On the scaling range of power-laws originated from fluctuation analysis

TL;DR

This paper investigates the scaling range properties of various fluctuation analysis techniques, introduces a new method called MDMA, and compares their power law dependence on data length and accuracy, with implications for analyzing short time series.

Contribution

The study introduces MDMA, compares its scaling range properties with DFA and DMA, and generalizes the power law relation for autocorrelated data.

Findings

DMA and MDMA exhibit power law dependence of scaling range on data length and fit accuracy.

The power law relation applies to both uncorrelated and autocorrelated data.

Relations between scaling ranges of different techniques are discussed.

Abstract

We extend our previous study of scaling range properties done for detrended fluctuation analysis (DFA) \cite{former_paper} to other techniques of fluctuation analysis (FA). The new technique called Modified Detrended Moving Average Analysis (MDMA) is introduced and its scaling range properties are examined and compared with those of detrended moving average analysis (DMA) and DFA. It is shown that contrary to DFA, DMA and MDMA techniques exhibit power law dependence of the scaling range with respect to the length of the searched signal and with respect to the accuracy $R^2$ of the fit to the considered scaling law imposed by DMA or MDMA schemes. This power law dependence is satisfied for both uncorrelated and autocorrelated data. We find also a simple generalization of this power law relation for series with different level of autocorrelations measured in terms of the Hurst exponent. Basic relations between scaling ranges for different techniques are also discussed. Our findings should be particularly useful for local FA in e.g., econophysics, finances or physiology, where the huge number of short time series has to be examined at once and wherever the preliminary check of the scaling range regime for each of the series separately is neither effective nor possible.