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Taken theorem gives us a one-to-one mapping between the original manifold and reconstructed shadow manifold. Here we will explain how this important aspect of attractor reconstruction can be used to determine if two time series variables belong to the same dynamic system and are thus causally related. This particular reconstruction is based on lags of variable x if we now do the same for variable y we find something similar. Here, we see the original manifold M as well as the shadow manifold Mx and My created from lags of x and y respectively.

Because both Mx and My map one to the original manifold M, they also map one to each other. This implies that the points that are nearby on the manifold My correspond to points that are also nearby on Mx. We can demonstrate this principle by finding the nearest neighbors in My and using their time indices to find the corresponding points in Mx, these points will be nearest neighbors on Mx only if x and y are causally related. Thus, we can use nearby points. em... Why? To identify nearby points on Mx this allows us to use the historical record of y to estimate the states of x and vice versa, a technique we call cross mapping.

With longer time series, the reconstructed manifold are denser, nearest neighbors are closer, and the cross map estimates increase in precision. We call this phenomenon convergent cross mapping(CCM), and use this convergence as a practical criterion for detecting causation.

翻译

Taken定理给出了原始流形（manifold）与重构阴影流形之间的一对一映射。这里我们将解释吸引子重构的这一重要方面如何被用来确定两个时间序列变量是否属于同一个动态系统以及是否因果相关的。这个特殊的重建是基于变量x的滞后，如果我们现在对变量y做同样的事情，我们会发现一些相似的东西。在这里我们看到原始流形M，以及阴影流形Mx和My分别从x和y的滞后创建。

因为Mx和My都将一个映射到原始流形M，它们也是互相映射。这意味着流形My上的邻近点对应于Mx上的邻近点。我们可以通过找到My上的最近邻来证明这一原理，并使用它们的时间指数来找到Mx上的对应点。只有当x和y是因果相关的时，这些点才是Mx上的最近邻。因此，我们可以利用电磁波附近的点。为什么？为了识别Mx上的附近点，这允许我们使用y的历史记录来估计x的状态，反之亦然，我们称之为交叉映射（cross mapping）。

时间序列越长，重建的流形越密集，最近邻越近，交叉映射估计精度越高。我们称这种现象为收敛交叉映射（convergent cross mapping），并将这种收敛性用作检测因果关系的实用准则。