This paper introduces a structural equation formulation that gives rise to a new family of quasi-periodic Gaussian processes, useful to process a broad class of natural and physiological signals. The proposed formulation simplifies generation and forecasting, and provides hyperparameter estimates, which we exploit in a convergent and consistent iterative estimation algorithm. A bootstrap approach for standard error estimation and confidence intervals is also provided. We demonstrate the computational and scaling benefits of the proposed approach on a broad class of problems, including water level tidal analysis, CO\textsubscript{2} emission data, and sunspot numbers data. By leveraging the structural equations, our method reduces the cost of likelihood evaluations and predictions from $\mathcal{O}(k^2 p^2)$ to $\mathcal{O}(p^2)$, significantly improving scalability.

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