A Comparison of Different Representations of Ordinal Patterns and Their Usability in Data Analysis
Alexander Schnurr, Angelika Silbernagel
TL;DR
The paper addresses how to represent ordinal patterns in time-series analysis and compares three core representations—permutation, rank, and inversion—across digital implementation, inverse (space/time) transformations, and ties handling for ordinal patterns of length $d$. It formalizes these representations, introduces explicit numeric encodings such as the Lehmer-like $n_{LC}$ and KSE-based $n_{KSE}$ codes, and analyzes computational costs to guide practical choice. A case study on SPX and VIX demonstrates the trade-offs in readability and efficiency, showing that rank representations offer intuitive interpretation while inversion-based encodings excel for digital processing, and that generalized rank representations are advantageous when ties are frequent. The work provides concrete recommendations and guidelines for selecting ordinal-pattern representations in real-world time-series analyses, including scenarios with categorical data or many ties.
Abstract
We describe and analyze different approaches to represent ordinal patterns. All of these can be found in the literature. The most important representations (plus sub-classes) are compared in terms of their applicability from different angles. Namely we consider digital implementation, inverse patterns and ties between values. At the end we provide a guideline on which occasions which representation should be used.