论文标题
关于因果关系与相关性的注释
A note on causation versus correlation
论文作者
论文摘要
最近，已经表明，可以从严格而定量的意义上推断两个时间序列之间的因果关系和信息流，此外，可以将所产生的因果关系标准化。在线性极限中，随后的推论意味着相关性，而相关性并不意味着因果关系。现在假设有一个事件$ a $采用谐波表格（正弦/余弦），它通过某个过程产生另一个事件$ b $，因此$ b $总是落后于$π/2 $的阶段。在这里，显然可以看到因果关系，而通过计算，相关性为零。这种看似矛盾的根源在于，谐波系统总是在庞加莱部分留下一个点。它不添加信息。也就是说，尽管绝对信息从$ a $流向$ b $为零，即$ t_ {a \ to b} = 0 $，但总信息增加的总信息增加也为零，因此标准化的$ t_ {a \ to b} $，表示为$τ_{a \ a \ a \ a \ y the $ $ $ $ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00士表。通过用一些噪声稍微扰动系统，求解随机微分方程，并让扰动到零，可以证明$τ_{a \ to b} $接近100 \％，就像人们预期的那样。
Recently, it has been shown that the causality and information flow between two time series can be inferred in a rigorous and quantitative sense, and, besides, the resulting causality can be normalized. A corollary that follows is, in the linear limit, causation implies correlation, while correlation does not imply causation. Now suppose there is an event $A$ taking a harmonic form (sine/cosine), and it generates through some process another event $B$ so that $B$ always lags $A$ by a phase of $π/2$. Here the causality is obviously seen, while by computation the correlation is, however, zero. This seemingly contradiction is rooted in the fact that a harmonic system always leaves a single point on the Poincaré section; it does not add information. That is to say, though the absolute information flow from $A$ to $B$ is zero, i.e., $T_{A\to B}=0$, the total information increase of $B$ is also zero, so the normalized $T_{A\to B}$, denoted as $τ_{A\to B}$, takes the form of $\frac 0 0$. By slightly perturbating the system with some noise, solving a stochastic differential equation, and letting the perturbation go to zero, it can be shown that $τ_{A\to B}$ approaches 100\%, just as one would have expected.
