Regression with compositional responses is challenging due to the nonlinear geometry of the simplex and the limitations of Euclidean methods. We propose a regression framework for manifold-valued data based on mappings to statistically tractable intermediate spaces. For compositional data, responses are embedded in the positive orthant of the sphere and analysed using Principal Nested Spheres (PNS), yielding a cylindrical intermediate space with a circular leading score and Euclidean higher-order scores. Regression is performed in this intermediate space and fitted values are mapped back to the simplex. A simulation study demonstrates good performance of PNS-based regression. An application to environmental chemical exposure data illustrates the interpretability and practical utility of the method.

翻译：成分响应变量的回归因单纯形的非线性几何结构及欧氏方法的局限性而颇具挑战。我们提出一种基于中间空间映射的流形值数据回归框架，该中间空间具有统计可处理性。对于成分数据，响应变量嵌入球面正象限，通过主嵌套球面（PNS）进行分析，生成具有圆形主得分与欧氏高阶得分的圆柱状中间空间。在此中间空间执行回归后，将拟合值映射回单纯形。模拟研究表明基于PNS的回归性能良好，环境化学暴露数据实例展示了该方法的可解释性与实用价值。