On Means and Their Asymptotics: Circles and Shape Spaces
On Means and Their Asymptotics: Circles and Shape Spaces
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论均值及其渐进：圆和形状空间
DOI：
10.1007/s10851-013-0462-3
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发表时间：
2013
影响因子：
2
通讯作者：
Huckemann
中科院分区：
文献类型：
--
作者：
Huckemann
我们调查的一些影响，奇异地层可能有正曲率的情况下，圆和形状空间进行（半）内在的统计分析。这里，分层空间上的数据分析基于在可能不同的分层空间中定义的统计描述符。例如，在形状空间的测地线主成分分析中，形状数据由广义测地线描述，广义测地线自然形成它们自己的形状空间，不同于原始的形状空间。在一般情况下，如果描述符是作为广义Fréchet平均获得的，在相当一般的情况下，强大数定律是有效的。此外，如果描述符是足够好的行为，一个经典的中心极限定理可以采用。关键条件之一是，必须控制单一地层和切割轨迹（如果存在）的撞击。我们审查的统计作用的削减轨迹的内在手段圆以及奇异地层的形状空间（发生在群体行动是退化），并得出结论与识别潜在的研究方向。
We survey some effects that singular strata may have in the positive curvature context of circles and shape spaces when conducting (semi-)intrinsic statistical analyses. Here, the analysis of data on a stratified space is based on statistical descriptors defined in a possibly different stratified space. E.g. in geodesic principal component analysis for shape spaces, shape data are described by generalized geodesics which naturally form a shape space of their own, different from the original one. In a general context, if the descriptors are obtained as generalized Fréchet means, under rather general circumstances, a strong law of large numbers is valid. If furthermore the descriptors are sufficiently well behaved, a classical central limit theorem can be adopted. One of the crucial conditions is that hitting of singular strata as well as of cut loci, if present, must be controlled. We review the statistical role of the cut locus of intrinsic means for circles as well as that of singular strata for shape spaces (occurring where the group action is degenerate) and conclude with an identification of potential research directions.