Permutation-Based Distances for Groups and Group-Valued Time Series

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Abstract
Permutations on a set, endowed with function composition, build a group called a symmetric group. In addition to their algebraic structure, symmetric groups have two metrics that are of particular interest to us here: the Cayley distance and the Kendall tau distance. In fact, the aim of this paper is to introduce the concept of distance in a general finite group based on them. The main tool that we use to this end is Cayley’s theorem, which states that any finite group is isomorphic to a subgroup of a certain symmetric group. We also discuss the advantages and disadvantage of these permutation-based distances compared to the conventional generator-based distances in finite groups. The reason why we are interested in distances on groups is that finite groups appear in symbolic representations of time series, most notably in the so-called ordinal representations, whose symbols are precisely permutations, usually called ordinal patterns in that context. The natural extension from groups to group-valued time series is also discussed, as well as how such metric tools can be applied in time series analysis. Both theory and applications are illustrated with examples and numerical simulations.
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