Abstract
This chapter considers two unexpected applications of the fractional moments. The first application is associated with the distribution of stable points, which are described by defining the quantitative universal label (QUL). The “universal” distribution of stable points that exists in some random sequence can be expressed in the form of the generalised Gaussian distribution (GGD). This GGD is tightly associated with the distribution of the coefficients of some polynomial describing the stable points. If this distribution is general, then a reduced number (six) of stable parameters can describe a broad set of complex phenomena. The QUL is applied for quantitative description of famous transcendental numbers such as π and E (Euler constant). Then examples are provided for economic data and earthquake data. Besides, it is shown how to code valuable information by quantum dots.
Another application of the fractional moments regards the beta-distribution. It is proven that a part of the so-called sequence of the range amplitudes (SRA) after subtraction of the mean value and integration turns out to a bell-like curve that can be fitted by the beta-distribution. It means that the new set of obtained stable parameters can characterise a wide set of fluctuations of trendless sequences. This beta-distribution has remarkable scaling properties and can describe a broad set of long-time series. A nontrivial example is considered in details based on long-time membrane current fluctuations. The application of the fractional moments statistics helps to find the differences between the series containing the hidden output.
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Nigmatullin, R.R., Lino, P., Maione, G. (2020). The Quantitative “Universal” Label and the Universal Distribution Function for Relative Fluctuations. Qualitative Description of Trendless Random Functions. In: New Digital Signal Processing Methods. Springer, Cham. https://doi.org/10.1007/978-3-030-45359-6_4
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