Direct Calculation of the f(α) Fractal Dimension Spectrum from High-Dimensional Correlation-Integral Partitions

Fractal dimension spectra have been used to characterize the complexity of dynamical time series since the 1980s. Calculation of these spectra are traditionally based on fixed-size methods that are grid-based, such as the histogram technique, or sample-based, such as the correlation-integral method. This paper extends the Chhabra and Jensen direct approach on histogram-binned data by deriving the direct calculation of the f(α) spectrum of scaling indices from correlation-integral based partition functions. That is, the canonical correlation-integral approach to f(α) is defined. The benefit of this novel method is that the extended dynamical range of the correlation-integral can be used to generate the compact f(α) spectrum from high-dimensional embeddings without resorting to the Legendre transform. A comparison of spectra results on the Ikeda attractor are presented.