Comparative regression analysis to degree distributions of visibility graph
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摘要: 可视图(Visibility Graph,VG)算法为研究时间序列的动力学特性提供了复杂网络的思想.网络的度分布反映了时间序列的动力学特征.通过自回归随机过程和分数布朗运动两种不同数据,分别构建可视图.对比结果表明,在自回归随机过程中,度分布可以用指数函数刻画;而在分数布朗运动中,度分布用幂律函数刻画更为合适.这一结论不但适用于VG算法,同时也适用于水平可视图(Horizontal Visibility Graph,HVG)算法.Abstract: Visibility graph has provided much insight to study the dynamics of time series from the perspective complex network. We construct visibility graphs for time series from both auto-regressive stochastic and fractional Brownian motions. Our results suggest that degree distributions of the resulted complex networks of auto-regressive processes are characterized by exponential forms, while that of fractional Brownian motions obey power-law forms. Our conclusions hold for both the traditional visibility graph and its variant horizontal visibility graph.
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图 1 图 1(a)-图 1(d) 半对数坐标下的度累积分布的指数拟合, 虚线表示率参数 $\lambda$ ; 图 1(e)-图 1(h) 双对数坐标下的度累积分布的幂率拟合, 虚线表示幂律指数 $\gamma$
Fig. 1 (a)-(d) Exponential fitting to the cumulative degree distributions in semi-log plot, and dashed lines are slopes for $\lambda $ ; (e)-(h) Similar to (a)-(d) but power law forms in double log plots
表 1 AR (1) 在VG算法下指数与幂率拟合比较
Tab. 1 Comparison between exponential and power law scaling factors of VG graphs of AR (1)
φ=0 φ=0.3 φ=0.9 φ=-0.5 λ γ λ γ λ γ λ γ 数值 0.13 4.21 0.12 3.98 0.10 4.09 0.12 4.22 R2 0.998 0.957 0.999 0.919 0.986 0.953 0.998 0.959 SSE 0.655 2.575 0.123 9.074 3.829 5.290 0.440 2.458 φ=0 φ=0.3 φ=0.9 φ=-0.5 λ γ λ γ λ γ λ γ 数值 0.45 7.61 0.50 6.54 0.76 8.96 0.43 7.37 R2 0.995 0.947 0.993 0.911 0.994 0.955 0.996 0.952 SSE 1.515 1.599 1.839 3.273 0.951 1.227 0.6017 1.199 表 3 FBM在VG算法下指数与幂率拟合比较
Tab. 3 Comparison between exponential and power law scaling factors VG graphs of FBM
H=0.3 H=0.5 H=0.7 H=-0.9 λ γ λ γ λ γ λ γ 数值 0.07 2.47 0.03 1.97 0.01 1.56 0.004 1.12 R2 0.989 0.990 0.965 0.995 0.939 0.990 0.962 0.986 SSE 0.527 0.121 6.826 0.105 15.520 0.429 11.770 0.452 -
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