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The Statistics of Fractional Moments and its Application for Quantitative Reading of Real Data

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New Digital Signal Processing Methods

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Abstract

This chapter analyses in detail the concept of fractional moments. The modern mathematical statistics widely uses only four integer moments: the arithmetic mean, the standard deviation (based on the second moment), the measure of asymmetry, and the measure of skewness associated with the value of the fourth moment. Why not increase the moments concept and create a space of real moments, including not only the whole set of integer moments but also the fractional moments? This space of moments will serve as a source of additional information in analysing random functions and sequences in time or frequency domains. The concept of fractional moment is quite new for most of the readers, and, therefore, the description will start from scratch and consider: (a) the definitions of the integer moments; (b) the reconsideration of a random sequence stability problem; (c) the definition of the generalised mean value function; (d) the generalised Pearson correlation function. These new definitions will be helpful to solve the problem of correlations. After reading this chapter, the reader will understand the difference between external and internal correlations and grasp the concept and the directions of application of fractional moments. The examples given in the chapter show the directions of unexpected applications, and more are outlined in the cited authors’ publications. The exercises given in this chapter should help to fasten and consolidate the read text.

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Author information

Authors and Affiliations

  1. Radioelectronics and Informative-Measurement Technics Department, Kazan National Research Technical University named by A.N. Tupolev (KNRTU-KAI), Kazan, Tatarstan Republic, Russia

    Raoul R. Nigmatullin

  2. Department of Electrical and Information Engineering, Polytechnic University of Bari, Bari, Italy

    Paolo Lino & Guido Maione

Authors
  1. Raoul R. Nigmatullin
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  2. Paolo Lino
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  3. Guido Maione
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Nigmatullin, R.R., Lino, P., Maione, G. (2020). The Statistics of Fractional Moments and its Application for Quantitative Reading of Real Data. In: New Digital Signal Processing Methods. Springer, Cham. https://doi.org/10.1007/978-3-030-45359-6_3

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  • DOI: https://doi.org/10.1007/978-3-030-45359-6_3

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