We propose a novel nonparametric regression framework subject to the positive definiteness constraint. It offers a highly modular approach for estimating covariance functions of stationary processes. Our method can impose positive definiteness, as well as isotropy and monotonicity, on the estimators, and its hyperparameters can be decided using cross validation. We define our estimators by taking integral transforms of kernel-based distribution surrogates. We then use the iterated density estimation evolutionary algorithm, a variant of estimation of distribution algorithms, to fit the estimators. We also extend our method to estimate covariance functions for point-referenced data. Compared to alternative approaches, our method provides more reliable estimates for long-range dependence. Several numerical studies are performed to demonstrate the efficacy and performance of our method. Also, we illustrate our method using precipitation data from the Spatial Interpolation Comparison 97 project.

翻译：我们提出了一种满足正定性约束的新型非参数回归框架，为平稳过程的协方差函数估计提供了一种高度模块化的方法。该方法能够对估计量施加正定性、各向同性和单调性约束，其超参数可通过交叉验证确定。我们通过核基分布代理的积分变换定义估计量，并采用迭代密度估计进化算法（一种分布估计算法的变体）来拟合这些估计量。此外，我们将该方法扩展至点参考数据的协方差函数估计。与现有方法相比，本方法在长程依赖场景下能提供更可靠的估计。通过多项数值实验验证了方法的有效性与性能，并利用"空间插值比较97"项目的降水数据进行了实例分析。