A New Symbolic Time Series Analysis Method Based on Time-to-Space Mapping, through a Symmetric Magnetic Field, Quantized by Prime Numbers

Abstract

This work presents a new analysis method for two-symbol symbolic time series based on the time-to-space mapping achieved through a device of current carrying circular rings. An algorithm based on the theory of prime numbers is proposed for the approximate estimation of the stratified magnetic field produced by the aforementioned device. The main property of the specific algorithm is that it quantizes the stratified magnetic field. If a two-symbol symbolic time series is used to determine the flow directions of the rings’ currents, a time-to-space mapping of the dynamics of the system producing the time series is observed. A unique “fingerprint” of the symbolic dynamics is shaped by the spatial allocation of the values of the six-valued symmetric quantized magnetic field produced by the device. This allows for the quantitative evaluation of the original system’s dynamics by analyzing the resultant quantized magnetic field values space allocation, in a spectrum ranging from the lack of dynamics (randomness) to the presence of dynamics at all scales (criticality). Two examples of application–corresponding to the extremes of the dynamics spectrum, specifically, for symbolic time series resulting from (a) a random numbers generator and (b) the spin alternation of 2D-Ising in its critical state, verify the reliable time-to-space mapping of the involved symbolic dynamics. Moreover, an application to the symbolic sequence produced by the DNA of the GAPDH (Glyceraldehyde-3-Phosphate Dehydrogenase) human gene is presented as a real-world, intermediate dynamics case. The proposed symbolic time series analysis method presents the advantage that can take into account information related to both symbols, which is particularly useful in analyzing two-symbol time series of relatively short length where the probabilities of occurrence of the two symbols are not equal. By inferring the universality class of an artificial-neural-network-based hybrid spin model through the value of the critical exponent , it is shown that for such time series, the proposed method provides a unique way to expose the real dynamics of the underlying complex system, in contrast to the analysis of waiting times in the time domain that leads to an ambiguous quantitative result.

Spontaneous Symmetry Breaking in Systems Obeying the Dynamics of On–Off Intermittency and Presenting Bimodal Amplitude Distributions

“…It is known that this model undergoes a second-order phase transition when the temperature drops below a critical value [10,16]. Thus, for a lattice of 20 3 nodes in three dimensions (3D Ising model), the critical (or pseudocritical for finite-size lattices) temperature has been found to be T c = 4.545 (J ij =1). Considering the sweep of the whole lattice as an algorithmic time unit, the possible values that spin can have in the model are ±1.…”

“…In a series of works [1,2], it has been shown that the use of a device with circular rings carrying electric currents of the same intensity but in a random flow direction results in a symmetrical stratified magnetic field, i.e., a magnetic field that presents a zonal structure around an axis of symmetry. Furthermore, using the most recently introduced primenumbers-based algorithm (PNA) to simulate the magnetic field of the specific device [3], the quantization of the above-mentioned stratified magnetic field is achieved, i.e., the magnetic field values' zones can be converted to quantized magnetic field values. Thus, if a two-symbol symbolic time series (using, for example, the symbols "+1", "−1") is employed to determine the flow directions of the rings' currents, a time-to-space mapping of the dynamics of the system producing the time series, is achieved [3].…”

“…Furthermore, using the most recently introduced primenumbers-based algorithm (PNA) to simulate the magnetic field of the specific device [3], the quantization of the above-mentioned stratified magnetic field is achieved, i.e., the magnetic field values' zones can be converted to quantized magnetic field values. Thus, if a two-symbol symbolic time series (using, for example, the symbols "+1", "−1") is employed to determine the flow directions of the rings' currents, a time-to-space mapping of the dynamics of the system producing the time series, is achieved [3]. This mapping enables the calculation of the exponents of the "waiting times" distributions in the space of Symmetry 2023, 15, 1448 2 of 17 the magnetic field positions, "k," leading to the development of the method of symbolic time-series analysis referred to as the PNA-based symbolic-time-series-analysis method (PNA-STSM) [3]).…”

“…Thus, if a two-symbol symbolic time series (using, for example, the symbols "+1", "−1") is employed to determine the flow directions of the rings' currents, a time-to-space mapping of the dynamics of the system producing the time series, is achieved [3]. This mapping enables the calculation of the exponents of the "waiting times" distributions in the space of Symmetry 2023, 15, 1448 2 of 17 the magnetic field positions, "k," leading to the development of the method of symbolic time-series analysis referred to as the PNA-based symbolic-time-series-analysis method (PNA-STSM) [3]). The PNA-STSM can be applied to any two-symbol symbolic time series that is produced directly or indirectly (after coarse-graining) by any real or numerical dynamic system and presents the advantage that it can reliably expose the real dynamics of a complex system, even if a relatively short time series is available and the probabilities of the occurrence of each of the two symbols are not equal [3].…”

“…Waiting times are essential because their distribution can reveal the dynamics of the system generating the time series under analysis [3]. For example, if the distribution of the waiting times follows a power law with the exponent p ∈ [1, 2), this is a strong indication that the system is in its critical state [4].…”