分形无标度区的一种自动识别方法

摘要

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 correlation-dimension of Lorenz attractors based on this method is very close to the theoretical value. The numerical results show that the scaling region can be identified accurately and automatically by this method.%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.