标题： 基于排列的群和群值时间序列的距离 显示英文标题

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

摘要： 在一个集合上的排列，赋予函数复合运算，构成一个称为对称群的群。 除了它们的代数结构外，对称群还有两种度量，这两种度量在此特别引起我们的兴趣：Cayley距离和Kendall tau距离。 事实上，本文的目的是基于这些概念，在一般的有限群中引入距离的概念。 为此，我们使用的主要工具是Cayley定理，该定理指出任何有限群都同构于某个对称群的子群。 我们还讨论了基于排列的距离与有限群中传统的生成器基距离相比的优势和劣势。 我们之所以对群上的距离感兴趣，是因为有限群出现在时间序列的符号表示中，最著名的是所谓的序表示，其符号正是排列，通常在该上下文中称为序模式。 还讨论了从群到群值时间序列的自然扩展，以及如何将这些度量工具应用于时间序列分析。 理论和应用都通过示例和数值模拟进行了说明。 显示英文摘要

摘要： 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.