This article gives a comprehensive theoretical framework to the modeling, inference, and applications of Gaussian random fields using what we term the hypertorus as an index set. The hypertorus is obtained through a product of hyperspheres. We envision the following as appropriate settings for random fields on the hypertorus: continuous-time data with multiple sources of seasonality, directional data with seasonality or over the globe, and global spatiotemporal data with temporal seasonality. We propose modeling strategies for such data through covariance structures over the hypertorus. We develop various families of covariance functions over the hypertorus and discuss how to construct random fields using these covariance functions. We show the utility of our findings on three datasets. Our first example is a dataset of ozone concentrations from Mexico City that exhibits multiple sources of seasonality. Our second dataset is a wind speed dataset, where the data show daily seasonality and are indexed by wind direction. Our third illustration considers a global space-time dataset of cloud coverage, demonstrating strong seasonality. In all analyses, we compare the predictive performance of random fields specified through various covariance structures and examine the results of the best predictive model.

中文翻译：

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hypertorus 上的随机场：协方差建模和应用

本文使用我们所说的 hypertorus 作为索引集，为高斯随机场的建模、推理和应用提供了一个全面的理论框架。hypertorus 是通过超球体的乘积获得的。我们设想以下作为 hypertorus 上随机场的适当设置：具有多个季节性来源的连续时间数据、具有季节性或全球范围的定向数据，以及具有时间季节性的全球时空数据。我们通过超曲面上的协方差结构为此类数据提出建模策略。我们在超曲面上开发了各种协方差函数族，并讨论了如何使用这些协方差函数构造随机场。我们在三个数据集上展示了我们的发现的实用性。我们的第一个示例是来自墨西哥城的臭氧浓度数据集，该数据集展示了多种季节性来源。我们的第二个数据集是风速数据集，其中数据显示每日季节性并按风向索引。我们的第三个插图考虑了云覆盖的全球时空数据集，展示了强烈的季节性。在所有分析中，我们比较了通过各种协方差结构指定的随机场的预测性能，并检查了最佳预测模型的结果。