Embedding

The EDM Framework is based on a multidimensional representation of system dynamics, colloquially referred to as an embedding. Given a dynamical system of dimension D, the Whitney Embedding Theorem establishes limits on the embedding dimension, E, needed to completely represent the dynamics. Generally, EDM uses the parameter E to represent the dimension of the embedding, however, to support multivariate embeddings there are deviations from the above definitions as noted below.

The combination of the columns and embedded parameters control what variables are included in the embedding, and, whether a time-delay embedding is created. The default embedded = false instructs EDM to create a time-delay embedding using each variable in columns. If embedded = true, the set of variables specified in columns is assumed to be a valid embedding, no time-delay embedding is performed.

These functions accept multiple values for the columns parameter: Embed() Simplex() SMap() Multiview() PredictInterval() PredictNonlinear()

These functions accept the embedded parameter: Simplex() SMap() PredictInterval() PredictNonlinear()

  • If embedded = false (default):

    • If columns is a single value, then an E-dimensional time-delay embedding is created.
    • If columns has N values, then each of the N column variables is time-delay embedded to dimension E. The EDM state-space has overall dimension E*N.

  • If embedded = true:

    • The EDM state-space has dimension E = N, the number of columns.

Examples

  1. Simplex( ..., columns = 'x', E = 2 ) : 2-dimensional univariate embedding.
  2. Simplex( ..., columns = 'x y', E = 2 ) : 4-dimensional multivariate embedding.
  3. Simplex( ..., columns = 'x y', E = 2, embedded = True ) : 2-dimensional multivariate embedding.

CCM()

CCM() normally operates on a single column cross-mapped to a single target. It is possible to cross map multivariate embeddings, however, full application of convergent cross mapping has not been validated in the literature. See the CCM() function for embedding details.

Multiview()

Multiview() has parameters D, E, multiview.

  • D is the dimension of the state-space processed by simplex in each multivariate evaluation. D defaults to the number of columns.

  • E is the embedding dimension of each variable. If E = 1, no time delay embedding is done, the state-space is the D columns. If E > 1, then each of the D columns are embedded to E, however, the state-space remains D-dimensional as combinations of the embeddings are evaluated D-at-a-time.

  • multiview is the number of top-ranked D-dimensional predictions to average for the final prediction. Default multiview = sqrt(C) where C is the number of combinations C(n,D) available from the n = D * E variables taken D at-a-time.

SMap()

SMap() should be called with columns explicity corresponding to dimensions E. In the multivariate case (number of columns > 1) use embedded = true with E equal to the number of columns. See the embedding note in SMap() documentation.