Global Permutation Entropy (GPE)

• Global Permutation Entropy (GPE) is a complexity measure quantifying uncertainty and temporal structure in sequences by analyzing the distribution of ordinal patterns.
• It employs a robust, model-free approach that remains invariant under monotonic transformations, making it suitable for both time series and graph data.
• Advanced algorithmic techniques enable efficient extraction of permutation profiles, facilitating applications in biosignal analysis, chaotic systems, and networked data.

Global Permutation Entropy (GPE) is a family of complexity measures that quantifies the uncertainty and temporal structure of sequences by analyzing the distribution of ordinal patterns (permutations) derived from subsequences. Initially conceptualized as an analogue of excess entropy for finite-state stationary processes, GPE has evolved to encompass a broad spectrum of methods for real-valued series, random processes, and structured data on graphs. Central to GPE is the property of invariance under monotonic transformations, making it robust to amplitude perturbations and suitable as a model-free complexity index for dynamical and stochastic systems.

1. Formal Definitions and Generalizations

1.1 Permutation Patterns and Entropies

Given a finite-state stationary process $S = \{S_1, S_2, \ldots\}$ over an alphabet $A_n=\{1, 2, \ldots, n\}$, a length-$L$ word $s_1 \ldots s_L \in A^L$ maps to an ordinal pattern $\pi \in S_L$ under a ‘coarse-graining’ map $\phi: A^L \to S_L$. This construction generalizes naturally to real-valued processes by embedding subsequences of length $L$ (with or without time delay) and ranking their entries to record permutations.

The permutation entropy of order $L$ is defined as

$H^*(L) = -\sum_{\pi \in S_L} p(\pi) \log_2 p(\pi),$

where $p(\pi)$ is the empirical (or stationary) probability of pattern $\pi$. The corresponding permutation entropy rate is

$h^*(S) = \lim_{L \to \infty} \frac{H^*(L)}{L}.$

1.2 Permutation Excess Entropy (GPE)

The global or excess permutation entropy—hereafter GPE—is

$$E^*(S) = \lim_{L\to\infty} \left( H^*(L) - h^*(S)\, L \right),$$

whenever the limit exists. $E^*(S)$ characterizes global temporal correlations in the process, directly paralleling classical excess entropy defined from block Shannon entropy rates (Haruna et al., 2011).

1.3 Global Ordinal Pattern Entropy in Real-Valued Series

For real-valued time series $X = (X_1, ..., X_n)$, GPE can be formulated by aggregating statistics over all (possibly nonconsecutive) strictly increasing index subsequences of size $k$: $$\mathcal{I}_k = \left\{ (i_1, ..., i_k) : 1 \le i_1 < \dots < i_k \le n \right\}$$

$p(\sigma) = \frac{N(\sigma)}{\binom{n}{k}}$

$$H_{\mathrm{GPE}}(k) = -\sum_{\sigma \in S_k} p(\sigma) \log p(\sigma)$$

This “global” construction distinguishes GPE from classical permutation entropy, which counts only consecutive or fixed-delay patterns (Avhale et al., 27 Aug 2025).

2. The Duality Framework and Theoretical Underpinnings

A key theoretical mechanism is the duality between value sequences and their induced orderings (Galois connection via maps $\phi$ and $\mu$). For finite-state processes, this underpins the exact relationship between permutation and Shannon entropies: $$0 \le H(L) - H^*(L) \le \sum_{\pi: |\phi^{-1}(\pi)|>1} p(\pi) \log_2 |\phi^{-1}(\pi)|,$$ with the preimage size given by $|\phi^{-1}(\pi)| = \binom{L+n-i}{n-i}$, where $i$ is the position of the first appearance of a symbol in $\pi$ (Haruna et al., 2011).

For ergodic Markov processes, this yields the fundamental result: $h^*(S) = h(S), \qquad E^*(S) = E(S),$ making GPE a fully ordinal-based surrogate for classical excess entropy. In non-ergodic cases, $E^* < E$ can occur due to the existence of persistent ambiguities in the symbol-to-permutation coarse-graining (Haruna et al., 2011).

