Regression models, where the response variable is circular, are common in areas such as biology, geology and meteorology. A typical model assumes that the conditional distribution of the response follows a von-Mises distribution. However, this assumption is inadequate when the response variable is multimodal. For this reason, in this paper, a finite mixture of regressions model is proposed for the case of a circular response variable and a set of circular and/or linear covariates. Mixture models are very useful when the underlying population is multimodal. Despite the prevalence of multimodality in regression modelling of circular data, the use of mixtures of regressions has received no attention in the literature. This paper aims to close this knowledge gap. To estimate the proposed model, we develop a maximum likelihood estimation procedure via the Expectation-Maximization algorithm. An extensive simulation study is used to demonstrate the practical use and performance of the proposed model and estimation procedure. In addition, the model is shown to be useful as a model-based clustering tool. Lastly, the model is applied to a real dataset from a wind farm in South Africa.

翻译：回归模型在生物学、地质学和气象学等领域中应用广泛，其中响应变量为圆形数据的情况尤为常见。典型模型通常假设响应变量的条件分布服从冯·米塞斯分布。然而，当响应变量呈现多模态特性时，这一假设便不再适用。为此，本文针对圆形响应变量与一组圆形及/或线性协变量的情况，提出了一种有限混合回归模型。当基础总体呈现多模态分布时，混合模型具有显著优势。尽管圆形数据回归建模中多模态现象普遍存在，但回归混合模型在现有文献中尚未得到关注。本文旨在填补这一知识空白。为估计所提出的模型，我们通过期望最大化算法开发了最大似然估计程序。通过广泛的模拟研究，验证了所提模型及估计方法的实际效用与性能。此外，该模型被证明可作为基于模型的聚类工具发挥重要作用。最后，我们将模型应用于南非某风电场的实际数据集进行实证分析。