We investigate a class of models for non-parametric estimation of probability density fields based on scattered samples of heterogeneous sizes. The considered SLGP models are Spatial extensions of Logistic Gaussian Process models and inherit some of their theoretical properties but also of their computational challenges. We introduce SLGPs from the perspective of random measures and their densities, and investigate links between properties of SLGPs and underlying processes. Our inquiries are motivated by SLGP's abilities to deliver probabilistic predictions of conditional distributions at candidate points, to allow (approximate) conditional simulations of probability densities, and to jointly predict multiple functionals of target distributions. We show that SLGP models induced by continuous GPs can be characterized by the joint Gaussianity of their log-increments and leverage this characterization to establish theoretical results pertaining to spatial regularity. We extend the notion of mean-square continuity to random measure fields and establish sufficient conditions on covariance kernels underlying SLGPs for associated models to enjoy such regularity properties. From the practical side, we propose an implementation relying on Random Fourier Features and demonstrate its applicability on synthetic examples and on temperature distributions at meteorological stations, including probabilistic predictions of densities at left-out stations.

翻译：本文研究一类基于异质大小散点样本对概率密度场进行非参数估计的模型。所考虑的SLGP模型是逻辑斯蒂高斯过程模型的空间扩展，继承了部分理论性质及其计算挑战。我们从随机测度及其密度的角度引入SLGP，并探讨SLGP性质与底层过程之间的联系。本研究的动机在于SLGP能够提供候选点条件分布的概率预测、实现概率密度（近似）条件模拟，以及联合预测目标分布的多个泛函。我们证明，由连续高斯过程诱导的SLGP模型可通过其对数增量的联合高斯性进行刻画，并利用该刻画建立关于空间正则性的理论结果。我们将均方连续性概念推广至随机测度场，并建立SLGP底层协方差核的充分条件，使相关模型具备此类正则性质。在实践层面，我们提出一种基于随机傅里叶特征的实现方案，并通过合成示例及气象站温度分布数据（包括留出站密度概率预测）验证其适用性。