# Ordinal Spectrum: Mapping Ordinal Patterns into Frequency Domain

## TL;DR

This paper introduces a new method called ordinal spectrum to better understand complex time series data by capturing nonlinear patterns in the frequency domain.

## Contribution

The novel contribution is the development of the ordinal spectrum, a frequency-domain approach for analyzing nonlinear temporal structures in time series.

## Key findings

- The ordinal spectrum successfully identifies temporal scales of chaotic behavior in diverse datasets.
- It complements traditional spectral methods by revealing nonlinear dynamics that classical approaches may miss.
- The method is applicable across physical, biological, and astronomical time series data.

## Abstract

Classical spectral analysis characterizes linear systems effectively but often fails to reveal the nonlinear temporal structure of chaotic dynamics. We introduce the ordinal spectrum, a frequency-domain characterization derived from the ordinal-pattern representation of a time series. Applied to both synthetic and real-world datasets—including periodic, stochastic, and chaotic signals from physical, biological, and astronomical sources—the ordinal spectrum identifies the temporal scales implied in a possible chaotic behavior. By providing an interpretable, data-driven view of symbolic dynamics in the frequency domain, this approach complements state–space reconstructions and enhances the detection of nonlinear temporal organization that classical spectra may obscure. Its ability to distinguish between qualitatively different dynamics make it a useful tool for exploring complex time series across diverse scientific domains.