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 work is to present techniques using multivariate mixtures for establishing validity that are simultaneously simplified and comprehensive. This is accomplished using results on conditionally negative semidefinite matrices and the Schur product theorem. For illustration, we use the recently-introduced Confluent Hypergeometric (CH) class of covariance functions. In addition, we establish the spectral density of the Confluent Hypergeometric covariance and use this to construct valid multivariate models as well as propose new cross-covariances. Our approach leads to valid multivariate cross-covariance models that inherit the desired marginal properties of the Confluent Hypergeometric 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 the new models, including results on equivalence of Gaussian measures. We demonstrate the new model's use for multivariate oceanography dataset consisting of temperature, salinity and oxygen, as measured by autonomous floats in the Southern Ocean.

翻译：通过高斯过程对多元随机场进行建模，需要构建有效的互协方差函数来描述不同空间位置上任意两个分量过程之间的依赖关系。所需的有效性条件往往带来挑战，导致参数空间存在复杂的限制。本研究的目的是提出利用多元混合建立有效性的技术，这些技术既简化又全面。这是通过利用条件负半定矩阵和Schur乘积定理的结果来实现的。为了说明，我们使用了最近引入的汇合超几何（CH）协方差函数类。此外，我们建立了汇合超几何协方差的光谱密度，并利用它来构建有效的多元模型以及提出新的互协方差函数。我们的方法产生了有效的多元互协方差模型，这些模型继承了汇合超几何模型所需的边缘特性，并且在基础多元随机场具有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。我们还建立了新模型的性质，包括高斯测度等价性的结果。我们通过一个由南大洋自主浮标测量的温度、盐度和氧气组成的多元海洋学数据集，展示了新模型的应用。