Exact results of the limited penetrable horizontal visibility graph associated to random time series and its application

Minggang Wang,H. Eugene Stanley,Lixin Tian,Longfeng Zhao,Ruijin Du,Gaogao Dong,André L. M. Vilela

doi:10.1038/s41598-018-23388-1

Minggang Wang, H. Eugene Stanley + Show 5 more

Open Access

https://doi.org/10.1038/s41598-018-23388-1

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Abstract

The limited penetrable horizontal visibility algorithm is an analysis tool that maps time series into complex networks and is a further development of the horizontal visibility algorithm. This paper presents exact results on the topological properties of the limited penetrable horizontal visibility graph associated with independent and identically distributed (i:i:d:) random series. We show that the i.i.d: random series maps on a limited penetrable horizontal visibility graph with exponential degree distribution, independent of the probability distribution from which the series was generated. We deduce the exact expressions of mean degree and clustering coefficient, demonstrate the long distance visibility property of the graph and perform numerical simulations to test the accuracy of our theoretical results. We then use the algorithm in several deterministic chaotic series, such as the logistic map, H´enon map, Lorenz system, energy price chaotic system and the real crude oil price. Our results show that the limited penetrable horizontal visibility algorithm is efficient to discriminate chaos from uncorrelated randomness and is able to measure the global evolution characteristics of the real time series.

Highlights

Several methodologies for understanding the complicated behavior of nonlinear time series have been recently developed, including chaos analysis[1,2], fractal analysis[3,4], and complexity measurement[5,6]
We prove that an i.i.d. random series can be mapped on a limited penetrable horizontal visibility graph with exponential degree distribution, which is an extension of the result presented by Luque et al.[14]
We show several exact results of limited penetrable horizontal visibility graph (LPHVG) associated with i.i.d. random time series and apply them to the deterministic chaotic series of a logistic map, a Hénon map, a Lorenz system and energy price chaotic system and a crude oil price series

Summary

Introduction

Several methodologies for understanding the complicated behavior of nonlinear time series have been recently developed, including chaos analysis[1,2], fractal analysis[3,4], and complexity measurement[5,6]. The fourth one is recurrence networks method[20,21] This method uses the concept of recurrences in phase space and the recurrence matrix of a time series is interpreted as the adjacency matrix of an associated complex network. Researchers have used complex network theories to study multivariate time series[26,27,28,29] These methods all effectively maintain most of the properties of different types of time series, and they have been successfully used in many different fields[30,31,32,33,34,35,36,37]. We derive exact results on the properties of the limited penetrable horizontal visibility graphs associated with independent and identically distributed (i.i.d.) random series. To verify our theoretical solution, we acquire simulation results by using several deterministic chaotic series and a real crude oil price series that confirms the accuracy and usability of our results

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