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Advances in Numerical Analysis
Volume 2012 (2012), Article ID 579050, 15 pages
http://dx.doi.org/10.1155/2012/579050
Research Article

Interpreting the Phase Spectrum in Fourier Analysis of Partial Ranking Data

School of Computer Engineering, Nanyang Technological University, Singapore 637665

Received 28 September 2011; Revised 14 February 2012; Accepted 23 February 2012

Academic Editor: Mustapha Ait Rami

Copyright © 2012 Ramakrishna Kakarala. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Whenever ranking data are collected, such as in elections, surveys, and database searches, it is frequently the case that partial rankings are available instead of, or sometimes in addition to, full rankings. Statistical methods for partial rankings have been discussed in the literature. However, there has been relatively little published on their Fourier analysis, perhaps because the abstract nature of the transforms involved impede insight. This paper provides as its novel contributions an analysis of the Fourier transform for partial rankings, with particular attention to the first three ranks, while emphasizing on basic signal processing properties of transform magnitude and phase. It shows that the transform and its magnitude satisfy a projection invariance and analyzes the reconstruction of data from either magnitude or phase alone. The analysis is motivated by appealing to corresponding properties of the familiar DFT and by application to two real-world data sets.