Big review of works by our group

This summer we (all active members of the group) have contributed to a pretty big paper [1]. In the said paper we have reviewed all of our varied approaches to the modeling of the long-range memory (which we understand as 1/f noise). The core difference of our approach, from the approaches taken by other groups, is that we use Markovian models without embedding actual memory into our models.

Title page

The paper also includes a new result - application of burst statistics (note that the paper uses another term - burst and inter-burst duration analysis) to understanding the nature of long-range memory of the fractional Levy stable motion. Fractional Levy stable motion is an interesting generalization of the Brownian motion in two regards. First of all the driving noise is not normally distributed, but is instead distributed according to the stable distribution, which makes large jumps quite likely. Also, the time series is integrated using fractional integral and thus possesses true long-range memory (one embedded into the model). We have shown that burst statistics is a good tool for this particular task (at least it has some advantages over other alternatives).

In the upcoming posts we will talk about ARFIMA process, which was instrumental in our analysis as it helped us to generate fractional Levy stable motion.

Disclaimer: While I was one of the authors of the paper, my impact on Section 4 was rather limited. I have written implementation of ARFIMA in Python and Mathematica. I have also validated that figures can be reproduced following the instructions given in Section 4. But this is fine as here we will focus on ARFIMA itself.