The EDM Framework
The EDM framework is predicated on the availability of a multidimensional representation of the system dynamics. For example, the well-known Rössler attractor is a 3-dimensional (3-D) system that can express chaotic dynamics. The set of 3 equations that define the Rössler system define an attractor or manifold defining the state-space or phase-space of the system. Often, one does not have complete information regarding the system dynamics, in which case we can invoke Takens embedding theorem.
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.
Embedding
The process of creating this representation is termed embedding.
Embedding is performed implicitly in GMN/EDM functions unless the embedded
argument is set True indicating that the data are already embedded.
Default embeddings are time-delay (lagged) with τ = -1.
Nearerst Neighbor Forecasting: Simplex and S-map
GMN/EDM implements two timeseries prediction algorithms:
Simplex(),
and SMap().
Both operate in the embedding state-space, using nearest neighbors
of a query point (location in the state-space from which a prediction is
desired) to project a new estimate along the manifold.
Simplex()
uses the centroid of the k-nearest neighbors (knn) of the query point as the
estimate. The number of neighbors, knn, is conventionally set as the number
of dimensions plus one: knn = E + 1.
SMap()
uses a linear regression of query point neighbors to project a new estimate
along the manifold. By default, the number of
neighbors knn in the regression are set to the total number of state-space
observation points. An exponential localisation function F(θ) = exp(-θd/D)
is used to selectively ignore neighbors beyond the localisation radius.
θ is the localisation parameter, d a neighbor distance, and D the mean
distance to all neighbors. This allows one to vary the
extent to which local neighbors are considered in the linear projection,
effectively modeling different local "resolutions" on the attractor.
The effect of this knn localisation can be assessed with the EDM
PredictNonlinear() function that
evaluates S-map predictive skill over a range of localisations θ.