L\'{e}vy scaling: the Diffusion Entropy Analysis applied to DNA sequences

TL;DR

This paper introduces the Diffusion Entropy Analysis (DEA), a novel method for accurately detecting scaling in complex time series, and applies it to DNA sequences to reveal Levy statistics at large scales.

Contribution

The paper demonstrates that DEA reliably identifies true scaling in time series and distinguishes Levy from Gaussian statistics, especially applied to DNA sequences.

Findings

DEA always yields correct scaling values when applicable.

DNA sequences exhibit Levy statistics at large scales.

DEA confirms the dynamic approach to DNA sequence analysis.

Abstract

We address the problem of the statistical analysis of a time series generated by complex dynamics with a new method: the Diffusion Entropy Analysis (DEA) (Fractals, {\bf 9}, 193 (2001)). This method is based on the evaluation of the Shannon entropy of the diffusion process generated by the time series imagined as a physical source of fluctuations, rather than on the measurement of the variance of this diffusion process, as done with the traditional methods. We compare the DEA to the traditional methods of scaling detection and we prove that the DEA is the only method that always yields the correct scaling value, if the scaling condition applies. Furthermore, DEA detects the real scaling of a time series without requiring any form of de-trending. We show that the joint use of DEA and variance method allows to assess whether a time series is characterized by L\'{e}vy or Gauss statistics. We apply the DEA to the study of DNA sequences, and we prove that their large-time scales are characterized by L\'{e}vy statistics, regardless of whether they are coding or non-coding sequences. We show that the DEA is a reliable technique and, at the same time, we use it to confirm the validity of the dynamic approach to the DNA sequences, proposed in earlier work.