Abstract
While models are great, and easy to tackle from many different directions, the real world is not so kind. When describing and measuring real systems, one often has access only to a single timeseries that characterizes the system. Fortunately for us, in most cases even from a single timeseries one can squeeze out knowledge about the original system, in a way reconstructing the original dynamical properties. In this chapter we discuss the way to do this, using delay coordinate embeddings. We particularly focus on successfully applying this tool to real world data, discussing common pitfalls and some nonlinear timeseries analysis techniques based on delay embedding.
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Notes
- 1.
Notice that now that we are back discussing practical considerations, we use \(\tau \) as the delay time, which is always in multiples of the sampling time \(\delta t\).
- 2.
The mutual information is defined properly in Chap. 7 along with a sketch of how to compute it. The self-mutual information  specifically has an optimized implementation in DynamicalSystems.jl as
. - 3.
Nearest neighbors are points in the state space that are close to each other according to some metric, typically Euclidean or Chebyshev. There are many algorithms to find such neighbors, e.g., KD-trees.
- 4.
In the case of equality, best practice is to choose randomly between < or >.
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Datseris, G., Parlitz, U. (2022). Delay Coordinates. In: Nonlinear Dynamics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-91032-7_6
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DOI: https://doi.org/10.1007/978-3-030-91032-7_6
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