Comparative analysis of the original and amplitude permutations

TL;DR

This paper compares the original and amplitude permutations, revealing their distinct roles in characterizing vector structures and their relation to time- and amplitude-symmetric vectors, with implications for symbolic and topological data analysis.

Contribution

It clarifies the relationship between original and amplitude permutations and their connection to symmetric vectors, enhancing understanding in symbolic and topological data analysis.

Findings

Amplitude permutation reflects the vector's temporal structure.

Original permutation conveys the structural pattern of the reorganized vector.

The study clarifies the association with time- and amplitude-symmetric vectors.

Abstract

The original and amplitude permutations are two basic ordinal patterns; however, their relationship has received little attention. This paper compares the original and amplitude permutations used to characterize vector structures. To accurately convey the vector structure, we modify indexes of equal values in the permutations to be the same ones in each group of equalities. Comparative analysis suggests that the amplitude permutation, comprising the positions of the original values in the reordered vector, directly reflects the vector's temporal structure, whereas the original permutation, consisting of the indexes of reorganized values in the original vector, conveys the structural pattern of the reorganized vector. Moreover, we clarify the association of the original and amplitude permutations with timeand amplitude-symmetric vectors, thus contributing to the fields of symbolic analysis, topological data analysis, and so on.