On the Cyclostationary Linear Inverse Models: A Mathematical Insight and Implication

TL;DR

This paper explores the mathematical foundations of cyclostationary linear inverse models (CS-LIMs), introduces two variants with improved properties, and demonstrates their effectiveness in analyzing complex stochastic processes and climate data.

Contribution

It provides a mathematical analysis of CS-LIMs, introduces e-CS-LIM and l-CS-LIM variants, and evaluates their performance on synthetic and real ENSO data.

Findings

e-CS-LIM converges to l-CS-LIM under certain conditions

Both models effectively reveal temporal structures in data

CS-LIMs applied to ENSO data align with current climate understanding

Abstract

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.