It is known that the shapes of planar triangles can be represented by a set of points on the surface of the unit sphere. On the other hand, most of the objects can easily be triangulated and so each triangle can accordingly be treated in the context of shape analysis. There is a growing interest to fit a smooth path going through the cloud of shape data available in some time instances. To tackle this problem, we propose a longitudinal model through a triangulation procedure for the shape data. In fact, our strategy initially relies on a spherical regression model for triangles, but is extended to shape data via triangulation. Regarding modeling of directional data, we use the bivariate von Mises–Fisher distribution for density of the errors. Various forms of the composite likelihood functions, constructed by altering the assumptions considered for the angles defined for each triangle, are invoked. The proposed regression model is applied to rat skull data. Also, some simulations results are presented along with the real data results.

中文翻译：

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通过三角剖分的形状的纵向模型

众所周知，平面三角形的形状可以由单位球面上的一组点表示。另一方面，大多数对象可以轻松地进行三角剖分，因此可以在形状分析的上下文中相应地处理每个三角形。人们越来越感兴趣的是，在某些情况下，适合通过形状数据云的平滑路径。为了解决这个问题，我们通过三角剖分程序为形状数据提出了一个纵向模型。实际上，我们的策略最初依赖于三角形的球面回归模型，但后来扩展到通过三角测量来塑造数据的形状。关于方向数据的建模，我们使用双变量冯·米塞斯·费舍尔分布来确定误差的密度。各种形式的复合似然函数，通过改变为每个三角形定义的角度所考虑的假设而构造的结构被调用。拟议的回归模型应用于大鼠颅骨数据。此外，还提供了一些模拟结果以及实际数据结果。