I consider the Lerch-Hurwitz or periodic zeta function as covariance function of a periodic continuous-time stationary stochastic process. The function can be parametrized with a continuous index $\nu$ which regulates the continuity and differentiability properties of the process in a way completely analogous to the parameter $\nu$ of the Mat\'ern class of covariance functions. This makes the periodic zeta a good companion to add a power-law prior spectrum seasonal component to a Mat\'ern prior for Gaussian process regression. It is also a close relative of the circular Mat\'ern covariance, and likewise can be used on spheres up to dimension three. Since this special function is not generally available in standard libraries, I explain in detail the numerical implementation.

中文翻译：

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高斯过程回归的周期性 zeta 协方差函数

我将 Lerch-Hurwitz 或周期性 zeta 函数视为周期性连续时间平稳随机过程的协方差函数。该函数可以用一个连续索引 $\nu$ 参数化，它以完全类似于 Mat\'ern 类协方差函数的参数 $\nu$ 的方式调节过程的连续性和可微性属性。这使得周期性 zeta 成为将幂律先验频谱季节性分量添加到 Mat\'ern 先验以进行高斯过程回归的良好伴侣。它也是圆形 Mat\'ern 协方差的近亲，同样可用于高达 3 维的球体。由于这个特殊函数在标准库中一般不可用，所以我详细解释了数值实现。