Embedded Dimension and Time Series Length. Practical Influence on Permutation Entropy and Its Applications

Area of Science:

• Time Series Analysis
• Complexity Science
• Biomedical Signal Processing

Background:

• Permutation Entropy (PE) is a vital time series complexity measure, widely applied in medicine.
• PE calculation typically requires time series length (N), embedded dimension (m), and delay (τ).
• Existing guidelines for parameter selection are general, lacking specific recommendations for optimal N, m, and τ.

Purpose of the Study:

• To investigate the practical implications of the N >> m! rule for Permutation Entropy.
• To analyze how Permutation Entropy varies with series length (N) and embedded dimension (m).
• To evaluate classification performance using PE with varying N and m across diverse datasets.

Main Methods:

• Analysis of Permutation Entropy variations across synthetic (random, spikes, logistic) and real-world (climatology, seismic, financial, biomedical) time series.
• Systematic variation of time series length (N) and embedded dimension (m) to assess their impact on PE.
• Classification performance assessment using PE-derived features with different N and m values.

Main Results:

• Shorter time series lengths than suggested by N >> m! are adequate for stable PE calculation.
• Robust classification is achievable with very short time series using PE measurements.
• Differences among time series classes become apparent even at minimal lengths.

Conclusions:

• The N >> m! guideline for Permutation Entropy may be overly stringent.
• Shorter time series lengths offer a viable alternative for stable PE computation and effective classification.
• The inherent properties of time series, such as forbidden patterns, facilitate early differentiation between classes.