Optimizing Time Delays for Reconstructing Attractors of Dynamical Systems from Time Series Signals

从基于非线性动力系统理论的时间序列分析的角度来看，确定性混沌的识别及其定量表征是非常重要的。对于确定性混沌的定量描述，有很多统计量，如分形维数、李雅普诺夫指数、度量熵等，为了估计这些非线性统计量，首先要从观测的时间序列中重构吸引子。为了解决这一问题，Takens（1981）的嵌入定理和Sauer等人的嵌入论被证明是有效的。对于从观测到的单变量时间序列重构动力系统的实际问题，嵌入理论证明了，如果重构吸引子的维数至少大于原动力系统盒维数的2倍，则时滞坐标变换是嵌入.虽然在理论上，这种重构方法中的时延值是任意的，但在分析真实的时间序列的情况下，由于时间序列的长度和测量精度都是有限的，因此确定一个合适的时延值是很重要的，特别是对于连续的时间序列，本文提出了一种新的时延判定准则，用于从单变量时间序列重构吸引子。我们的准则考虑高阶相关函数，并找到极值的收敛值。为了验证该方法的有效性，我们通过观察重构吸引子的形状和计算相空间的连续性，分析了两个数值例子，结果表明，该方法可以作为在重构吸引子上设置合适的时间延迟的一个很好的判据.

Identifying deterministic chaos and its quantitative characterization are very important from the viewpoint of time series analysis based on nonlinear dynamical systems theory. For quantitative characterization of deterministic chaos, there are many statistics, for example, the fractal dimensions, the Lyapunov exponents, the metric entropies and so on.In order to estimate these nonlinear statistics, the first step is the reconstructior of attractors from observed time series'. In order to accomplish the above issue, the embedding theorems by Takens (1981) and Embedology theory by Sauer et al.(1991) are lmportant from theoretical justification.For practical reconstruction of dynamical systems from an observed single variable time series, the theory of embedding proved that the transformation by the time delay coordinates is an embedding if the dimension of reconstructing attractors is at least larger than twice of the box counting dimension of an underlying dynamical system. Although the time delay value in this reconstrution is an arbitrary value under the theoretical situation, in the case of analyzing real time series, which is corrupted by noise and whose length and measurement precision are finite, it is important to set an appropriate value, particularly for continuous data.We propose a novel criterion of deciding time delays for reconstructing attractors from a single variable time series. Our criterion considers higher order correlation functions, and finds the convergence values of the extrema. In order to see how our method works well, we analyze two numerical examples by observing the shapes of reconstructed attractors and calculating phase space continuities.Our method can be a good criterion to set appropriate time delays on reconstructing attractors as a result.