal predictability for the target variable.