In this paper, we propose two multimodal circular distributions which are suitable for modeling circular data sets with two or more modes. Both distributions belong to the regular exponential family of distributions and are considered as extensions of the von Mises distribution. Hence, they possess the highly desirable properties, such as the existence of non-trivial sufficient statistics and optimal inferences for their parameters. Fine particulates (PM2.5) are generally emitted from activities such as industrial and residential combustion and from vehicle exhaust. We illustrate the utility of our proposed models using a real data set consisting of fine particulates (PM2.5) pollutant levels in Houston region during Fall season in 2019. Our results provide a strong evidence that its diurnal pattern exhibits four modes; two peaks during morning and evening rush hours and two peaks in between.

在本文中，我们提出了两种多模态圆形分布，它们适用于对具有两种或多种模态的圆形数据集进行建模。这两个分布都属于正则指数分布族，被认为是 von Mises 分布的扩展。因此，它们具有非常理想的特性，例如存在非平凡的充分统计数据和对其参数的最佳推断。细颗粒物 (PM2.5) 通常由工业和住宅燃烧等活动以及汽车尾气排放。我们使用由 2019 年秋季休斯顿地区细颗粒物 (PM2.5) 污染物水平组成的真实数据集来说明我们提出的模型的效用。我们的结果有力地证明了其昼夜模式表现出四种模式；