收敛交叉映射算法

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Convergent cross mapping (CCM) is a statistical test for a cause-and-effect relationship between two time series variables that, like the Granger causality test, seeks to resolve the problem that correlation does not imply causation.[1][2] While Granger causality is best suited for purely stochastic systems where the influences of the causal variables are separable (independent of each other), CCM is based on the theory of dynamical systems and can be applied to systems where causal variables have synergistic effects. The fundamental idea of this test was first published by Cenys et al. in 1991[3] and used in a series of statistical approaches (see for example,[4][5][6]). It was then further elaborated in 2012 by the lab of George Sugihara of the Scripps Institution of Oceanography.[7]

Convergent cross mapping (CCM) is a statistical test for a cause-and-effect relationship between two time series variables that, like the Granger causality test, seeks to resolve the problem that correlation does not imply causation. While Granger causality is best suited for purely stochastic systems where the influences of the causal variables are separable (independent of each other), CCM is based on the theory of dynamical systems and can be applied to systems where causal variables have synergistic effects. The fundamental idea of this test was first published by Cenys et al. in 1991 and used in a series of statistical approaches (see for example,). It was then further elaborated in 2012 by the lab of George Sugihara of the Scripps Institution of Oceanography.

收敛交叉映射(CCM,Convergent cross mapping )是一种统计检验方法,主要用于检验两个时间序列变量之间的因果关系,就像格兰杰因果关系一样,它试图解决相关不蕴含因果的问题。虽然格兰杰因果关系最适合于纯随机系统,因果变量的影响是可分离的(相互独立) ,CCM 是基于动态系统理论,可以应用于有协同效应因果变量的系统。这个检验的基本思想最早由 Cenys 等人发表。在1991年被用于一系列的统计方法(例如,见。2012年,斯克里普斯海洋研究所的 George Sugihara 的实验室对此进行了进一步的研究。


Theory

Convergent cross mapping is based on Takens' embedding theorem, which states that generically the attractor manifold of a dynamical system can be reconstructed from a single observation variable of the system, [math]\displaystyle{ X }[/math]. This reconstructed or shadow attractor [math]\displaystyle{ M_X }[/math] is diffeomorphic (has a one-to-one mapping) to the true manifold, [math]\displaystyle{ M }[/math]. Consequently, if two variables X and Y belong to the same dynamics system, the shadow manifolds [math]\displaystyle{ M_X }[/math] and [math]\displaystyle{ M_Y }[/math] will also be diffeomorphic. Time points that are nearby on the manifold [math]\displaystyle{ M_X }[/math] will also be nearby on [math]\displaystyle{ M_Y }[/math]. Therefore, the current state of variable [math]\displaystyle{ Y }[/math] can be predicted based on [math]\displaystyle{ M_X }[/math].

Convergent cross mapping is based on Takens' embedding theorem, which states that generically the attractor manifold of a dynamical system can be reconstructed from a single observation variable of the system, [math]\displaystyle{ X }[/math]. This reconstructed or shadow attractor [math]\displaystyle{ M_X }[/math] is diffeomorphic (has a one-to-one mapping) to the true manifold, [math]\displaystyle{ M }[/math]. Consequently, if two variables X and Y belong to the same dynamics system, the shadow manifolds [math]\displaystyle{ M_X }[/math] and [math]\displaystyle{ M_Y }[/math] will also be diffeomorphic. Time points that are nearby on the manifold [math]\displaystyle{ M_X }[/math] will also be nearby on [math]\displaystyle{ M_Y }[/math]. Therefore, the current state of variable [math]\displaystyle{ Y }[/math] can be predicted based on [math]\displaystyle{ M_X }[/math].

收敛交叉映射是基于 Takens 的嵌入定理,该定理指出,一般来说,一个动力系统的吸引子流形可以从系统的一个观测变量中重构。这个重构或阴影吸引子[math]\displaystyle{ M_X }[/math]是微分同构于[math]\displaystyle{ M }[/math](有一个一对一的映射)。因此,如果两个变量 x 和 y 属于同一个动力系统,那么流形[math]\displaystyle{ M_X }[/math][math]\displaystyle{ M_Y }[/math]也是微分同胚的。在流形[math]\displaystyle{ M_X }[/math].附近的时间点也会在[math]\displaystyle{ M_Y }[/math]附近。因此,可以基于[math]\displaystyle{ M_X }[/math]来预测变量[math]\displaystyle{ M_X }[/math]的当前状态。


Cross mapping need not be symmetric. If [math]\displaystyle{ X }[/math] forces [math]\displaystyle{ Y }[/math] unidirectionally, variable [math]\displaystyle{ Y }[/math] will contain information about [math]\displaystyle{ X }[/math], but not vice versa. Consequently, the state of [math]\displaystyle{ X }[/math] can be predicted from [math]\displaystyle{ M_Y }[/math], but [math]\displaystyle{ Y }[/math] will not be predictable from [math]\displaystyle{ M_X }[/math].

Cross mapping need not be symmetric. If [math]\displaystyle{ X }[/math] forces [math]\displaystyle{ Y }[/math] unidirectionally, variable [math]\displaystyle{ Y }[/math] will contain information about [math]\displaystyle{ X }[/math], but not vice versa. Consequently, the state of [math]\displaystyle{ X }[/math] can be predicted from [math]\displaystyle{ M_Y }[/math], but [math]\displaystyle{ Y }[/math] will not be predictable from [math]\displaystyle{ M_X }[/math].

