Time series characterization via horizontal visibility graph and Information Theory
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 -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 , be a time series of data. Two nodes and in the graph are connected if it is possible to
Shannon entropy
When considering discrete probability distributions () the Shannon entropy [18] is defined as:
If we are in a position to predict with certainty which of the possible outcomes whose probabilities are given by 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 ,
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 (), that describes the raggedness of the motion. The Hurst’s parameter defines two distinct regions in the interval . For , 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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2021, Chaos, Solitons and FractalsCitation Excerpt :In recent years, some new methodologies which use complex networks to characterize complex systems have emerged [11–15], especially the complex network analysis of time series. Many efficient methods have been developed to transform time series into complex networks [16–20]. In particular, Lacasa et al. proposed visibility graph [16] and horizontal visibility graph (HVG) [17] as methods to transform time series into complex networks.
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2021, Advances in Applied MathematicsCitation Excerpt :An illuminating characterization of HVGs using “one-point compactified” times series and tools of algebraic topology is obtained in a recent work [29]. Theoretical body of work on the HVGs includes studies of their degree distributions [14,16], information-theoretic [9,15] and other [11] topological characteristics, motifs [12,31], spectral properties [6,17], and dependence of graph features on the parameter for a specific parametric family of chaotic [4] or stochastic processes [32,34]. For more, see a recent comprehensive survey [35] and an extensive review of earlier results [23].
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2019, Applied Mathematics and ComputationCitation Excerpt :Each edge represents the transition between patterns, thus the name “ordinal patterns transition graphs”, denoted here as Gπ. The study of time series via their transformation into graphs is a very successful strategy [16–24]. Some notable examples are the visibility graph (VG) [21,23,24] and horizontal visibility graph (HVG) [22].