Intrinsic MANOVA for Riemannian Manifolds with an Application to Kendall's Space of Planar Shapes
Intrinsic MANOVA for Riemannian Manifolds with an Application to Kendall's Space of Planar Shapes
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黎曼流形的本征多元方差分析及其在肯德尔平面形状空间中的应用
DOI：
10.1109/tpami.2009.117
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发表时间：
2010
影响因子：
23.6
通讯作者：
A. Munk
中科院分区：
文献类型：
--
作者：
S. Huckemann;T. Hotz;A. Munk
我们提出了一个内在的多因子模型的黎曼流形上的数据，通常发生在统计分析的形状。由于缺乏线性结构，线性模型通常无法定义;迄今为止，只有单向MANOVA可用。对于一般的多因子模型，我们假设模型无法解释的变化集中在定义效应的元素附近。通过确定平行运输下各自的样本协方差的渐近分布，我们表明，它们可以比较标准的MANOVA。通常在应用中，流形仅隐含地给出为平行分量，其中底部空间平行输运可以通过微分方程表示。对于平面形状的Kendall空间，我们提供了一个显式的解决方案。我们说明了我们的方法由一个内在的双向MANOVA的一组叶片形状。虽然生物学家可以通过视觉识别基因型效应，但我们可以检测到其他无法识别的身高效应。
We propose an intrinsic multifactorial model for data on Riemannian manifolds that typically occur in the statistical analysis of shape. Due to the lack of a linear structure, linear models cannot be defined in general; to date only one-way MANOVA is available. For a general multifactorial model, we assume that variation not explained by the model is concentrated near elements defining the effects. By determining the asymptotic distributions of respective sample covariances under parallel transport, we show that they can be compared by standard MANOVA. Often in applications manifolds are only implicitly given as quotients, where the bottom space parallel transport can be expressed through a differential equation. For Kendall's space of planar shapes, we provide an explicit solution. We illustrate our method by an intrinsic two-way MANOVA for a set of leaf shapes. While biologists can identify genotype effects by sight, we can detect height effects that are otherwise not identifiable.