Abstract
This paper outlines the foundations of an original theory related to the quantitative description of different scales fluctuations based on the verifiable ergodic hypothesis associated with the construction and subsequent analysis of a sequence of the ranged amplitudes (SRA). The integral calculated from this sequence leads to a fitting function of the form \(Bd(x)=A(x-x_{0})^{\alpha}(x_{N}-x)^{\beta}+B\) . This function besides 4 fitting parameters \(A,B,\alpha,\) and \(\beta\) characterizing the shape of the curve, contains 2 more important parameters related to the localization of the extrema of this function \(\bar{x}=x_{0}+w\Delta\) , \(\bar{y}=Aw^{\alpha}(1-w)^{\beta}\Delta^{\alpha+\beta}\) , where \(w=\alpha/(\alpha+\beta)\) , \(\Delta=x_{N}-x_{0}\) . The analysis shows that there are only 8 basic parameters (including the relative percentage fitting error) describing the fluctuation in the form of beta ‘‘finger’’ of a given width \(\Delta\) , which allows to describe in detail the set of fluctuations of a given width in the interval \([x_{0},x_{N}]\) forming a given trend-less sequence (TLS). This analysis allows us to propose an original fluctuation spectroscopy based on beta distribution (FSBoBD) for detailed description of fluctuations of different scales. This work makes up for these shortcomings and shows that the capabilities of FSBoBD are much broader and can be applied to describe a wide class of trend-less sequences. The proposed theory was tested on meteorological data (stream data of methane acceleration), which were kindly provided to the authors by the experimental group of Kazan Federal University.
