IJPAM: Volume 99, No. 3 (2015)

ON THE SYMMETRICAL PROPERTY OF
PROCRUSTES MEASURE OF DISTANCE

Toni Bakhtiar$^1$, Siswadi$^2$
$^{1,2}$Department of Mathematics
Bogor Agricultural University
Dramaga Campus, Bogor, 16680, INDONESIA


Abstract. 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. 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 measure 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.

Received: November 1, 2014

AMS Subject Classification: 00A35, 62H20

Key Words and Phrases: procrustes analysis, similarity transformation, singular value decomposition, symmetrical property

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DOI: 10.12732/ijpam.v99i3.7 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2015
Volume: 99
Issue: 3
Pages: 315 - 324


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