Metric used for the estimation of dimensions
heliosdrm opened this issue · 1 comments
There is an inconsistency in the metrics used for calculating the optimal dimensions of reconstruct / embed.
In the code of the internal function _average_a (used by estimate_dimension), the distance δ of nearest neighbors is calculated using an infinity norm, as in Cao's paper - e.g. δ = norm(R2[i]-R2[j], Inf). However, the KDTree that was used to find the nearest neighbors is calculated with the default options (tree2 = KDTree(R2)), including Euclidean distances (as said in https://github.com/KristofferC/NearestNeighbors.jl).
The KDTree should be calculated with the Chebyshev metric to be consistent. Another option (which I advocate) is let the user choose what Metric is used. PR #9 includes code to facilitate this (only for the calculation of δ, not for the KDTree).
The authors of the other methods (FNN, FFNN) used Euclidean distances in their papers. This is the only option consistent with Kennel's equations [4-5] (FNN), but in the case of the FFNN, other metrics could be used without problem.