Nonlinear Fractal Characteristics of Air Traffic Flow
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摘要: 为了给空中交通流建模与预测提供科学依据，建立4种时间尺度的时间序列，采用替代数据法进行了非线性检验，确定5 min尺度的时间序列作为后续研究对象；应用小波分解方法研究了时间序列的自相似性特征，通过R/S方法计算了全局和局部Hurst指数，研究了时间序列的长相关性特征；应用关联积分的二阶差分法计算了时间序列的无标度性区间；采用多重分形谱方法研究了时间序列的多重分形特征，应用Grassbeger-Procaccia （GP）算法计算了时间序列的关联维数. 研究结果表明：5 min时间尺度的时间序列具有非线性因素的概率为99.2%，其他3种时间序列的非线性尚不明确；从小波分解图可定性观察出时间序列具有较强的自相似性；全局Hurst指数为0.756 5，局部Hurst指数均大于0.5，表明时间序列具有长相关性；关联积分二阶差分法能够有效识别出无标度区间，说明时间序列具有无标度性，不同嵌入维数对应的无标度区间是不同的；多重分形谱图呈现钟型，说明具有多重分形特征；关联维数为6.89，表明至少需要7个变量才能清晰描述本时间序列对应的空中交通流.Abstract: To provide scientific evidence for traffic flow modeling and prediction, the nonlinear characteristics of air traffic flow were studied based on fractal. First, 4 time series were constructed, and their nonlinearities were tested by the surrogate data method, and the 5-minute-scale time series was determined as the subsequent research object. Then, the wavelet decomposition method was used to study the self-similarity of time series. The global and local Hurst exponents were calculated by R/S method to study the long-range correlation characteristics. Next, scale-free ranges of time series were calculated using second-order difference of correlation integral. Then, the multi-fractal characteristics of time series were studied by multi-fractal spectrum method. Finally, the correlation dimensions of time series were calculated by Grassbeger-Procaccia method. The results show that the probability of the nonlinearity of 5-min-scale time series is 99.2%, and the nonlinearities of the other 3 time series are not clear. It is qualitatively observed that the time series has strong self-similarity. The global Hurst exponent is 0.756 5, and the local Hurst exponents are all more than 0.5, which indicate that the time series has a long-range correlation. The second-order difference of the correlation integral can effectively identify scale-free ranges, which shows that the time series has scale-free property and the scale-free ranges are different corresponding to different embedding dimensions. The bell-shaped multi-fractal spectrum shows the time series has multi-fractal characteristics. The correlation dimension is 6.89, indicating that at least 7 variables are needed to clearly describe the corresponding air traffic flow.
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Key words:
- air traffic management /
- air traffic flow /
- nonlinear /
- fractal /
- time series
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图 1 不同时间尺度的空中交通流量时间序列
Figure 1. Air traffic flow time series based on different time scales
图 5 双对数关联积分和二阶差分曲线
Figure 5. Curves of double-logarithmic correlation integral and second-order difference
图 8 关联维数随嵌入维数的变化曲线
Figure 8. Curve of the correlation dimension D2 with the increasing embedding dimension m
表 1 时间序列非线性检验结果
Table 1. Nonlinear test results of time series
时间尺度/
min非线性判定
次数/次非线性
概率非线性检验
结果30 217 0.217 小概率 15 354 0.354 小概率 10 742 0.742 大概率 5 992 0.992 确定 
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表 2 不同嵌入维度下的无标度区间
Table 2. Non-scale range under different embedding dimensions
嵌入维数 无标度区间 嵌入维数 无标度区间 2 [1.054 4，1.305 7] 9 [1.252 4，1.763 2] 3 [1.542 7，1.725 0] 10 [1.429 0，1.680 3] 4 [1.115 8，1.521 3] 11 [1.968 8，2.123 0] 5 [1.171 7，1.577 2] 12 [1.990 3，2.144 5] 6 [1.363 3，1.614 6] 13 [1.937 4，2.104 5] 7 [1.645 7，1.846 4] 14 [1.949 8，2.191 0] 8 [1.646 7，1.847 3] 15 [1.871 8，2.054 1]
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