Quasi-periodicity refers to a pattern in a function where it appears periodic but has evolving amplitudes over time. This is often the case in practical settings such as the modeling of case counts of infectious disease or the carbon dioxide (CO2) concentration over time. In this paper, we introduce a class of Gaussian processes, called seasonal Gaussian Processes (sGP), for model-based inference of such quasi-periodic behavior. We illustrate that the exact sGP can be efficiently fit within $O(n)$ time using its state space representation for equally spaced locations. However, for large datasets with irregular spacing, the exact approach becomes computationally inefficient and unstable. To address this, we develop a continuous finite dimensional approximation for sGP using the seasonal B-spline (sB-spline) basis constructed by damping B-splines with sinusoidal functions. We prove that the proposed approximation converges in distribution to the true sGP as the number of basis functions increases, and show its superior approximation quality through numerical studies. We also provide a unified and interpretable way to define priors for the sGP, based on the notion of predictive standard deviation (PSD). Finally, we implement the proposed inference method on several real data examples to illustrate its practical usage.

翻译：准周期性是指函数中存在一种模式，其看似具有周期性但振幅随时间演变。这在实践场景中较为常见，例如对传染病病例数或二氧化碳（CO2）浓度随时间变化的建模。本文引入一类名为季节性高斯过程（sGP）的高斯过程，用于对此类准周期行为进行基于模型的推断。我们展示了对于等间距数据，利用其状态空间表示方法，精确的sGP可在$O(n)$时间复杂度内高效拟合。然而，对于非等间距的大规模数据集，精确方法会变得计算效率低下且不稳定。为解决此问题，我们通过阻尼正弦函数调制的B样条构建季节性B样条（sB-spline）基，开发了sGP的连续有限维近似。我们证明随着基函数数量增加，所提近似在分布上收敛至真实sGP，并通过数值研究展示其优越的近似质量。此外，基于预测标准差（PSD）概念，我们提供了统一且可解释的sGP先验定义方法。最后，我们在多个真实数据示例上实现所提推断方法以说明其实用价值。