Análise de séries temporais via grafo de visibilidade horizontal e teoria da informação

Abstract

The method called Horizontal Visibility Graph (Horizontal Visibility Graph - HVG) [B Luque et al., Phys. Rev. E 80: 046103 (2009)], has the function of converting a time series into a graph. This method has been used to study several dynamical systems and as a distinction tool between chaotic and stochastic systems [L Lacasa, R Toral, Phys. Rev. E. 82: 036120 (2010)]. Specifically, the authors propose the degree distribution of the resulting degrees follows an exponential function P(k) exp(lk), wherein k and the node degree and l is a positive parameter to distinguish between stochastic dynamics and chaotic dynamics. In this paper, we first investigated the distribution characteristics of the degree using the HVG to several chaotic and stochastic systems and observe that, even if this methodology works, several examples are found wherein results differ from those expected. Then we propose a methodology that combines HVG with Information Theory quantifiers, in order to distinguish the deterministic nature of the stochastic behind the systems under study. Specifically, we showed that, by using the causal Shannon-Fisher plane, it is possible to characterize, by the positions on the plane, thenature of the systems. After that, we analyzed the stochastic nature systems using traditional quantifiers of networks, through which it was possible to distinguish among the different degrees of correlation structures. We also showed, through the degree distribution function bymapping with HVG of Fractional Brownian Motion (fBm) time series, a possible methodology for approaching the value of the Hurst exponent. Then we propose two adapted versions of the HVG method. The first one, called HVG-Windows, is a computationally faster version, which uses point windows to perform the analysis of time series. After extensive experimentation, we showed that, in this new version, time series does not lose information with the inclusion of time windows. In the second one, called HVG-weight, the way of mapping is identical to the HVG. However, we also calculate the edge weight, defined by the amplitude between two points that meet the visibility criteria, which allows the extraction of an alternative distribution (distribution of the edge weight). Thus, we propose a different way to extract series informationfrom a built network, compared to the distance distribution and the usual way through the degree distribution. Finally, we showed the weight distribution efficiency by studying here the Fractional Brownian Motion (fBm) time series and paleoclimatic data of temporal changesduring the Holocene age by Millennial Proxy Record of ENSO, comparing the results with the distance and degree distribution.