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首页> 外文期刊>The Open Statistics & Probability Journal >Closeness Coefficients between Euclidean-Embeddable HomologousConfigurations
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Closeness Coefficients between Euclidean-Embeddable HomologousConfigurations

机译：欧几里德可嵌入的同源配置之间的紧密度系数

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摘要

Measurement of closeness between homologous configurations is often of interest. For configurations that canbe embedded onto the Euclidean space, we attempted to develop closeness coefficients between corresponding Euclideancoordinate matrices. A suitable closeness coefficient was required to satisfy the following five properties: 1) It must rangebetween 0 and 1; 2) It must be invariant over translation, rotation and dilation of coordinate matrices, namely, TRDinvariance;3) It must be one between equivalent coordinate matrices; 4) It must be zero between coordinate matriceswhose corresponding configurations are orthogonal; and 5) It must be symmetric between any pair of coordinate matrices.We showed that the following two closeness coefficients derived based on different approaches were equivalent and bothsatisfied the five required properties: 1) a goodness of fit coefficient GF based on minimum distance fitting of coordinatematrices by translation, rotation and dilation; and 2) the Gower-Lingoes-Sch?nenman coefficient RGLS based on themaximum of correlations of coordinate matrices over rotation. In addition, the Escoufier’s RV coefficient was also shownto satisfy all the five properties. Finally, RGLS, or equivalently GF, and RV were all shown to be a function of centeredforms or singular values of coordinate matrices.
机译：同源构型之间的紧密度的测量通常是令人感兴趣的。对于可以嵌入到欧几里得空间上的配置，我们尝试在相应的欧几里得坐标矩阵之间建立紧密度系数。需要满足以下五个特性的合适的闭合系数：1）必须在0到1之间; 2）必须在坐标矩阵的平移，旋转和扩张上不变，即TRD不变性; 3）必须在等效坐标矩阵之间是一个; 4）坐标矩阵之间的正交配置必须为零; 5）在任何一对坐标矩阵之间必须是对称的。我们表明，基于不同方法得出的以下两个接近系数是等效的，并且都满足了五个必需的特性：1）基于最小距离拟合的拟合系数GF的良好性通过平移，旋转和膨胀的坐标矩阵; 2）基于坐标矩阵相对于旋转的最大相关性的高尔-林格斯-舒曼曼系数RGLS。此外，Escoufier的RV系数也显示满足所有这五个属性。最后，RGLS或等效的GF和RV均显示为坐标矩阵的中心形式或奇异值的函数。

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