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

• Raoul R. Nigmatullin
• Paolo Lino
• Guido Maione
Chapter

## 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.

## Keywords

Fractional and complex moments The generalised mean function (GMV) The generalised Pearson correlation function (GPCF) The complete correlation factor (CCF) Internal and external correlations.

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## Authors and Affiliations

• Raoul R. Nigmatullin
• 1
• Paolo Lino
• 2
• Guido Maione
• 2
1. 1.Radioelectronics and Informative-Measurement Technics DepartmentKazan National Research Technical University named by A.N. Tupolev (KNRTU-KAI)KazanRussia
2. 2.Department of Electrical and Information EngineeringPolytechnic University of BariBariItaly