The periodic Gaussian process (PGP) has been increasingly used to model periodic data due to its high accuracy. Yet, computing the likelihood of PGP has a high computational complexity of $\mathcal{O}\left(n^{3}\right)$ ($n$ is the data size), which hinders its wide application. To address this issue, we propose a novel circulant PGP (CPGP) model for large-scale periodic data collected at grids that are commonly seen in signal processing applications. The proposed CPGP decomposes the log-likelihood of PGP into the sum of two computationally scalable composite log-likelihoods, which do not involve any approximations. Computing the likelihood of CPGP requires only $\mathcal{O}\left(p^{2}\right)$ (or $\mathcal{O}\left(p\log p\right)$ in some special cases) time for grid observations, where the segment length $p$ is independent of and much smaller than $n$. Simulations and real case studies are presented to show the superiority of CPGP over some state-of-the-art methods, especially for applications requiring periodicity estimation. This new modeling technique can greatly advance the applicability of PGP in many areas and allow the modeling of many previously intractable problems.

翻译：周期高斯过程（PGP）因其高精度而越来越多地被用于对周期数据进行建模。然而，PGP似然计算具有$\mathcal{O}\left(n^{3}\right)$的高计算复杂度（$n$为数据规模），这阻碍了其广泛应用。为解决此问题，我们针对信号处理应用中常见的网格采集的大规模周期数据，提出了一种新颖的循环周期高斯过程（CPGP）模型。所提出的CPGP将PGP的对数似然分解为两个计算可扩展的复合对数似然之和，且不涉及任何近似。对于网格观测数据，CPGP的似然计算仅需$\mathcal{O}\left(p^{2}\right)$（或在特殊情况下为$\mathcal{O}\left(p\log p\right)$）的时间，其中分段长度$p$独立于$n$且远小于$n$。我们通过仿真和实际案例研究展示了CPGP相对于某些最先进方法的优越性，尤其在需要周期性估计的应用中。这一新型建模技术可极大提升PGP在众多领域的适用性，并使得许多以往难以处理的问题得以建模。