Invariance, Identifiability, and Morphometrics
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不变性、可识别性与形态测量
- 研究问题
- 在平移、旋转及可能的反射不改变对象形态时，如何确保形态统计分析使用不变量，并针对可识别参数开展推断？
- 方法与思路
- 作者从群不变性出发推导最大不变量及其分布，分析单样本和双样本可识别性，并比较矩估计与最大似然估计。
- 材料陈述的贡献
- 摘要称部分常用方法依赖不可识别参数，而所讨论的矩估计和似然方法满足不变性；矩估计可简化计算且效率损失较小。
- 阅读关注点
- 应核对形态与形状的定义、变换群和扰动模型，以及可识别参数的刻画及两类估计的效率比较适用范围。
- 阅读边界
- 仅据版本未注明的索引摘要；摘要未列出问题方法、分布公式或效率数值，不能将形态测量结论推广到其他识别问题。
原文摘要
The form of an object is that characteristic that remains invariant under a group of transformations comprising translation, rotation, and possibly reflection. Group invariance thus naturally plays an important role in the statistical analysis of forms. We examine the existing methods for the statistical analysis of form from the invariance perspective. We begin with a review of the important basic ideas behind invariance and derive a maximal invariant under the group of transformations consisting of rotation, reflection, and translation and its distribution under the Gaussian and elliptically symmetric perturbation models. We first consider the single-sample case and discuss the issue of identifiability of the parameters. We show that method-of-moments estimators based on the distances between landmarks and maximum likelihood estimators (MLEs) based on the size and shape coordinates are invariant and estimate identifiable parameters. However, a number of commonly used methods do not. We compare the statistical and computational efficiencies of the method-ofmoments estimator and MLE and show that the method of moments substantially simplifies the estimation procedure computationally with only a small loss of statistical efficiency. We then extend the discussion of invariance to the comparison of two forms. We discuss the relationship between identifiability and invariance in the two-sample case and again show that many commonly used methods base inference on nonidentifiable parameters and discuss the scientific implications of basing inferences on nonidentifiable parameters using a biological example. We provide a brief summary of a method for shape analysis that is invariant.