This paper introduces a novel regression model designed for angular response variables with linear predictors, utilizing a generalized M\"{o}bius transformation to define the regression curve. By mapping the real axis to the circle, the model effectively captures the relationship between linear and angular components. A key innovation is the introduction of an area-based loss function, inspired by the geometry of a curved torus, for efficient parameter estimation. The semi-parametric nature of the model eliminates the need for specific distributional assumptions about the angular error, enhancing its versatility. Extensive simulation studies, incorporating von Mises and wrapped Cauchy distributions, highlight the robustness of the framework. The model's practical utility is demonstrated through real-world data analysis of Bitcoin and Ethereum, showcasing its ability to derive meaningful insights from complex data structures.

翻译：本文提出了一种新颖的回归模型，用于处理具有线性预测变量的角度响应变量。该模型利用广义莫比乌斯变换定义回归曲线，通过将实轴映射到圆上，有效捕捉线性分量与角度分量之间的关系。一个关键创新是引入基于面积的损失函数，其灵感来源于弯曲环面的几何特性，以实现高效的参数估计。该模型的半参数特性避免了对角度误差的特定分布假设，从而增强了其适用性。包含冯·米塞斯分布与环绕柯西分布的广泛模拟研究，突显了该框架的鲁棒性。通过对比特币和以太坊的实际数据分析，展示了模型从复杂数据结构中提取有意义信息的能力，验证了其实际应用价值。