空中交通流非线性分形特征

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. 

Nonlinear Fractal Characteristics of Air Traffic Flow

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.