Delay coordinates in practice

Even though Takens' theorem does not give any guarantees of the success of the embedding procedure in the noisy case, the delay coordinate embedding has been found useful in practice and is used widely [22]. In the noisy case, having more than one-dimensional measurements of the same process can help very much in the reconstruction even though Takens' theorem achieves the goal with just a single time series.

The embedding dimension must also be chosen carefully. Too low dimensionality may cause problems with noise amplification but using too high dimensionality will inflict other problems and is computationally expensive [9]. Assuming there is access to the original continuous measurement stream there is also another free parameter in the procedure, namely choosing the embedding delay $\tau$. Takens' theorem applies for almost all delays, at least as long as an infinitely long series of noiseless observations is used. According to Haykin and Principe [22] the delay should be chosen to be long enough for the consecutive observations to be essentially, but not too, independent. In practice a good value can often be found at the first minimum of the mutual information between the consecutive samples.