Constraint-based and noise-based methods have been proposed to discover summary causal graphs from observational time series under strong assumptions which can be violated or impossible to verify in real applications. Recently, a hybrid method (Assaad et al, 2021) that combines these two approaches, proved to be robust to assumption violation. However, this method assumes that the summary causal graph is acyclic, but cycles are common in many applications. For example, in ecological communities, there may be cyclic relationships between predator and prey populations, creating feedback loops. Therefore, this paper presents two new frameworks for hybrids of constraint-based and noise-based methods that can discover summary causal graphs that may or may not contain cycles. For each framework, we provide two hybrid algorithms which are experimentally tested on simulated data, realistic ecological data, and real data from various applications. Experiments show that our hybrid approaches are robust and yield good results over most datasets.

翻译：约束基与噪声基方法已被提出，用于在强假设下从观测时间序列中发现摘要因果图，但这些假设在实际应用中可能被违反或难以验证。最近，一种结合这两种方法的混合方法（Assaad 等，2021）被证明对假设违反具有鲁棒性。然而，该方法假设摘要因果图是无环的，而环在许多应用中很常见。例如，在生态群落中，捕食者和猎物种群之间可能存在循环关系，从而形成反馈回路。因此，本文提出了两种新的约束基与噪声基混合方法框架，能够发现可能包含或不包含环的摘要因果图。针对每个框架，我们提供了两种混合算法，并在模拟数据、真实生态数据以及来自不同应用的真实数据上进行了实验测试。实验表明，我们的混合方法具有鲁棒性，并在大多数数据集上取得了良好结果。