论文标题
圆形的母体协方差函数及其与圆上马尔可夫随机场的链接
The Circular Matern Covariance Function and its Link to Markov Random Fields on the Circle
论文作者
论文摘要
高斯随机场与马尔可夫随机场之间的联系是基于欧几里得空间中随机部分微分方程的良好建立的，其中Matérn协方差函数至关重要。但是，Matérn协方差函数并不总是在圆形和球体上确定的。在此手稿中，我们专注于将此链接扩展到圆圈，并证明圆圈上高斯随机字段和马尔可夫随机字段之间的链接是基于圆形Matérn协方差函数有效的。首先，我们表明，这种圆形的Matérn函数是固定解的固定解对圆上的随机微分方程的协方差，并具有正式定义的白噪声空间度量。然后，对于相应的条件自回归模型，我们为其协方差函数得出了一个封闭式公式。连同圆形Matérn协方差函数的封闭形式公式，可以明确建立这两个随机场之间的联系。此外，众所周知，平均值的估计器在圆圈上不一致，我们为这种非恋性问题提供了同等的高斯度量解释。
The link between Gaussian random fields and Markov random fields is well established based on a stochastic partial differential equation in Euclidean spaces, where the Matérn covariance functions are essential. However, the Matérn covariance functions are not always positive definite on circles and spheres. In this manuscript, we focus on the extension of this link to circles, and show that the link between Gaussian random fields and Markov random fields on circles is valid based on the circular Matérn covariance function instead. First, we show that this circular Matérn function is the covariance of the stationary solution to the stochastic differential equation on the circle with a formally defined white noise space measure. Then, for the corresponding conditional autoregressive model, we derive a closed form formula for its covariance function. Together with a closed form formula for the circular Matérn covariance function, the link between these two random fields can be established explicitly. Additionally, it is known that the estimator of the mean is not consistent on circles, we provide an equivalent Gaussian measure explanation for this non-ergodicity issue.
