How is the ‘DistanceThreshold’ influencing the embedding dimension in the Phase Space reconstruction calculation?

I am calculating the Lyapunov-Exponent of a time series and for this I need a reconstructed phase space. I reconstruct it with the ‘phaseSpaceReconstruction’ function from Matlab (Predictive Maintenance Toolbox). One input parameter is the 'DistanceThreshold'. If I understand correctly it determines the minimal distance of nearest neighbors in the reconstructed phase space.

In the ‘phaseSpaceReconstruction’ the function ‘localFalseNeighbors’ computes the embedding dimension. The ‘DistanceThreshold’ is named ‘RT’ here.

Searching the closest point is done in

‘predmaint.internal.NonlinearFeatures.getNearestNeighbors(Mdl,X,np,1)‘

The 4th input parameter is for MinSeparation. Should this be not the ‚DistanceThreshold‘ Parameter, rather than set to 1?

Cause when reconstructing the Phase space it is now possible that the nearest neighbor is only 1 index away from the original time series. While I set my Threshold in the beginning to 20.

Answer 1

[Ayush Modi]

Hi Frederik,

As per my understanding, you would like to understand how "DistanceThreshold" argument is affecting embedding dimension(dim) argument in ‘phaseSpaceReconstruction’ function.

• The embedding dimension refers to the number of variables or dimensions used to reconstruct the phase space from a time series. It determines the complexity and dimensionality of the reconstructed phase space.

• The 'DistanceThreshold' parameter is used to determine the nearest neighbors in the time series data. It specifies the maximum distance or similarity allowed between points to be considered neighbors. Points that are closer than the 'DistanceThreshold' are considered neighbors, while points farther apart are not.

• In the context of Phase Space reconstruction, the 'DistanceThreshold' influences the selection of neighbors for each point in the reconstructed phase space. If the 'DistanceThreshold' is too small, it may result in fewer neighbors being identified, potentially leading to an inaccurate representation of the underlying dynamics. This can result in a lower embedding dimension, as the reconstructed phase space may not capture the full complexity of the system.

• On the other hand, if the 'DistanceThreshold' is too large, it may include more distant points as neighbors, resulting in a higher embedding dimension. Including more neighbors can increase the complexity of the reconstructed phase space and capture fine-grained details of the system's dynamics. However, if the 'DistanceThreshold' is set too large, it may also introduce noise or unrelated points, leading to a distorted representation.

Note: The expected value for “DistanceThresold” argument is between 10 and 50.

Please refer to the following MathWorks documentation for more information on “phaseSpaceReconstruction” function and various arguments:

https://www.mathworks.com/help/predmaint/ref/phasespacereconstruction.html

I hope this resolves the issue you were facing.