On cyclostationary linear inverse models: a mathematical insight and implication

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, namely, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:math> -CS-LIM and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:math> -CS-LIM. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:math> -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 robustness against noise but is affected by the Nyquist issue, similar to the ST-LIM. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:math> -CS-LIM, on the other hand, estimates the time-dependent Jacobian of the underlying system. Although more sensitive to noise, this method is free from the Nyquist issue. Numerical experiments demonstrate that both CS-LIM variants effectively capture the temporal structure of the underlying system using synthetic observational data. Moreover, when applied to real-world ENSO data, CS-LIMs yield consistent results that align well with the observations and current El Niño–Southern Oscillation (ENSO) understanding.