Computing correlation integral with the Euclidean distance normalized by the embedding dimension

The Grassberger-Procaccia method is revisited in this paper with a modified approach to compute the correlation integral through a Euclidean distance measure normalized by the embedding dimension. The performance of the suggested modification is assessed using three different types of signals, including Lorenz attractor, mechanical vibrations of helicopter flight, and biological data of animal sleep EEG. Results have shown consistent improvements over the original approach when the normalized Euclidean distance measure is used-correlation integrals for different embedding dimensions not only converge faster in scaling radius but also are more uniformly clustered within the same region. The implementation of the suggested modification is straightforward and resultant correlation integrals and linearly scaling regions for correlation dimension estimation are less sensitive to the varying embedding dimension.