Renewal, modulation, and superstatistics in times series

Area of Science:

• Statistical Physics
• Time Series Analysis
• Complex Systems

Background:

• Generating time series with non-exponential waiting time distributions is crucial for modeling various complex systems.
• Understanding the statistical equivalence and physical properties of different time series generation methods is essential.

Purpose of the Study:

• To compare two distinct approaches, renewal and modulation, for generating time series with non-exponential waiting times.
• To investigate whether different time series exhibiting the same waiting time distribution yield equivalent physical properties.
• To analyze the impact of each method on aging, scaling, and diffusion processes.

Main Methods:

• The study employs two distinct methods: renewal and modulation, to generate synthetic time series.
• Analysis involves examining the statistical properties and resulting physical behaviors, including aging and diffusion characteristics.
• Crucial events within the modulation approach are identified and analyzed for their impact on scaling.

Main Results:

• Renewal method generates aging and anomalous scaling in time series.
• Modulation method can result in ordinary or anomalous diffusion, with aging effects being insignificant.
• Modulation generates numerous 'exponential' events and rare 'crucial' events that govern diffusion scaling.

Conclusions:

• Different time series with identical waiting time distributions can exhibit distinct physical properties.
• Modulation generates 'crucial events' that, though rare, determine the diffusion process scaling.
• Identifying these crucial events within modulation is challenging due to their concealment by numerous exponential events.