On the cyclostationary linear inverse models: a mathematical insight and implication

Justin Lien ^1* Yan-Ning Kuo ² Hiroyasu Ando ¹ Shoichiro Kido ³

• ¹ Tohoku University, Sendai, Japan
• ² Cornell University, Ithaca, New York, United States
• ³ Japan Agency for Marine-Earth Science and Technology (JAMSTEC), Yokosuka, Kanagawa, Japan

Cyclostationary linear inverse models (CS-LIMs) are advanced data-driven techniques for extracting first-order time-dependent dynamics and random forcing information from cyclostationary observational data. This study focuses on the mathematical perspective of CS-LIMs and presents two variants: e-CS-LIM and l-CS-LIM. The e-CS-LIM, improved from the original CS-LIM, constructs the first-order dynamics through the interval-wise application of the stationary LIM (ST-LIM), capturing the integrated effect of each interval where similar cyclostationary dependencies are present. This approach provides a robust method against noise, while it is affected by the Nyquist issue, similar to the ST-LIM. The l-CS-LIM, on the other hand, estimates the time-dependent Jacobian of the underlying system. While more sensitive to noise, this method is free from the Nyquist issue. The numerical experiments demonstrate that both CS-LIMs effectively capture the temporal structure of the underlying system from synthetic observational data. Moreover, CS-LIMs are applied to real-world ENSO data, yielding a consistent result aligning well with the observations and the current ENSO understanding.