Cyclostationary linear inverse models (CS-LIMs), generalized versions of the classical (stationary) LIM, are advanced data-driven techniques for extracting the first-order time-dependent dynamics and random forcing relevant information from complex non-linear stochastic processes. Though CS-LIMs lead to a breakthrough in climate sciences, their mathematical background and properties are worth further exploration. This study focuses on the mathematical perspective of CS-LIMs and introduces two variants: e-CS-LIM and l-CS-LIM. The former refines the original CS-LIM using the interval-wise linear Markov approximation, while the latter serves as an analytic inverse model for the linear periodic stochastic systems. Although relying on approximation, e-CS-LIM converges to l-CS-LIM under specific conditions and shows noise-robust performance. Numerical experiments demonstrate that each CS-LIM reveals the temporal structure of the system. The e-CS-LIM optimizes the original model for better dynamics performance, while l-CS-LIM excels in diffusion estimation due to reduced approximation reliance. Moreover, CS-LIMs are applied to real-world ENSO data, yielding a consistent result aligning with observations and current ENSO understanding.

翻译：循环平稳线性逆模型（CS-LIMs）是经典（平稳）线性逆模型的推广版本，作为先进的数据驱动技术，可用于从复杂的非线性随机过程中提取一阶时间依赖动力学及随机强迫相关信息。尽管CS-LIMs在气候科学领域带来了突破，其数学背景与性质仍值得进一步探究。本研究聚焦于CS-LIMs的数学视角，并引入两种变体：e-CS-LIM与l-CS-LIM。前者通过区间线性马尔可夫近似对原始CS-LIM进行改进，后者则为线性周期随机系统提供了一种解析逆模型。尽管依赖于近似，e-CS-LIM在特定条件下收敛于l-CS-LIM，并表现出噪声鲁棒性能。数值实验表明，每种CS-LIM均能揭示系统的时间结构。e-CS-LIM通过优化原始模型获得了更优的动力学性能，而l-CS-LIM因减少了对近似的依赖，在扩散估计方面表现卓越。此外，将CS-LIMs应用于实际ENSO数据，得到了与观测结果及当前ENSO认知相一致的有效结论。