Discrete Geometrical Invariants in 3D Space: How Three Random Sequences Can Be Compared in Terms of “Universal” Statistical Parameters - Nigmatullin, Raoul R. (Frontiers in Physics, 2020)
Previous work has determined the discrete geometrical invariant (DGI) in 2D space, which allows the analysis of a pair of random sequences ([r1k, r2k], k = 1,2,…N) containing an equal number of data points to be reduced to eight “universal” parameters that present different inter-correlations between the two compared sequences. These eight parameters can serve as a “universal” platform for comparison of various random sequences of different natures. In this paper, we derive mathematical expressions for the DGI in 3D space, which represent three random sequences in the form of a “trajectory” of an “imaginary” particle in 3D space. The DGI is of the fourth order in 3D space and allows three random sequences (r1k, r2k, r3k, k = 1,2,…N) to be compared with one another. This unified and “universal” platform identifies (in total) six surfaces and 13 reduced and compact parameters obtained from 28 basic moments and their intercorrelations up to the fourth order, inclusively. The transcendental numbers π and E (Euler constant) are considered as examples of the use of the DGI method, and their 3D images are derived together with values for the 13 quantitative parameters that differentiate them from each other. An application of the method, identifying two different classes of earthquake data, is also presented to illustrate the potentially wide application of the approach to the identification and classification of nominally random data sequences.