Lecture 3 (part1)
THE STATISTICS OF THE FRACTIONAL
MOMENTS: IS THERE ANY CHANCE
TO ‘READ QUANTITATIVELY’
ANY RANDOMNESS?
by Prof. R.R. Nigmatullin
Kazan National Research Technical University Karl Marx str. 10, Kazan,
Tatarstan, Russian Federation
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The basic questions:
3.1. Evaluation of statistical stability of random sequences based on higher moments
3.2The GMV-function and its basic properties
3.3The approximate expression for the GMV-function. Fractional and complex moments
3.4Different generalizations of the GMV-function and basic inequalities. Relationship of the fractional moments with the nonextensive Tsallis entropy
3.5The Generalized Pearson’s Correlation Function (GPCF) in the space of the moments. How to compare a part of a randomness with the whole one?
3.6Reduced of the fractional modelling in video-streams. FERMA approach
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The basic motivation:
is it possible to transform any random sequence to a smooth and “quantitatively readable” curve?
The answer can be positive if one can use the properties
of the so-called generalized mean value (GMV)-function
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3.1 Evaluation of statistical stability of random sequences based on higher moments
j = 1,2,…,N |
where p = 1,2,…,k. |
One can formulate the following question: to what system of equations these new set of added points should satisfy in order to keep invariant the values of the first k moments previously belonging to the initial segment?
where p = 1,2,…,k.
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The last requirement is equivalent to solution of the following nonlinear system of equations for the given set of stable k points
For k = 1
,
For k = 1
Coincides with Arithmetic Mean!
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For k = 2 the result is also interesting!
The case of three roots: r = 1, 2, 3 (k = 3)
The case of four roots: r = 1, 2, 3, 4 (k = 4)
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3.2 The GMV-function and its basic properties
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Detection of statistically stable points located inside a random sequence
For p =1,2 – the traditional statistics is recovered !
Comparison of desired samplings having different number of points !
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3.3 The approximate expression for the GMV- function. Fractional and complex moments
Calculation of the fitting parameters of the GMV-function with the help of the ECs method! – This is the solution!
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ECs method helps to transform the function with nonlinear fitting parameters to the linear combination of new parameters. They can be found by the LLSM.
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