Rényi divergence as an improved generalized fractal dimension for signal and fractal analysis

Abstract

The Generalized Fractal Dimension (GFD) is widely used to analyze the scaling properties of multi-fractal signals. However, its reliance on a uniform distribution baseline limits its sensitivity to subtle structural variations. This study introduces Rényi divergence as an enhanced alternative, providing greater sensitivity by quantifying deviations from uniformity. A mathematical relationship between GFD and Rényi divergence is established, demonstrating how the latter offers a more detailed characterization of multi-fractal structures. To improve computational efficiency, the Rényi divergence spectrum is approximated using a generalized logistic function, ensuring analytical tractability without compromising accuracy. The method\RL{'}s effectiveness is illustrated through an example showing that Rényi divergence can distinguish signals that appear nearly identical under GFD analysis. These findings position Rényi divergence as a valuable extension of GFD, with potential applications in anomaly detection, signal classification, and multi-fractal analysis.