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

Area of Science:

• Group Theory
• Time Series Analysis
• Computational Mathematics

Background:

• Symmetric groups are formed by permutations under function composition.
• These groups possess algebraic structures and relevant metrics like Cayley and Kendall tau distances.
• Finite groups are fundamental in symbolic time series representations, particularly ordinal representations.

Purpose of the Study:

• To introduce a general concept of distance in finite groups.
• To leverage permutation-based distances (Cayley, Kendall tau) for group analysis.
• To explore applications of these distances in group-valued time series analysis.

Main Methods:

• Utilizing Cayley's theorem to establish group isomorphism to symmetric group subgroups.
• Defining and applying permutation-based distances within finite groups.
• Extending metric concepts to group-valued time series.

Main Results:

• A framework for defining distances on general finite groups is presented.
• Comparison of permutation-based distances against conventional generator-based distances.
• Demonstration of the utility of these metric tools in time series analysis.

Conclusions:

• Permutation-based distances offer a novel approach to quantifying relationships within finite groups.
• The developed methods provide valuable tools for analyzing complex time series data.
• The study bridges abstract group theory with practical data analysis applications.