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:定期 Gausian 进程( PGP) 因其高度精准而越来越多地用于模拟定期数据。 然而, 计算PGP的可能性的计算复杂性很高, 其计算复杂性为$mathcal{Oá3 ⁇ right (n ⁇ 3 ⁇ right) 美元, 妨碍其广泛应用。 为了解决这一问题, 我们提议为在信号处理应用程序中常见到的电网收集的大规模定期数据建立一个新的 Circurant PGP (CPGP) 模式。 拟议的CPGP 模式将PGP 的日志相似性分解成两种可计算可扩展的复合日志相似性的总和, 后者不包含任何近似值。 计算CPGP 的可能性只需要$gal{Oleft (p ⁇ 2 ⁇ right) 美元( 美元), 或 $maxcalcal left (p\log p\right) 模式观测时间, 其分解成多个段长度, 大大小于$n。 。 模拟和真实的案例研究研究显示, 特别需要周期性GPGPP 应用的高级方法。