平面形状流形上的函数可加模型
Functional additive models on manifolds of planar shapes and forms
摘要：
平面曲线和/或标志配置的“形状”被认为是其在平移、旋转和缩放下的等价类，其“形式”被认为是其在平移和旋转下的等价类，同时保持缩放。我们扩展广义添加剂回归模型的形状/形式的响应尊重所得到的商几何采用平方测地距离作为损失函数和测地响应函数映射到形状/形式空间的添加剂预测。为了拟合模型，我们提出了一个黎曼$L_2$ -Boosting算法，非常适合于一个潜在的大量的可能的参数密集型模型条款，这也产生自动模型选择。我们提供了新的直观的可解释的可视化（甚至非线性）协变量的影响，在形状/形式空间通过适当的张量积分解。所提出的框架的有用性说明在分析1）黄芪形状的野生和驯养的绵羊和2）细胞形式中产生的生物物理模型，以及3）在现实的模拟研究与响应形状和形式的动机从数据集上的瓶子轮廓。
英文摘要：
The "shape" of a planar curve and/or landmark configuration is considered its equivalence class under translation, rotation and scaling, its "form" its equivalence class under translation and rotation while scale is preserved. We extend generalized additive regression to models for such shapes/forms as responses respecting the resulting quotient geometry by employing the squared geodesic distance as loss function and a geodesic response function to map the additive predictor to the shape/form space. For fitting the model, we propose a Riemannian $L_2$-Boosting algorithm well suited for a potentially large number of possibly parameter-intensive model terms, which also yields automated model selection. We provide novel intuitively interpretable visualizations for (even non-linear) covariate effects in the shape/form space via suitable tensor-product factorization. The usefulness of the proposed framework is illustrated in an analysis of 1) astragalus shapes of wild and domesticated sheep and 2) cell forms generated in a biophysical model, as well as 3) in a realistic simulation study with response shapes and forms motivated from a dataset on bottle outlines.
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