Procrustes Methods in the Statistical Analysis of Shape
形状统计分析中的 Procrustes 方法AI 解读 ↓
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形状统计分析中的 Procrustes 方法
- 研究问题
- 在旋转、平移与等比例缩放不改变形状的设定下，如何为标志点数据建立形状模型并进行估计与差异检验？
- 方法与思路
- 作者直接处理标志点坐标矩阵，允许相似变换，以高斯模型、最小二乘叠合及 QR 和奇异值分解发展估计与推断。
- 材料陈述的贡献
- 摘要称，方法扩展到两样本、方差分析和回归模型，允许非各向同性误差，并讨论度量匹配下的有效估计与形状检验。
- 阅读关注点
- 应核对拟合度量与检验协方差度量的关系、均值估计一致性条件、奇异误差变换及检验分布，并查原 PDF 确认符号。
- 阅读边界
- 仅据所给 Crossref 摘要。具体条件与应用数值未提供，检验记号及维度排版需查原 PDF；不能将度量匹配下的效率结论普遍化。
原文摘要
SUMMARY Two geometrical figures, X and Y, in RK, each consisting of N landmark points, have the same shape if they differ by at most a rotation, a translation and isotropic scaling. This paper presents a model-based Procrustes approach to analysing sets of shapes. With few exceptions, the metric geometry of shape spaces is quite complicated. We develop a basic understanding through the familiar QR and singular value decompositions of multivariate analysis. The strategy underlying the use of Procrustes methods is to work directly with the N x K co-ordinate matrix, while allowing for an arbitrary similarity transformation at all stages of model formulation, estimation and inference. A Gaussian model for landmark data is defined for a single population and generalized to two-sample, analysis-of-variance and regression models. Maximum likelihood estimation is by least squares superimposition of the figures; we describe generalizations of Procrustes techniques to allow non-isotropic errors at and between landmarks. Inference is based on an N x K linear multivariate Procrustes statistic that, in a double-rotated co-ordinate system, is a simple but singular linear transformation of the errors at landmarks. However, the superimposition metric used for fitting, and the model metric, or covariance, used for testing, may not coincide. Estimates of means are consistent for many reasonable choices of superimposition metric. The estimates are efficient (maximum likelihood estimates) when the metrics coincide. F-ratio and Hotelling's R 2-tests for shape differences in one- and two-sample data are derived from the distribution of the Procrustes statistic. The techniques are applied to the shapes associated with hydrocephaly and nutritional differences in young rats.