Abstract
The prediction of nonlinear and non-stationary systems is a research topic of great scientific significance. In some recent work the convergent cross mapping (CCM) algorithm is used to detect the causal relationship between variables. In the CCM algorithm, the points close to each other in the phase space have similar trends and trajectories in time. Therefore, this method can be applied to the prediction of experimental researches of nonlinear and non-stationary systems. Therefore, in this paper the CCM algorithm is applied to the prediction of the Lorenz system and the actual climate time series, and the effects of different phase space reconstruction methods on the prediction skill are investigated. The preliminary results are as follows. 1) No matter whether the ideal Lorenz model or the actual climate series, of the three reconstruction phase space methods of univariate, multivariate, and multiview embedding method, the multiview embedding method is the best predictive skill, indicating that for a given length of time series, the more the information contained in the reconstructed phase space, the stronger its predictive ability is. 2) Adding the data of NAM (northern hemisphere annular mode) to the reconstructed phase space of SAT (surface air temperature) can improve the prediction effect on prediction of SAT. Using the univariable, multivariable, and multiview embedding method for implementing prediction, the characteristics of common information in the complex system are considered. On condition that the length of the time series is fixed, the complexity of the dynamic system can be used to increase the information of the system. Based on causality detection, through the extraction of quantitative information of data, a novel idea for the improvement of predictive skills in nonlinear and non-stationary systems can be obtained.
Highlights
algorithm to detect the causal relationship between variables
this method can try to be applied to the prediction experiment research
in this paper the Convergent Cross Mapping (CCM) algorithm was applied to the prediction of the Lorenz system
Summary
非线性、非平稳系统的预测是一个具有重要科学意义的研究课题。最近一些 工作已将收敛交叉映射算法(Convergent Cross Mapping ,CCM)用于检验变量 之间的因果关系,由于在 CCM 算法中,相空间中相互靠近的点在时间上具有相 似的发展趋势和运动轨迹,因此该方法可以尝试应用于非线性、非平稳系统的预 测试验研究中。鉴于此,本文将 CCM 算法分别应用于 Lorenz 系统和实际气候 时间序列的预测中,并检测不同相空间重构方法对预测效果的影响。主要结果如 下:(1)不论是理想 Lorenz 模型还是实际气候序列,对于单变量、多变量、和 多视角嵌入法三种重构相空间方法而言,多视角嵌入法对变量的预测效果最好, 表明对于给定长度的时间序列,重构相空间中包含的信息越多,其预测能力越强。 (2)将 NAM(Northern Hemisphere Annular Mode)加入到 SAT(Surface Air Temperature)的重构相空间中可以改善 SAT 的预测效果。在使用单变量、多变 量和多视角嵌入法进行预测时,利用复杂系统中变量中共有信息的特性,在时间 序列长度一定的情况下,可以利用动力系统的复杂性来增加系统内的信息。基于 因果检验的预测建模方法,通过挖掘数据中定量信息的提取,对非线性、非平稳 系统预测技巧的改进提供了一个新颖的思路。 此外,建立在因果关系上的驱动力分析,近年来也得到了长足的发展。Wiener 提出了一种因果关系的哲学概念,即因必须有助于改善果的预测[20]。在此概念基 础上,格兰杰提出了著名的格兰杰因果关系(Granger Causality)[21],然而此种方 法并不适用于复杂的非线性系统[22,23]。2012 年,生物学家 Sugihara 等[24]提出了基 于相空间重构和 Takens 定理的收敛交叉映射算法(Convergent Cross Mapping , CCM),该方法可以检验自然界中非线性动力系统中的因果关系,并已得到广泛 的应用[25,26,27]。例如,Zhang 等[28]利用 CCM 算法,探讨了北半球环状模(Northern Hemisphere Annular Mode ,NAM)与东北亚地区冬季地面气温(Surface Air Temperature,SAT)的信息传递,结果表明,二者存在单向因果关系,NAM 作为 驱动力因子影响东北亚地区冬季 SAT。 同时,由于在 CCM 算法中相空间中相互靠近的点在时间上具有相似的发展 趋势和运动轨迹,我们还可以尝试利用此方法对变量进行预测。因此,本文运用 CCM 方法建立预测模型,并以 Lorenz 系统以及东北亚地区冬季地面温度时间序 列为例,将 NAM 信号加入到 SAT 的重构相空间中,检验对 SAT 的预测效果。借 助因果检验的手段识别影响气候变化要素的外强迫因子,并将其应用在实际的气 候预测中,检验预测建模效果。论文章节安排如下:在第二节简要介绍收敛交叉 映射算法以及预测建模的思路,第三节给出理想序列的预测检验,第四节给出包 含 NAM 信息的东北亚地区冬季 SAT 时间序列的预测分析,最后给出结果和讨论。 为了讨论 CCM 算法在预测上的应用以及上述三种嵌入方法的异同,本文将 Lorenz 系统[34]作为理想模型来进行实验设计。Lorenz 模型是描述大气对流等问
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