Effective temperature of self--similar time series: analytical and numerical developments

TL;DR

This paper develops a thermodynamic framework for self-similar time series, introducing effective temperature linked to fractal dimension, and tests it through numerical models based on non-extensive statistics.

Contribution

It presents a novel thermodynamic approach to analyze self-similar time series using effective temperature and non-extensive statistics, supported by analytical and numerical methods.

Findings

Effective temperature correlates with fractal dimension.

Numerical models confirm temperature as average energy per degree of freedom.

Predictability limits are derived from thermodynamic considerations.

Abstract

Within both slightly non--extensive statistics and related numerical model, a picture is elaborated to treat self--similar time series as a thermodynamic system. Thermodynamic--type characteristics relevant to temperature, pressure, entropy, internal and free energies are introduced and tested. Predictability conditions of time series analysis are discussed on the basis of Van der Waals model. Maximal magnitude for time interval and minimal resolution scale of the value under consideration are found and analyzed in details. The statistics developed is shown to be governed by effective temperature being exponential measure of the fractal dimension of the time series. Testing of the analytical consideration is based on numerical scheme of non--extensive random walk. A statistical scheme is introduces to present numerical model as a grand canonical ensemble for which entropy and internal energy are calculated as functions of particle number. Effective temperature is found numerically to show that its value is reduced to averaged energy per one degree of freedom.