Time series characterization via horizontal visibility graph and Information Theory

https://doi.org/10.1016/j.physa.2016.07.063Get rights and content

Highlights

  • This work deals with the characterization of dynamical systems using Horizontal Visibility Graphs (HVG) and Information Theory quantifiers.

  • We propose the use of the weight distribution, which is based on the difference of the time series values of connected points.

  • We study fractional Brownian motion time series and a paleoclimatic proxy record of ENSO taken from Pallcacocha Lake.

  • The weight distribution allows a better characterization of the studied systems, using considerable shorter time series.

Abstract

Complex networks theory have gained wider applicability since methods for transformation of time series to networks were proposed and successfully tested. In the last few years, horizontal visibility graph has become a popular method due to its simplicity and good results when applied to natural and artificially generated data. In this work, we explore different ways of extracting information from the network constructed from the horizontal visibility graph and evaluated by Information Theory quantifiers. Most works use the degree distribution of the network, however, we found alternative probability distributions, more efficient than the degree distribution in characterizing dynamical systems. In particular, we find that, when using distributions based on distances and amplitude values, significant shorter time series are required. We analyze fractional Brownian motion time series, and a paleoclimatic proxy record of ENSO from the Pallcacocha Lake to study dynamical changes during the Holocene.

Introduction

In the last few years, methods to transform time series into networks have been proposed, and with them, novel ways to analyze and characterize time series, have been developed. Among others, these novel methodologies include the use of disjoint cycles and their distances in the phase space to generate the links in the corresponding network  [1], [2]. Li and Wang  [3], [4] introduce a method based on n-tuples. Donner et al.  [5], [6] work with recurrence networks. There are also methods based on the phase space reconstruction of the time series  [2], [7], [8]. Latora et al.  [9] propose a graph based on the recurrence of time series motifs. Other methods take into account the visibility of elements in a time series, like the Visibility Graphs or the Horizontal Visibility Graphs  [10], [11]. Our article focuses on the use of the latter.

Following previous works  [12], [13], we extract probability distribution functions (PDFs) from the constructed networks to characterize the topological structure and to capture the dynamics of the transformed time series, using Information Theory quantifiers. Related works have primarily focused on the network’s degree distribution. We investigate in this work, alternative probability distributions and we compare their performance with the usual degree distribution. Specifically, we explore the distance distribution, that despite being poorly explored, it was shown to be efficient in capturing network’s topological changes  [14]. We also propose a PDF based on the difference of the time series values (amplitudes) between the nodes connected by the horizontal visibility algorithm. We find the distance distribution and the one based on amplitude differences more efficient in characterizing the studied systems as they require significantly shorter time series than the degree distribution.

We study fractional Brownian motion (fBm) time series generated with different degrees of correlations (different Hurst exponents), and a paleoclimatic proxy record of the Laguna Pallcacocha used to study the millennial El Niño/Southern Oscillation (ENSO) dynamic.

Section snippets

Horizontal visibility graph and associated PDFs

The horizontal visibility graph (HVG) is a methodology that transforms a time series into a graph maintaining the inherent characteristics of the transformed time series  [11]. The HVG consists in a geometrical simplification of the firstly proposed visibility graph (VG)  [10]. It considers each point in the time series, a node in the network, connected by the following consideration: Let {xi,i=1,,N}, be a time series of N data. Two nodes i and j in the graph are connected if it is possible to

Shannon entropy

When considering discrete probability distributions (P={pj:j=1,,M}) the Shannon entropy S[P]   [18] is defined as: S[P]=j=1Mpjlnpj.

If S[P]=0 we are in a position to predict with certainty which of the possible outcomes j whose probabilities are given by pj will actually take place. Our knowledge of the underlying process described by the probability distribution is, in this instance, maximal. On the contrary, our ignorance is maximal for a uniform distribution. For a given distribution P,

Characterization of fBm

The first experiment evaluates the performance of the methodology over artificially created time series. We study fractional Brownian motion (fBm) time series, that are continuous-time Gaussian processes, self-similar, and endowed with stationary increments  [26]. Motion and noise are characterized by the Hurst exponent (H), that describes the raggedness of the motion. The Hurst’s parameter defines two distinct regions in the interval (0,1). For H>1/2, consecutive increments tend to have the

Discussion and conclusions

In this work, we analyze the performance of a methodology that combines Horizontal Visibility Graph and Information Theory quantifiers to characterize dynamical systems. Most works rely on the use of the HVG degree distribution, however, we show through extensive experimentation, that the weight distribution based on amplitude differences, allows a better characterization with considerable shorter time series, relevant fact when analyzing real systems. Persistent processes usually require very

Acknowledgments

This research has been partially supported by CNPq and FAPEMIG. O.A. Rosso acknowledges partial support from the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina.

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