3. Modern Extensions: Group Entropy, Complexity Classes, and Rates

Traditional permutation metrics diverge for many stochastic processes due to factorial growth of allowed patterns ($\sim L!$). The permutation group entropy framework extends GPE by replacing classical Shannon or Rényi entropies with group entropic functionals $Z_{g,\alpha}(p)$, tailored to the process’s combinatorial complexity class:

• Exponential class: deterministic/chaotic, pattern growth $g(L) \sim c L$
• Factorial class: random/forbidden-pattern-free, $g(L) \sim L \ln L$

The permutation group entropy of order $L$ is

$$Z_{g,\alpha}^*\bigl(X_t^L\bigr) = g^{-1}(R_\alpha(p)) - g^{-1}(0), \quad R_\alpha(p)=\frac1{1-\alpha}\ln\sum_{r}p(r)^\alpha$$

and the Global Permutation Entropy rate is

$$z_{g,\alpha}^*(\mathbf X) = \lim_{L\to\infty} \frac{1}{L} Z_{g,\alpha}^*(X_t^L).$$

This construction ensures extensivity and resolves divergence issues, with rates finite and meaningful for both deterministic and random processes (Amigó et al., 20 Jan 2024).

4. Algorithmic Aspects and Computational Complexity

Efficient extraction of full permutation profiles for GPE requires counting all order-$k$ patterns among $\binom{n}{k}$ subsequences. Naïve enumeration is $\mathcal O(n^k)$, but advanced algorithms utilizing combinatorial data structures (corner trees, posets, etc.) achieve subquadratic or quadratic complexities for $k\le 7$ (Avhale et al., 27 Aug 2025). For graph signals, the computational cost is $O(N\,m\log m)$ for sorting steps, plus sparse matrix operations for neighborhood averaging (Fabila-Carrasco et al., 2021).

Typical parameter regimes restrict $k$ (or embedding dimension $L$) to $\le 7$ due to factorial growth of permutation classes. For real-time or high-throughput applications, efficient implementations via specialized libraries (e.g., Julia packages for sliding-window GPE computation) are available (Avhale et al., 27 Aug 2025).

5. Comparative Properties and Theoretical Guarantees

GPE exhibits several invariant and robust properties:

• Invariance: Under strictly monotone (order-preserving) transformations of the input. For graph GPE, invariance holds under affine amplitude changes.
• Noise robustness: GPE is insensitive to small additive noise, as ordinal structure typically persists.
• Parameter-free: No need for amplitude thresholds (contrast with Sample Entropy, ApEn).
• Extremal Behavior: $GPE(k) = 0$ for monotone signals; $GPE(k) \to 1$ for i.i.d. random sequences.
• Faster convergence: GPE converges to its limiting value more rapidly with $n$ than classical permutation entropy at $k > 2$ (Avhale et al., 27 Aug 2025, Amigó et al., 20 Jan 2024).

The table below summarizes the distinctions between classical permutation entropy, GPE for time series, and GPE for graph data:

Metric	Pattern Sampling	Domain
Classical PE	Consecutive/fixed-delay subseries	1D time series
GPE (Avhale et al.)	All strictly increasing index subsequences	1D time series
Graph GPE	Neighborhood-averaged vectors, vertex permutations	Arbitrary graphs

6. Applications and Empirical Performance

GPE and its variants are applied to a wide range of domains:

• Complexity detection: Distinguishing periodic, chaotic, and stochastic regimes in synthetic data (e.g., logistic map, MIX$_{2D}$ processes) (Fabila-Carrasco et al., 2021, Amigó et al., 20 Jan 2024).
• Biomedical signal analysis: GPE discriminates between physiological states (e.g., young vs elderly from heart-rate data) and is robust to measurement noise and nonstationarity (Fabila-Carrasco et al., 2021, Amigó et al., 20 Jan 2024).
• Communication signals: Permutation entropy features (often global) significantly improve classification of radio-frequency modulations in noisy environments compared to raw waveform or spectrogram-based approaches (Kay et al., 2023).
• Graph-structured data: GPE generalizes to sensor networks, social, infrastructure, and spatially embedded graphs, providing a model-free nonlinear complexity tool (Fabila-Carrasco et al., 2021).

Synthetic experiments demonstrate that GPE detects regime changes, identifies noise bursts with higher sensitivity than conventional PE, and exhibits superior noise-to-randomization convergence (Avhale et al., 27 Aug 2025).

7. Open Directions, Limitations, and Outlook

Key limitations include the factorial increase in required sample size and computational demands with embedding dimension ($L$ or $k$), constraining practical applications to moderate order. For non-ergodic or highly nonstationary processes, GPE and excess entropy can diverge, and selecting optimal parameters remains problem-specific (Haruna et al., 2011, Amigó et al., 20 Jan 2024). For graph signals, the construction of “optimal” graphs to best reveal dynamic correlations is unresolved (Fabila-Carrasco et al., 2021). Generalization to multivariate series, statistical significance assessments, and connections with information geometry and formal-group theory are active areas of research (Amigó et al., 20 Jan 2024).

A plausible implication is that GPE offers a unified ordinal-complexity framework, capable of bridging deterministic, stochastic, and networked signal analysis, with robust, model-free, and interpretable measures of global structure and unpredictability.