发表机构
Saitama University（埼玉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出随机系统中延迟坐标重构的条件矩解释，通过有限矩闭合和库普曼算子理论论证，并应用于收敛交叉映射以诊断随机系统的因果关系。
AI 中文摘要
部分观测和延迟坐标重构根植于确定性动力系统理论，而许多系统具有内在随机性。我们提出了随机系统中延迟坐标重构的条件矩解释，其中延迟向量用于重构未来分布的条件矩，而非唯一的未来样本路径。两个互补的论据支持这一观点。首先，随机微分方程的概率密度遵循确定性的福克-普朗克方程，并在某些假设下，由无限确定性的矩层级表示。因此，有限矩闭合提示了类似塔肯斯的有限维近似。其次，基于库普曼算子理论的讨论阐明，在森-兹万齐格形式中，可观测量随时间演化在随机系统中产生条件期望。然后，在系数空间的森-兹万齐格方程中，正交的“噪声”项在随机情形下消失；这一结果与基于矩的论据一致。作为这一随机延迟重构观点的应用，我们重新审视了用于诊断某些因果关系的收敛交叉映射（CCM）。尽管基于嵌入定理的CCM通常不能应用于随机系统，但利用条件矩可以检验某些类型的因果关系。使用具有加性和乘性耦合机制的耦合逻辑斯蒂系统，我们讨论了因果关系如何嵌入随机系统。
英文摘要
Partial observation and delay-coordinate reconstruction are rooted in deterministic dynamical-systems theory, whereas there are many systems with intrinsic stochasticity. We propose a conditional-moment interpretation of delay-coordinate reconstruction for stochastic systems, in which a delay vector is used to reconstruct conditional moments of a future distribution rather than a unique future sample path. Two complementary arguments motivate this viewpoint. First, the probability density of a stochastic differential equation obeys a deterministic Fokker-Planck equation and, under certain assumptions, is represented by an infinite deterministic hierarchy of moments. Hence, a finite-moment closure suggests a Takens-like finite-dimensional approximation. Second, a discussion based on the Koopman operator theory clarifies that the time evolution of an observable in the Mori-Zwanzig formalism yields a conditional expectation in stochastic systems. Then, the orthogonal "noise" term in the coefficient-space Mori-Zwanzig equation vanishes in the stochastic cases; this result is consistent with the moment-based argument. As an application of this stochastic delay-reconstruction viewpoint, we revisit convergent cross mapping (CCM) for diagnosing certain causal relationships. Although CCM based on the embedding theorem cannot generally be applied to stochastic systems, it is possible to examine certain types of causal relationships by using conditional moments. Using coupled logistic systems with additive and multiplicative coupling mechanisms, we discuss how causal relationships are embedded in stochastic systems.
Comments20 pages, 9 figures
