Luque B, Lacasa L, Ballesteros F, Luque J
Departamento Matemática Aplicada y Estadística, ETSI Aeronáuticos, Universidad Politécnica de Madrid, Madrid, Spain.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Oct;80(4 Pt 2):046103. doi: 10.1103/PhysRevE.80.046103. Epub 2009 Oct 7.
The visibility algorithm has been recently introduced as a mapping between time series and complex networks. This procedure allows us to apply methods of complex network theory for characterizing time series. In this work we present the horizontal visibility algorithm, a geometrically simpler and analytically solvable version of our former algorithm, focusing on the mapping of random series (series of independent identically distributed random variables). After presenting some properties of the algorithm, we present exact results on the topological properties of graphs associated with random series, namely, the degree distribution, the clustering coefficient, and the mean path length. We show that the horizontal visibility algorithm stands as a simple method to discriminate randomness in time series since any random series maps to a graph with an exponential degree distribution of the shape P(k)=(1/3)(2/3)(k-2), independent of the probability distribution from which the series was generated. Accordingly, visibility graphs with other P(k) are related to nonrandom series. Numerical simulations confirm the accuracy of the theorems for finite series. In a second part, we show that the method is able to distinguish chaotic series from independent and identically distributed (i.i.d.) theory, studying the following situations: (i) noise-free low-dimensional chaotic series, (ii) low-dimensional noisy chaotic series, even in the presence of large amounts of noise, and (iii) high-dimensional chaotic series (coupled map lattice), without needs for additional techniques such as surrogate data or noise reduction methods. Finally, heuristic arguments are given to explain the topological properties of chaotic series, and several sequences that are conjectured to be random are analyzed.
可见性算法最近被引入作为时间序列与复杂网络之间的一种映射。该过程使我们能够应用复杂网络理论的方法来表征时间序列。在这项工作中，我们提出了水平可见性算法，它是我们之前算法在几何上更简单且在解析上可求解的版本，重点关注随机序列（独立同分布随机变量序列）的映射。在介绍了该算法的一些性质之后，我们给出了与随机序列相关的图的拓扑性质的精确结果，即度分布、聚类系数和平均路径长度。我们表明，水平可见性算法是一种区分时间序列随机性的简单方法，因为任何随机序列都映射到一个具有形状为P(k)=(1/3)(2/3)(k - 2)的指数度分布的图，与生成该序列所依据的概率分布无关。因此，具有其他P(k)的可见性图与非随机序列相关。数值模拟证实了有限序列定理的准确性。在第二部分中，我们表明该方法能够区分混沌序列与独立同分布（i.i.d.）理论，研究以下情况：（i）无噪声低维混沌序列，（ii）低维有噪声混沌序列，即使存在大量噪声时，以及（iii）高维混沌序列（耦合映射格子），无需诸如替代数据或降噪方法等额外技术。最后，给出了启发式论证来解释混沌序列的拓扑性质，并分析了几个被推测为随机的序列。
