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表征，有许多统计量，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，第一步是从观测到的时间序列中重建吸引子。为了解决上述问题，Takens（1981）的嵌入定理和Sauer等人（1991）的嵌入理论在理论论证上很重要。对于从观测到的单变量时间序列重构动力系统的实际情况，嵌入理论证明，如果重构吸引子的维数至少大于底层动力系统的盒计数维数的两倍，那么由时滞坐标进行的变换就是嵌入。虽然在理论情况下，这种重构中的时间延迟值是任意值，但是在分析实时时间序列的情况下，由于实时时间序列受到噪声的破坏，并且其长度和测量精度是有限的，因此设置一个合适的值是很重要的，特别是对于连续数据。本文提出了一种新的确定单变量时间序列中吸引子重构时滞的判据。我们的准则考虑了高阶相关函数，并找到了极值的收敛值。通过观察重构吸引子的形状和计算相空间连续性，对两个数值例子进行了分析。我们的方法可以作为一个很好的准则来设定适当的时间延迟来重建吸引子。

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.