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

Within both slightly non-extensive statistics and a 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 an 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 introduced to present the 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.