Department of Computer Science, University of Colorado, Boulder, Colorado 80309-0430, USA and Santa Fe Institute, Santa Fe, New Mexico 87501, USA.
Max Planck Institute for the Physics of Complex Systems, Noethnitzer Str. 38 D, 01187 Dresden, Germany.
Chaos. 2015 Sep;25(9):097610. doi: 10.1063/1.4917289.
In 1980 and 1981, two pioneering papers laid the foundation for what became known as nonlinear time-series analysis: the analysis of observed data-typically univariate-via dynamical systems theory. Based on the concept of state-space reconstruction, this set of methods allows us to compute characteristic quantities such as Lyapunov exponents and fractal dimensions, to predict the future course of the time series, and even to reconstruct the equations of motion in some cases. In practice, however, there are a number of issues that restrict the power of this approach: whether the signal accurately and thoroughly samples the dynamics, for instance, and whether it contains noise. Moreover, the numerical algorithms that we use to instantiate these ideas are not perfect; they involve approximations, scale parameters, and finite-precision arithmetic, among other things. Even so, nonlinear time-series analysis has been used to great advantage on thousands of real and synthetic data sets from a wide variety of systems ranging from roulette wheels to lasers to the human heart. Even in cases where the data do not meet the mathematical or algorithmic requirements to assure full topological conjugacy, the results of nonlinear time-series analysis can be helpful in understanding, characterizing, and predicting dynamical systems.
在 1980 年和 1981 年，两篇开创性的论文为后来被称为非线性时间序列分析奠定了基础：通过动力系统理论分析观测数据——通常是单变量。基于状态空间重构的概念，这组方法使我们能够计算出 Lyapunov 指数和分形维数等特征量，预测时间序列的未来走势，甚至在某些情况下重建运动方程。然而，在实践中，有许多问题限制了这种方法的能力：例如，信号是否准确而全面地采样了动力学，以及它是否包含噪声。此外，我们用于实现这些想法的数值算法并不完美；它们涉及到近似、比例参数和有限精度的算术等。即便如此，非线性时间序列分析已经在从轮盘赌到激光到人类心脏等各种系统的数千个真实和合成数据集上得到了很好的应用。即使在数据不符合数学或算法要求以确保完全拓扑共轭的情况下，非线性时间序列分析的结果也有助于理解、描述和预测动力系统。
