分形无标度区的一种自动识别方法

Automatic identification method of fractal scaling region

GP算法求分形关联维数时,双对数曲线的线性区间(无标度区)的识别十分关键。经典的GP算法中无标度区的识别主要依靠人工经验完成,同一条曲线,不同的人可能得到不同的无标度区,从而导致估算的关联维数存在较大差别。根据无标度区范围内的双对数曲线近似为一条直线段,其二阶导数应在0附近上下微幅波动的特点,提出了一种由计算机对无标度区进行自动识别的方法。该方法物理意义清晰,便于在计算机上编程实现。用Lorenz方程X轴的数据对方法进行了验证,计算结果表明,提出的方法可以有效地识别无标度区。

The identification of the linear segment in double logarithmic curves （or log-log curves）, also known as scaling region （or non-scale range in some papers）, is important in Grassberger-Procaccia （GP） algorithm. The scaling region is normally determined by experience in classical GP algorithm, which may lead the values of the correlation-dimension to be estimated different from person to person. In GP algorithm, the second-order derivative of log-log curves within the scaling region should be zero or slightly fluctuate near zero because the log-log curves are nearly straight lines in that scaling region. Based on this character of the log-log curves, a new method with clear physical meaning is presented to automatically identify the fractal scaling region. The process of this method is simple and can be realized easily on computer. A time series data of the Lorenz strange attractor are used to test the method. The estimated correla- tion-dimension of Lorenz attractors based on this method is very close to the theoretical value. The numerical results show that the scal- ing region can be identified accurately and automatically by this method.