GenML: A Python Library to Generate the Mittag-Leffler Correlated Noise

TL;DR

This work tackles the lack of practical tools for generating Mittag-Leffler correlated noise (M-L noise), which captures a broad spectrum of memory effects through its autocorrelation $C(t)=\frac{C}{\tau^{\lambda}} E_{\lambda}[-(t/\tau)^{\lambda}]$. It introduces GenML, a Python library that implements a Davies-Harte-based generation scheme with an FFT-based circulant approach and an optimization step to select an optimal sequence length $T_{\mathrm{opt}}$, enabling direct synthesis of M-L noise sequences. The authors validate GenML by comparing empirical autocorrelation functions to the theoretical Mittag-Leffler form and by examining diffusion via the Langevin equation, demonstrating close agreement and the expected long-time MSD slope of $2-\lambda$. GenML thus enables direct simulations and data-driven analyses of complex systems with memory, broadening applicability in physics, biology, and finance.

Abstract

Mittag-Leffler correlated noise (M-L noise) plays a crucial role in the dynamics of complex systems, yet the scientific community has lacked tools for its direct generation. Addressing this gap, our work introduces GenML, a Python library specifically designed for generating M-L noise. We detail the architecture and functionalities of GenML and its underlying algorithmic approach, which enables the precise simulation of M-L noise. The effectiveness of GenML is validated through quantitative analyses of autocorrelation functions and diffusion behaviors, showcasing its capability to accurately replicate theoretical noise properties. Our contribution with GenML enables the effective application of M-L noise data in numerical simulation and data-driven methods for describing complex systems, moving beyond mere theoretical modeling.

Figures (6)

• Figure 1: Comprehensive workflow of the GenML software for M-L noise generation.
• Figure 2: Representative examples of M-L noise generated by GenML with $N = 1$, $T = 500$, $C=1$, $\tau = 10$, where $\lambda$ = 0.6 (a), 1.2 (b), and 1.8 (c).
• Figure 3: Illustrative comparisons of calculated (dots) and theoretical (lines) normalized autocorrelation functions of M-L noise across various parameter sets: (a) $\lambda=0.6$, $\tau=20$; (b) $\lambda=1.2$, $\tau=20$; (c) $\lambda=1.8$, $\tau=20$; (d) $\lambda=0.6$, $\tau=100$; (e) $\lambda=1.2$, $\tau=100$; (f) $\lambda=1.8$, $\tau=100$. The calculated autocorrelation function of each parameter set is based on an average of 1000 independent noise sequences.
• Figure 4: Comparison of calculated (dots) and theoretical (lines) MSDs for the diffusion behavior driven by M-L noise. (a) and (b) show the MSDs in the log-log scale for $\lambda$ values of 0.6, 1.2, and 1.8 at $\tau$ values of 20 and 100, respectively.
• Figure : Selection of optimal sequence length