ON THE SYMMETRICAL PROPERTY OF PROCRUSTES MEASURE OF DISTANCE

T Bakhtiar,Siswadi Siswadi

doi:10.12732/ijpam.v99i3.7

Abstract

Measurement the degree of difference between two matrices by using Procrustes analysis is preceded by a series of Euclidean similarity trans- formations namely translation, rotation, and dilation, performed in respected order, for gaining maximal matching. It is easy, by a counter example, to show that Procrustes measure does not obey the symmetrical property, something should be owned by any distance function. In this paper we analytically proved that normalization over configuration matrices as an additional transformation results in the satisfaction of the symmetrical property by Procrustes analysis. We also proved that normalization can be undertaken prior to or after rotation to preserve symmetrical property. Moreover, we proved that Procrustes mea- sure can be expressed in term of singular values of the matrix. We here very much exploited the characteristic of full singular value decomposition under similarity transformations.

Highlights

Measurement the degree of difference between two matrices by using Procrustes analysis is preceded by a series of Euclidean similarity transformations namely translation, rotation, and dilation, performed in respected order, for gaining maximal matching
Procrustes analysis refers to a technique of comparing objects with different shapes and producing a measure of the match
If zero eigenvalues are included along with their corresponding eigenvectors, we have an full singular value decomposition (FSVD) of matrix A as given by Theorem 2

Summary

Procrustes Measure

Shape is all the geometrical information that remains when location, rotational, Received: November 1, 2014 §Correspondence author c 2015 Academic Publications, Ltd. url: www.acadpubl.eu. Procrustes analysis refers to a technique of comparing objects with different shapes and producing a measure of the match It eliminates possible incommensurability of variables within the individual data sets and size differences between data sets by employing data and configuration scalings in calculating the distance, respectively. Dilation on YTQ over XT is undertaken by multiplying YTQ by a scalar c ∈ R, where c It is proved in [1] that a series of similarity transformations in the order of translation-rotation-dilation provides the lowest possible distance as stated in the following theorem, where the distance is calculated according to (1) as follows d(XT, cYTQ) = tr(XT − cYTQ)′(XT − cYTQ). Given two matrices X and Y in Rm×n, the Procrustes measure p between X and Y after optimal translation-rotation-dilation process is provided by p(X,.

Singular Value Decomposition

Symmetrical Property

When Should We Normalize?

Conclusion