Wang Yan, Weng Tongfeng, Deng Shiguo, Gu Changgui, Yang Huijie
Business School, University of Shanghai for Science and Technology, Shanghai 200093, People's Republic of China.
Chaos. 2019 Feb;29(2):023109. doi: 10.1063/1.5074155.
Recent years have witnessed special attention on complex network based time series analysis. To extract evolutionary behaviors of a complex system, an interesting strategy is to separate the time series into successive segments, map them further to graphlets as representatives of states, and extract from the state (graphlet) chain transition properties, called graphlet based time series analysis. Generally speaking, properties of time series depend on the time scale. In reality, a time series consists of records that are sampled usually with a specific frequency. A natural question is how the evolutionary behaviors obtained with the graphlet approach depend on the sampling frequency? In the present paper, a new concept called the sampling frequency dependent visibility graphlet is proposed to answer this problem. The key idea is to extract a new set of series in which the successive elements have a specified delay and obtain the state transition network with the graphlet based approach. The dependence of the state transition network on the sampling period (delay) can show us the characteristics of the time series at different time scales. Detailed calculations are conducted with time series produced by the fractional Brownian motion, logistic map and Rössler system, and the empirical sentence length series for the famous Chinese novel entitled A Story of the Stone. It is found that the transition networks for fractional Brownian motions with different Hurst exponents all share a backbone pattern. The linkage strengths in the backbones for the motions with different Hurst exponents have small but distinguishable differences in quantity. The pattern also occurs in the sentence length series; however, the linkage strengths in the pattern have significant differences with that for the fractional Brownian motions. For the period-eight trajectory generated with the logistic map, there appear three different patterns corresponding to the conditions of the sampling period being odd/even-fold of eight or not both. For the chaotic trajectory of the logistic map, the backbone pattern of the transition network for sampling 1 saturates rapidly to a new structure when the sampling period is larger than 2. For the chaotic trajectory of the Rössler system, the backbone structure of the transition network is initially formed with two self-loops, the linkage strengths of which decrease monotonically with the increase of the sampling period. When the sampling period reaches 9, a new large loop appears. The pattern saturates to a complex structure when the sampling period is larger than 11. Hence, the new concept can tell us new information on the trajectories. It can be extended to analyze other series produced by brains, stock markets, and so on.
近年来，基于复杂网络的时间序列分析受到了特别关注。为了提取复杂系统的演化行为，一种有趣的策略是将时间序列分割成连续的片段，进一步将它们映射为作为状态代表的图元，并从状态（图元）链转移特性中提取，这被称为基于图元的时间序列分析。一般来说，时间序列的特性取决于时间尺度。在现实中，时间序列由通常以特定频率采样的记录组成。一个自然的问题是，通过图元方法获得的演化行为如何依赖于采样频率？在本文中，提出了一个名为采样频率相关可见性图元的新概念来回答这个问题。关键思想是提取一组新的序列，其中连续元素具有指定的延迟，并使用基于图元的方法获得状态转移网络。状态转移网络对采样周期（延迟）的依赖性可以向我们展示不同时间尺度下时间序列的特征。使用分数布朗运动、逻辑斯谛映射和罗斯勒系统生成的时间序列，以及著名中国小说《红楼梦》的经验句子长度序列进行了详细计算。发现具有不同赫斯特指数的分数布朗运动的转移网络都共享一种骨干模式。不同赫斯特指数运动的骨干中的连接强度在数量上有小但可区分的差异。这种模式也出现在句子长度序列中；然而，该模式中的连接强度与分数布朗运动的连接强度有显著差异。对于由逻辑斯谛映射生成的周期为八的轨迹，根据采样周期是八的奇数/偶数倍或两者都不是的情况，出现三种不同的模式。对于逻辑斯谛映射的混沌轨迹，当采样周期大于2时，采样1的转移网络的骨干模式迅速饱和到一种新结构。对于罗斯勒系统的混沌轨迹，转移网络的骨干结构最初由两个自环形成，其连接强度随着采样周期的增加而单调下降。当采样周期达到9时，出现一个新的大环。当采样周期大于11时，该模式饱和到一种复杂结构。因此，这个新概念可以告诉我们关于轨迹的新信息。它可以扩展到分析大脑、股票市场等产生的其他序列。
