PREDICTIVE ANALYSIS OF TIME SERIES WITH LONG-TERM MEMORY

Abstract

Predictive analysis with the use of nonlinear dynamics methods, chaos theory of real time series, characterizing the prevalence of a certain class of skin pathologies in Ukraine was carried out. The basis for such studies is the Tackens' theorem. The considered time series does not conform to the normal distribution law, and the hypothesis of a trend is not confirmed for them. According to previous studies, the authors calculated the fractality index \mu, the value of which indicates the state of relative stability of the studied process. The assessment of the correlation confirmed the practical absence of the influence of the present on the future in the studied numerical series. The bifurcation of the attractor revealed during the construction of the time series phase portrait allows the system to have such changes in its state that can be interpreted as jump-like or close to them. The value of the Lyapunov characteristic exponent confirms that the trajectory of the studied time series is chaotic. In the presented research, the procedure of qualitative analysis of the time series was carried out. Using the R/S procedure of fractal analysis, the effect of the time series long-term memory was revealed, the «depth of memory of the time series starting point» and the Hurst index were estimated. According to the performed calculations, the behavior of the H trajectory and R/S trajectory is such that it gives grounds to assert that the time series has a long-term memory. When carrying out the R/S procedure of fractal analysis of the time series as a whole, the time series of the family Q(X) of the initial time series X(t) were analyzed. The distribution of the memory depth estimates was constructed, the fuzzy set L(Q(X)) describing «memory depth of the time series X(t)» as a whole was formed, which is received from a sequence of pairs {1;\mu(1)}, where \mu(1) is the value of the membership function «depth l» of the fuzzy set L(Q(X)). The presence in the time series of the long-term memory effect makes it possible to apply the method of cellular automata in predicting its values.