## An entropic approach to the analysis of time series.

Description:
Statistical analysis of time series. With compelling arguments we show that the Diffusion Entropy Analysis (DEA) is the only method of the literature of the Science of Complexity that correctly determines the scaling hidden within a time series reflecting a Complex Process. The time series is thought of as a source of fluctuations, and the DEA is based on the Shannon entropy of the diffusion process generated by these fluctuations. All traditional methods of scaling analysis, instead, are based on the variance of this diffusion process. The variance methods detect the real scaling only if the Gaussian assumption holds true. We call H the scaling exponent detected by the variance methods and d the real scaling exponent. If the time series is characterized by Fractional Brownian Motion, we have HÂ¹d and the scaling can be safely determined, in this case, by using the variance methods. If, on the contrary, the time series is characterized, for example, by LÃ©vy statistics, H Â¹ d and the variance methods cannot be used to detect the true scaling. LÃ©vy walk yields the relation d=1/(3-2H). In the case of LÃ©vy flights, the variance diverges and the exponent H cannot be determined, whereas the scaling d exists and can be established by using the DEA. Therefore, only the joint use of two different scaling analysis methods, the variance scaling analysis and the DEA, can assess the real nature, Gauss or LÃ©vy or something else, of a time series. Moreover, the DEA determines the information content, under the form of Shannon entropy, or of any other convenient entopic indicator, at each time step of the process that, given a sufficiently large number of data, is expected to become diffusion with scaling. This makes it possible to study the regime of transition from dynamics to thermodynamics, non-stationary regimes, ...

Date:
December 2001

Creator:
Scafetta, Nicola