The geostatistical analysis of multivariate spatial data for inference as well as joint predictions (co-kriging) ordinarily relies on modeling of the marginal and cross-covariance functions. While the former quantifies the spatial dependence within variables, the latter quantifies the spatial dependence across distinct variables. The marginal covariance functions are always symmetric; however, the cross-covariance functions often exhibit asymmetries in the real data. Asymmetric cross-covariance implies change in the value of cross-covariance for interchanged locations on fixed order of variables. Such change of cross-covariance values is often caused due to the spatial delay in effect of the response of one variable on another variable. These spatial delays are common in environmental processes, especially when dynamic phenomena such as prevailing wind and ocean currents are involved. Here, we propose a novel approach to introduce flexible asymmetries in the cross-covariances of stationary multivariate covariance functions. The proposed approach involves modeling the phase component of the constrained cross-spectral features to allow for asymmetric cross-covariances. We show the capability of our proposed model to recover the cross-dependence structure and improve spatial predictions against traditionally used models through multiple simulation studies. Additionally, we illustrate our approach on a real trivariate dataset of particulate matter concentration ( $${\hbox {PM}}_{2.5}$$ ), wind speed and relative humidity. The real data example shows that our approach outperforms the traditionally used models, in terms of model fit and spatial predictions. Supplementary materials accompanying this paper appear on-line.

中文翻译：

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多元随机场互协方差函数中变量不对称性的灵活建模

用于推理和联合预测（共同克里金法）的多元空间数据的地统计分析通常依赖于边际和交叉协方差函数的建模。前者量化变量内的空间依赖性，后者量化不同变量之间的空间依赖性。边际协方差函数总是对称的；然而，交叉协方差函数在实际数据中经常表现出不对称性。非对称互协方差意味着在变量的固定顺序上互换位置的互协方差值的变化。交叉协方差值的这种变化通常是由于一个变量对另一个变量的响应的空间延迟引起的。这些空间延迟在环境过程中很常见，特别是当涉及盛行风和洋流等动态现象时。在这里，我们提出了一种新方法来在平稳多元协方差函数的交叉协方差中引入灵活的不对称性。所提出的方法涉及对受约束的交叉谱特征的相位分量进行建模以允许不对称的交叉协方差。我们展示了我们提出的模型能够恢复交叉依赖结构，并通过多次模拟研究改进对传统使用模型的空间预测。此外，我们在颗粒物浓度 ($${\hbox {PM}}_{2.5}$$)、风速和相对湿度的真实三变量数据集上说明了我们的方法。真实数据示例表明，我们的方法优于传统使用的模型，在模型拟合和空间预测方面。本文随附的补充材料已在线发布。