球密度估计的归一化流混合模型
Mixture Modeling with Normalizing Flows for Spherical Density Estimation
摘要：
规范化流是用于模拟复杂概率密度函数的对象，近年来引起了人们的极大兴趣。已经发展了许多正常化流动的灵活家庭。然而，迄今为止的重点主要集中在欧几里得域上的规范化流上;虽然已经为球形和其他非欧几里得域开发了规范化流，但这些规范化流通常不如欧几里得域灵活。为了解决这个问题，在这项工作中，我们引入了一个混合的归一化流模型来构建复杂的概率密度函数的球。该模型提供了一个灵活的替代现有的参数，半参数，非参数，有限混合模型。使用期望最大化算法及其变体来执行模型估计。该模型被应用到模拟数据，在传统的（单组分）归一化流程的好处进行了验证。然后，将该模型应用于地球表面发生的两个真实事件数据集;第一个与地震有关，第二个与恐怖活动有关。在这两种情况下，我们看到，混合的归一化流模型产生了一个很好的表示事件发生的密度。
英文摘要：
Normalizing flows are objects used for modeling complicated probability density functions, and have attracted considerable interest in recent years. Many flexible families of normalizing flows have been developed. However, the focus to date has largely been on normalizing flows on Euclidean domains; while normalizing flows have been developed for spherical and other non-Euclidean domains, these are generally less flexible than their Euclidean counterparts. To address this shortcoming, in this work we introduce a mixture-of-normalizing-flows model to construct complicated probability density functions on the sphere. This model provides a flexible alternative to existing parametric, semiparametric, and nonparametric, finite mixture models. Model estimation is performed using the expectation maximization algorithm and a variant thereof. The model is applied to simulated data, where the benefit over the conventional (single component) normalizing flow is verified. The model is then applied to two real-world data sets of events occurring on the surface of Earth; the first relating to earthquakes, and the second to terrorist activity. In both cases, we see that the mixture-of-normalizing-flows model yields a good representation of the density of event occurrence.
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