Ordinal Spectrum: Mapping Ordinal Patterns into Frequency Domain

Area of Science:

• Complex Systems Analysis
• Nonlinear Dynamics
• Time Series Analysis

Background:

• Classical spectral analysis excels with linear systems but struggles with nonlinear dynamics.
• Chaotic systems exhibit complex temporal structures often obscured by traditional methods.
• Understanding nonlinear temporal organization is crucial across diverse scientific fields.

Purpose of the Study:

• To introduce the ordinal spectrum, a new frequency-domain method for time series analysis.
• To demonstrate the ordinal spectrum's capability in identifying temporal scales within chaotic dynamics.
• To provide a data-driven approach for detecting nonlinear temporal organization.

Main Methods:

• Developed the ordinal spectrum based on the ordinal-pattern representation of time series data.
• Applied the ordinal spectrum to synthetic and real-world datasets (physical, biological, astronomical).
• Compared the ordinal spectrum's performance against classical spectral analysis and state-space reconstructions.

Main Results:

• The ordinal spectrum successfully identified temporal scales indicative of chaotic behavior.
• This method effectively distinguished between periodic, stochastic, and chaotic signals.
• The ordinal spectrum offers an interpretable, frequency-domain view of symbolic dynamics.

Conclusions:

• The ordinal spectrum is a valuable tool for exploring complex time series and detecting nonlinear temporal organization.
• It complements existing methods by revealing dynamics that classical spectra may miss.
• This approach enhances the analysis of chaotic dynamics across various scientific domains.