Modeling of multivariate random fields through Gaussian processes calls for the construction of valid cross-covariance functions describing the dependence between any two component processes at different spatial locations. The required validity conditions often present challenges that lead to complicated restrictions on the parameter space. The purpose of this paper is to present a simplified techniques for establishing multivariate validity for the recently-introduced Confluent Hypergeometric (CH) class of covariance functions. Specifically, we use multivariate mixtures to present both simplified and comprehensive conditions for validity, based on results on conditionally negative semidefinite matrices and the Schur product theorem. In addition, we establish the spectral density of the CH covariance and use this to construct valid multivariate models as well as propose new cross-covariances. We show that our proposed approach leads to valid multivariate cross-covariance models that inherit the desired marginal properties of the CH model and outperform the multivariate Mat\'ern model in out-of-sample prediction under slowly-decaying correlation of the underlying multivariate random field. We also establish properties of multivariate CH models, including equivalence of Gaussian measures, and demonstrate their use in modeling a multivariate oceanography data set consisting of temperature, salinity and oxygen, as measured by autonomous floats in the Southern Ocean.

翻译：多元随机场的高斯过程建模需要构建有效的交叉协方差函数，用以描述不同空间位置处任意两个分量过程之间的依赖关系。所需的有效性条件常带来挑战，导致参数空间受到复杂限制。本文旨在为近期提出的合流超几何（CH）协方差函数类建立多元有效性的简化技术。具体而言，我们基于条件负半定矩阵和Schur乘积定理的结果，利用多元混合方法提出简化的综合性有效性条件。此外，我们建立了CH协方差的谱密度，并据此构建有效的多元模型，同时提出新的交叉协方差函数。研究表明，所提方法能生成有效的多元交叉协方差模型，该模型继承了CH模型理想的边际性质，并在底层多元随机场呈现缓慢衰减相关性的情况下，其样本外预测性能优于多元Matérn模型。我们还建立了多元CH模型的性质（包括高斯测度的等价性），并展示了其在南大洋自主浮标测量的温度、盐度和溶解氧多元海洋数据集建模中的应用。