AI 中文总结
研究圆形数据的逻辑回归模型，核心方法是构建含特定线性预测器的圆形逻辑回归框架，用蒙特卡罗模拟评估，通过真实数据集验证，主要贡献是明确不同条件下对称与非对称链接函数的性能差异及给出实践指导。
AI 中文摘要
圆形（方向）数据如风向、时间或日历阶段等以单位圆上的角度测量，需要考虑其周期性的统计方法。本文开发并评估了一个圆形逻辑回归框架，其中线性预测器通过圆形协变量的余弦和正弦表示。通过蒙特卡罗模拟在两种集中状态下生成圆形预测器，并使用赤池信息准则（AIC）和偏差评估模型拟合。用两个真实数据集进行说明，结果表明当圆形预测器广泛分散且响应明显不平衡时，链接函数的选择最为重要；在预测器高度集中时，对称链接更优，非对称链接易不稳定。还讨论了实际指导方针和未来软件开发方向。
英文摘要
Circular (directional) data arise whenever observations are measured as angles on the unit circle, such as wind direction, time of day, or calendar phase, and require statistical methods that respect the periodicity of the domain $[0; 2π)$. While circular-linear and linear-circular regression models are well established, regression models for a binary or binomial response observed jointly with a circular predictor remain largely undeveloped, with the sole closely related study restricted to the symmetric logit link. This paper develops and evaluates a circular logistic regression framework in which the linear predictor is expressed through the cosine and sine of the circular covariate, and compares the performance of symmetric link functions (logit, probit) against asymmetric alternatives (complementary log-log, Cauchit, and a skew-logit power link) under a generalized linear model formulation. A Monte Carlo simulation generates circular predictors from the von Mises distribution under two concentration regimes and evaluates model fit using the Akaike Information Criterion (AIC) and deviance. The methodology is illustrated with two real data sets: daily rainfall occurrence and wind direction recorded in Macomb, Illinois, and monthly earthquake counts in Western Anatolia, Turkiye, the latter used to connect the binary circular model to the related circular Poisson regression framework for count outcomes. Results indicate that the choice of link function matters most when the circular predictor is broadly dispersed and the response is markedly unbalanced; under high concentration of the predictor, symmetric links are preferred and asymmetric links are prone to instability. Practical guidelines and directions for future software development are discussed.
