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Description
Sequences of small volume samples of $n \leq 10$ numbers of reference ${k_{j}}$ particle fluxes with mean $0 \leq \bar{k} \leq 5$ correspond to random vectors $(\mathrm{RV}-v(.)$ ) of frequencies $v_{j}\left(k_{j}\right)$ of values $k_{j}: v(.)=\left(v_{0}, v_{1}, \ldots, v_{l}\right), n=\sum_{j=0}^{i} v_{j}\left(k_{j}\right)$ and $\mathrm{RV}$ of relative frequencies $v_{j}^{\prime}(.)=v_{j} / n$. Analysis of homogeneity of individual $\mathrm{RV}$ pairs and their large $M>>1$ sequences remains a critical data handling procedure.
A method for evaluating homogeneity of random vectors $v(.)$ and $v^{\prime}(.)$ pregrouped in peaks with several fixed components $v_{j}(.)$ of multimodal distributions of functionals $ID(v(.))$ is proposed [1].
$ID(v(.))_{m}=a_{0} v_{0 . m}+a_{1} v_{1 . m}+\ldots+a_{l} v_{l, m}$, where $a=\left(a_{0}, \ldots, a_{l}\right)$ - is a defined vector $[1,2]$.
The method is based on an analysis of the $\rho\left(\mu(v, \ldots, S)_{m}, \mu(v, \ldots, S)\right)_{q}$ metric of the phase trajectory projections of the complex functions of the empirical central moments of $\mathrm{RV}$ of fractional orders $S>1$
$$
\mu(v(.), S)=\frac{1}{1-n} \sum_{j=1}^{n}\left(k_{j}-\bar{k}\right)^{s}=\text{Re}\left(\mu(v, S)+i \text{Im}(\mu(v, S)), \quad i^{2}=-1\right.
$$
$$
\begin{array}{|l|l|l|l|l|}
\hline \mathrm{m}, \mathrm{q} & \nu_{m} & P\left(\nu_{m}\right) & \rho(, m) & \rho(, m) \\
\hline 1 & 6031 & 0.0006 & 0.013 & 0.0087 \\
\hline 2 & 5131 & 0.0043 & 0.0092 & 0.0046 \\
\hline 3 & 4231 & 0.0123 & 0.0048 & 0 \\
\hline 4 & 3331 & 0.0193 & 0 & 0.0048 \\
\hline 5 & 2431 & 0.0169 & 0.0047 & 0.0092 \\
\hline 6 & 1531 & 0.0079 & 0.0094 & 0.013 \\
\hline 7 & 0631 & 0.0015 & 0.016 & 0.018 \\
\hline
\end{array}
$$
Contrary to [2] the proposed method takes into account besides the imaginary one also the real component $\text{Re}(\mu(v, S))$ of the momentum function $\mu(v, S)$. In particular, for RV, forming one of peaks in distributions $M(ID(v(.)))$ at $\bar{k}=1.176 \ldots ; n=10$ probabilities of their realization $P(v(.))$ and values of metrics $\rho(.)$ are $S_{\max }=4.9 \rho(m=3, m) \quad \rho(m=4, m)$ at $S_{0}=1$
References:
1. G. Babenko, V. M. Vakhtel, V. A. Rabotkin // Procelding of the “NUCLEUS – 2019”, Dubna, 2019, р.330.
2. N.M. Bliznyakov and others // Proceedings of the International Conference. Voronezh Winter Mathematical School, Voronezh, Voronezh State University Press, 2021, pp. 54 - 56.
