Closeness Coefficients between Euclidean-Embeddable Homologous Configurations

Measurement of closeness between homologous configurations is often of interest. For configurations that can be embedded onto the Euclidean space, we attempted to develop closeness coefficients between corresponding Euclidean coordinate matrices. A suitable closeness coefficient was required to satisfy the following five properties: 1) It must range between 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 matrices whose 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 both satisfied the five required properties: 1) a goodness of fit coefficient GF based on minimum distance fitting of coordinate matrices by translation, rotation and dilation; and 2) the Gower-Lingoes-Schönenman coefficient RGLS based on the maximum of correlations of coordinate matrices over rotation. In addition, the Escoufier’s RV coefficient was also shown to satisfy all the five properties. Finally, RGLS, or equivalently GF, and RV were all shown to be a function of centered forms or singular values of coordinate matrices.