On the meaning of mean shape: manifold stability, locus and the two sample test
On the meaning of mean shape: manifold stability, locus and the two sample test
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论平均形状的意义：流形稳定性、轨迹与二样本检验
DOI：
10.1007/s10463-012-0352-2
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发表时间：
2012
影响因子：
1
通讯作者：
Huckemann
中科院分区：
文献类型：
--
作者：
Huckemann
各种概念的平均形状以前在文献中无关的关系。特别是，对于非流形，如肯德尔的三维形状空间，本文回答了这个问题，这意味着可以应用两个样本的测试。如果使用固有或Ziezold方法，答案是肯定的。形状空间上的平均值的流形稳定性的基本一般结果，由于李群在黎曼流形上的适当和等距作用的商，将微分几何的切片定理与形状的统计融合在一起。对于3D Procrustes平均，给出了一个反例。为了进一步阐明的微妙之处的手段，为领域和肯德尔的形状空间，一个一阶关系的内在，残留/Procrustean和外在/Ziezold的手段，指出，对于高浓度的后者近似划分（广义）测地线段之间的前两者的比例1：3。这一事实，坐标选择的测试和其他细节的权力的后果，例如，外在勋伯格手段可能会增加维度进行了讨论和说明的模拟和示例性数据集。
Various concepts of mean shape previously unrelated in the literature are brought into relation. In particular, for non-manifolds, such as Kendall’s 3D shape space, this paper answers the question, for which means one may apply a two-sample test. The answer is positive if intrinsic or Ziezold means are used. The underlying general result of manifold stability of a mean on a shape space, the quotient due to an proper and isometric action of a Lie group on a Riemannian manifold, blends the slice theorem from differential geometry with the statistics of shape. For 3D Procrustes means, however, a counterexample is given. To further elucidate on subtleties of means, for spheres and Kendall’s shape spaces, a first-order relationship between intrinsic, residual/Procrustean and extrinsic/Ziezold means is derived stating that for high concentration the latter approximately divides the (generalized) geodesic segment between the former two by the ratio 1:3. This fact, consequences of coordinate choices for the power of tests and other details, e.g. that extrinsic Schoenberg means may increase dimension are discussed and illustrated by simulations and exemplary datasets.