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：成分响应回归由于单纯形的非线性几何以及欧几里得方法的局限性而面临挑战。我们提出了一种基于流形值数据的回归框架，通过映射到统计上易于处理的中间空间来实现回归。对于成分数据，响应变量嵌入球面的正象限，并利用主嵌套球面（PNS）进行分析，从而得到一个圆柱形中间空间，其中包含一个圆形的领先得分和欧几里得的高阶得分。在此中间空间中进行回归，并将拟合值映射回单纯形。模拟研究显示了基于PNS回归的良好性能。一项应用于环境化学暴露数据的实例说明了该方法的可解释性和实际效用。