Direct Calculation of the Fractal Singularity Spectrum

In the study of cognitive processes, quantitative time series are often recorded and subsequently analyzed using signal processing tools for characteristic features and dynamics. Since the 1980s, the signals from dynamical systems have been studied using fractal dimension spectra. Such spectra serve as a quantitative description of the complexity of the dynamical system's underlying attractor. This paper reviews the calculation of these spectra and reports on a recent extension to the Chhabra and Jensen direct calculation of the f(α) singularity spectrum. The new method overcomes the histogram-binning restriction of the traditional approach, and applies to correlation-integral based partition functions instead. 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 highly complex Ikeda attractor are presented.