Learning causal effects from data is a fundamental and well-studied problem across science, especially when the cause-effect relationship is static in nature. However, causal effect is less explored when there are dynamical dependencies, i.e., when dependencies exist between entities across time. Identifying dynamic causal effects from time-series observations is computationally expensive when compared to the static scenario. We demonstrate that the computational complexity of recovering the causation structure for the vector auto-regressive (VAR) model is $O(Tn^3N^2)$, where $n$ is the number of nodes, $T$ is the number of samples, and $N$ is the largest time-lag in the dependency between entities. We report a method, with a reduced complexity of $O(Tn^3 \log N)$, to recover the causation structure to obtain frequency-domain (FD) representations of time-series. Since FFT accumulates all the time dependencies on every frequency, causal inference can be performed efficiently by considering the state variables as random variables at any given frequency. We additionally show that, for systems with interactions that are LTI, do-calculus machinery can be realized in the FD resulting in versions of the classical single-door (with cycles), front and backdoor criteria. We demonstrate, for a large class of problems, graph reconstruction using multivariate Wiener projections results in a significant computational advantage with $O(n)$ complexity over reconstruction algorithms such as the PC algorithm which has $O(n^q)$ complexity, where $q$ is the maximum neighborhood size. This advantage accrues due to some remarkable properties of the phase response of the frequency-dependent Wiener coefficients which is not present in any time-domain approach.

翻译：从数据中学习因果效应是科学领域中一个基础且被广泛研究的问题，尤其当因果-效应关系本质上是静态时。然而，当存在动态依赖（即实体间存在跨时间依赖）时，因果效应的研究较少。与静态情形相比，从时间序列观测中识别动态因果效应的计算成本更高。我们证明，对于向量自回归（VAR）模型，恢复因果结构的计算复杂度为$O(Tn^3N^2)$，其中$n$为节点数，$T$为样本数，$N$为实体间依赖中的最大时间滞后。我们报告了一种复杂度降低至$O(Tn^3 \log N)$的方法，用于恢复因果结构，以获取时间序列的频域（FD）表示。由于FFT将所有时间依赖累积到每个频率上，通过将状态变量视为任意给定频率下的随机变量，可以高效地进行因果推断。我们还表明，对于具有线性时不变（LTI）交互的系统，do-演算机制可以在频域中实现，从而得到经典的单门（含循环）、前门和后门准则的频域版本。我们证明，对于一大类问题，使用多元Wiener投影进行图重构在计算上具有显著优势，其复杂度为$O(n)$，而PC算法等重构算法的复杂度为$O(n^q)$，其中$q$为最大邻域大小。这一优势源于频率依赖Wiener系数的相位响应的一些显著特性，这些特性在任何时域方法中都不存在。