A Signal Processing Approach to Fourier Analysis of Ranking Data: The Importance of Phase

Ranking data is a type of data obtained in some elections, in customer surveys, as well as from web search results. Such data may be considered as a type of signal defined on the group of permutations of n objects, denoted Sn. There exists a Fourier transform for Sn obtained from group representation theory, which is well known in the mathematics literature. However, previous work has not approached the transform from a signal processing perspective: in particular, there is no discussion of what constitutes “magnitude” and “phase,” nor any analysis of what phase information might tell us beyond a well-known connection to group translation. This paper explores the properties of the phase spectrum of ranking data; in particular, a novel contribution is the formulation of the bispectrum for ranking data, which may be used for studying phase linearity. Analysis of two well-known ranking data sets shows that they are surprisingly well fit by linear phase approximations.