交叉映射不一定是对称的。如果[math]\displaystyle{ M_X }[/math]单向的影响[math]\displaystyle{ M_Y }[/math],变量[math]\displaystyle{ M_Y }[/math]将包含[math]\displaystyle{ M_X }[/math]的信息,但反之则不然。因此,[math]\displaystyle{ M_X }[/math]的状态可以通过[math]\displaystyle{ M_Y }[/math]来预测,但是[math]\displaystyle{ M_Y }[/math]不能通过[math]\displaystyle{ M_X }[/math]来预测。


Algorithm

The basic steps of the convergent cross mapping test according to[8]

The basic steps of the convergent cross mapping test according to

根据交叉映射理论,给出了收敛交叉映射检验的基本步骤


  1. Create the shadow manifold for [math]\displaystyle{ X }[/math], called [math]\displaystyle{ M_X }[/math]
Create the shadow manifold for [math]\displaystyle{ X }[/math], called [math]\displaystyle{ M_X }[/math]

[math]\displaystyle{ M_X }[/math]创建阴影流形,称为[math]\displaystyle{ M_X }[/math]


  1. Find the nearest neighbors to a point in the shadow manifold at time t
Find the nearest neighbors to a point in the shadow manifold at time t

在t时刻在阴影流形中找到与此点最近的邻居

  1. Create weights using the nearest neighbors
Create weights using the nearest neighbors 

给最近的邻居创建权重

  1. Estimate Y using the weights; (this estimate is called [math]\displaystyle{ Y }[/math] | [math]\displaystyle{ M_X }[/math] )
Estimate Y using the weights; (this estimate is called [math]\displaystyle{ Y }[/math] | [math]\displaystyle{ M_X }[/math] )

使用权重估计 y; (这个估计被称为 [math]\displaystyle{ Y }[/math] | [math]\displaystyle{ M_X }[/math])

  1. Compute the correlation between [math]\displaystyle{ Y }[/math] and [math]\displaystyle{ Y }[/math] | [math]\displaystyle{ M_X }[/math]
Compute the correlation between [math]\displaystyle{ Y }[/math] and [math]\displaystyle{ Y }[/math] | [math]\displaystyle{ M_X }[/math]

计算 [math]\displaystyle{ Y }[/math][math]\displaystyle{ Y }[/math] | [math]\displaystyle{ M_X }[/math]之间的相关性

CCM算法步骤:

算法步骤.png

具体算法介绍:

参数说明:

M: 嵌入在 e 维状态空间(d ≤ e)中的 d 维流形 

{x}: M投影于x上产生的序列

{y}: M投影于y上产生的序列

L: 时间序列长度

E: 重构流形的维度

𝜏: 采样间隔

算法计算:

具体步骤.png

VS Granger causality test

CCM is not in competition with the many effective methods that use GC; rather, it is specifically aimed at a class of system not covered by GC

CCM不是和格兰杰因果检验竞争,而是针对格兰杰检验不包含的一类系统。如:含有弱耦合、中等耦合强度的系统、不可分离变量的系统、含有共同驱动变量影响的系统

Applications


References

  1. 1.0 1.1 Sugihara, George; et al. (26 October 2012). "Detecting Causality in Complex Ecosystems" (PDF). Science. 338 (6106): 496–500. Bibcode:2012Sci...338..496S. doi:10.1126/science.1227079. PMID 22997134. Retrieved 5 July 2013.
  2. "Cause test could end up in court". New Scientist. 28 September 2012. Opinion. Retrieved 5 July 2013.
  3. Čenys, A.; Lasiene, G.; Pyragas, K. (1991). "Estimation of interrelation between chaotic observables". Physica D: Nonlinear Phenomena. Elsevier BV. 52 (2–3): 332–337. doi:10.1016/0167-2789(91)90130-2. ISSN 0167-2789.
  4. Schiff, Steven J.; So, Paul; Chang, Taeun; Burke, Robert E.; Sauer, Tim (1996-12-01). "Detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble". Physical Review E. American Physical Society (APS). 54 (6): 6708–6724. doi:10.1103/physreve.54.6708. ISSN 1063-651X.
  5. Arnhold, J.; Grassberger, P.; Lehnertz, K.; Elger, C.E. (1999). "A robust method for detecting interdependences: application to intracranially recorded EEG". Physica D: Nonlinear Phenomena. Elsevier BV. 134 (4): 419–430. doi:10.1016/s0167-2789(99)00140-2. ISSN 0167-2789.
  6. Chicharro, Daniel; Andrzejak, Ralph G. (2009-08-27). "Reliable detection of directional couplings using rank statistics". Physical Review E. American Physical Society (APS). 80 (2): 026217. doi:10.1103/physreve.80.026217. hdl:10230/16204. ISSN 1539-3755.
  7. Michael Marshall in New Scientist magazine 2884: Causality test could help preserve the natural world, 28 September 2012
  8. McCracken, James (2014). "Convergent cross-mapping and pairwise asymmetric inference". Physical Review E. 90 (6): 062903. arXiv:1407.5696. Bibcode:2014PhRvE..90f2903M. doi:10.1103/PhysRevE.90.062903. PMID 25615160.


参考资料 /


External links

Animations:

Animations:

动画:

Category:Time series statistical tests

类别: 时间序列统计检验


This page was moved from wikipedia:en:Convergent cross mapping. Its edit history can be viewed at 收敛交叉映射算法/edithistory