Covariance functions are the core of spatial statistics, stochastic processes, machine learning as well as many other theoretical and applied disciplines. The properties of the covariance function at small and large distances determine the geometric attributes of the associated Gaussian random field. Having covariance functions that allow to specify both local and global properties is certainly on demand. This paper provides a method to find new classes of covariance functions having such properties. We term these models hybrid as they are obtained as scale mixtures of piecewise covariance kernels against measures that are also defined as piecewise linear combination of parametric families of measures. In order to illustrate our methodology, we provide new families of covariance functions that are proved to be richer with respect to other well known families that have been proposed by earlier literature. More precisely, we derive a hybrid Cauchy-Mat\'ern model, which allows us to index both long memory and mean square differentiability of the random field, and a hybrid Hole-Effect-Mat\'ern model, which is capable of attaining negative values (hole effect), while preserving the local attributes of the traditional Mat\'ern model. Our findings are illustrated through numerical studies with both simulated and real data.
翻译:共变函数是空间统计、 随机过程、 机器学习以及许多其他理论和应用学科的核心。 相变函数的大小和远距离特性决定了相关高斯随机字段的几何属性。 具有允许指定本地和全球属性的共变函数当然是需要的。 本文提供了一种方法, 以寻找具有这种属性的新的共变函数类别。 我们将这些模型混合起来, 因为它们是获得的成片共变内核的大小混合物, 相对于也定义为测量参数组合的细微线性组合的措施。 为了说明我们的方法, 我们提供了新的共变函数的组合, 这些函数对于先前文献中提议的其他众所周知的家族来说已经证明更为丰富。 更准确地说, 我们得出一个混合的 Cauchy- Mat\ ern 模型, 使我们能够将随机字段的长期记忆和平均可变性指数化, 以及一个混合的 Hole- Efect- Mat\ ERn 模型, 能够实现负值( 孔效应), 并同时保存传统 Mat\ “ ” 数字” 的本地模型研究, 和模拟的当地数据。