Christophe LETELLIER
03/04/2013

The embedding dimension is the smallest dimension required to embed an object (a chaotic attractor for instance). In other word, this is the minimum dimension of the space in which you reconstruct a phase portrait starting from your measurements and in which the trajectory does not cross itself, that is, in which the determinism is verified. Of course, this is a statistical measure, meaning that you may have some "rare" self-crossings. When a global model is attempted, this is the minimum dimension your model must have.

A practical method was proposed by Liangyue Cao to determine the minimum embedding dimension from a scalar time series [1]. Based on the method developed by Kennel and coworkers [2], it has the following advantages : (1) does not contain any subjective parameters except for the time-delay for the embedding ; (2) does not strongly depend on how many data points are available ; (3) can clearly distinguish deterministic signals from stochastic signals ; (4) works well for time series from high-dimensional attractors ; (5) is computationally efficient.

Thanks to Liangyue, this is the Fortran algorithm I used for years. You can dowload it and use it. Please, if so, quote Liangyue’s paper to thank him.

Thanks to Eric Foucault (Université de Poitiers) a c/Python code, based on Lyangyue’s one, is also available. Please, acknowledge them if you are using this code.

[1] **L. Cao**, Practical method for determining the minimum embedding dimension of a scalar time series, *Physica D*, **110**, 43-50, 1997.

[2] **M. Kennel, R. Brown & H. Abarbanel**, Determining embedding dimension for phase-space reconstruction using a geometrical construction, *Physical Review A*, **45**, 3403-3411, 1992.