Recent years have seen an increased interest in the application of methods and techniques commonly associated with machine learning and artificial intelligence to spatial statistics. Here, in a celebration of the ten-year anniversary of the journal Spatial Statistics, we bring together normalizing flows, commonly used for density function estimation in machine learning, and spherical point processes, a topic of particular interest to the journal’s readership, to present a new approach for modeling non-homogeneous Poisson process intensity functions on the sphere. The central idea of this framework is to build, and estimate, a flexible bijective map that transforms the underlying intensity function of interest on the sphere into a simpler, reference, intensity function, also on the sphere. Map estimation can be done efficiently using automatic differentiation and stochastic gradient descent, and uncertainty quantification can be done straightforwardly via nonparametric bootstrap. We investigate the viability of the proposed method in a simulation study, and illustrate its use in a proof-of-concept study where we model the intensity of cyclone events in the North Pacific Ocean. Our experiments reveal that normalizing flows present a flexible and straightforward way to model intensity functions on spheres, but that their potential to yield a good fit depends on the architecture of the bijective map, which can be difficult to establish in practice.

近年来，人们越来越关注将通常与机器学习和人工智能相关的方法和技术应用于空间统计。在这里，为了庆祝《空间统计》杂志创刊十周年，我们将机器学习中通常用于密度函数估计的归一化流和期刊读者特别感兴趣的主题球面点过程结合在一起，提出了一种在球面上建模非齐次泊松过程强度函数的新方法。这个框架的中心思想是建立和估计一个灵活的双射映射，它将球体上感兴趣的潜在强度函数转换为一个更简单的参考强度函数，也在球体上。使用自动微分和随机梯度下降可以有效地完成地图估计，并且可以通过非参数引导直接完成不确定性量化。我们在模拟研究中调查了所提出方法的可行性，并说明它在概念验证研究中的用途，我们在该研究中模拟北太平洋气旋事件的强度。我们的实验表明，归一化流提供了一种灵活且直接的方式来模拟球体上的强度函数，但它们产生良好拟合的潜力取决于双射图的架构，这在实践中可能难以建立。