Bandt–Pompe Symbolization Methodology

• Bandt–Pompe symbolization methodology is a framework that transforms time series into ordinal patterns capturing the causal structure and temporal ordering of data.
• It creates a probability distribution over these patterns to measure dynamical complexity and entropy, offering robust insights even under monotonic transformations.
• The method supports applications in chaos detection, time series clustering, and dynamical regime discrimination by distinguishing determinism from randomness.

The Bandt–Pompe symbolization methodology is a framework for transforming real-valued time series into symbolic sequences by focusing on ordinal patterns—the relative temporal ordering of data points. This procedure constructs a probability distribution over these patterns, enabling the rigorous quantification of system complexity, entropy, and underlying dynamical structures. The approach is robust to monotonic transformations, can faithfully represent causal information, and under specific conditions even allows estimation of Kolmogorov–Sinai entropy from coarse ordinal data alone. The methodology has found widespread application in discriminating determinism from randomness, time series clustering, dynamical reconstruction, and information-theoretic characterizations across diverse domains in science and engineering.

1. Extraction of Ordinal Patterns and Probability Distribution

Given a scalar time series $\{x_t\}_{t=1}^N$, the Bandt–Pompe method encodes the local dynamics into ordinal patterns. For each position $s$, consider the $d$-dimensional vector

$(x_{s-(d-1)T},\ x_{s-(d-2)T},\ ..., x_{s-T}, x_s),$

where $d$ is the embedding dimension and $T$ the delay (often $T=1$). The associated ordinal pattern, or permutation $\pi$, is defined by the ranking of these $d$ values in ascending order, with ties broken by the smallest index appearing first.

By sliding the window across the entire time series, one assigns to each segment a permutation $\pi_i \in S_d$ (the symmetric group). Let $P = \{p(\pi_i)\}_{i=1}^{d!}$ denote the distribution over ordinal patterns, with

$$p(\pi_i) = \frac{\text{number of segments with pattern }\pi_i}{N - (d-1)T}.$$

This ordinal probability distribution (henceforth BP-PDF) retains the causal, temporal ordering structure of the original process (1105.3927).

2. Information-Theoretic Quantifiers: Entropy and Statistical Complexity

Once the BP-PDF is defined, a range of quantifiers can be computed:

• Permutation Entropy:

$H[P] = -\sum_{i=1}^{d!} p(\pi_i) \ln p(\pi_i),$

frequently normalized by $\ln d!$ to yield $0 \leq H[P] \leq 1$.

• Statistical Complexity:

$C[P] = Q_J[P, P_e] \cdot H[P],$

where $Q_J$ is based on the Jensen–Shannon divergence between $P$ and the uniform distribution $P_e = \{\frac{1}{d!}, ..., \frac{1}{d!}\}$:

$$Q_J[P, P_e] = Q_0 \left(S\left[ \frac{P + P_e}{2} \right] - \frac{1}{2} S[P] - \frac{1}{2} S[P_e]\right),$$

with $Q_0$ normalizing $Q_J \in [0,1]$ (1105.3927, 1304.0399, Aquino et al., 2015). Complexity vanishes for completely ordered and completely random systems; it peaks at intermediate structural richness.

• Permutation Fisher Information:

Sensitive to local probability gradients, often computed as

$$F[P] = F_0 \sum_{i=1}^{d!-1} [\sqrt{p_{i+1}} - \sqrt{p_i}]^2,$$

with $F_0$ for normalization (Rosso et al., 2015, Rosso et al., 2016).

3. Forbidden and Missing Patterns: Signatures of Determinism

In deterministic chaotic systems certain ordinal patterns are "forbidden," never being realized due to the underlying dynamics. In contrast, for stochastic processes (white noise), as $N \to \infty$, all $d!$ patterns are observed with roughly equal frequency. The persistence of forbidden (or, for finite $N$, missing) patterns provides a diagnostic for determinism (1110.0776, Aronis et al., 2018). The decay of missing ordinal patterns $\mathcal{M}(N,d)$ as a function of series length under noise contamination indicates the relative strength of deterministic and stochastic components.

The analysis of the trajectory of the system's representative point in the complexity–entropy plane under increasing noise amplitude (and its correlation degree) discriminates between purely stochastic, deterministic, and mixed regimes (1110.0776).

4. Choice of Embedding Parameters and Sampling: Optimality Criteria

The selection of the embedding dimension $d$ and delay $T$ is critical. Typical values are $d \in [3,7]$ and $T=1$, though they may be optimized for problem-specific dynamics. Additionally, the method enables principled selection of an optimal sampling period for continuous-time dynamics:

• For chaotic attractors, the statistical complexity (as computed from BP-PDFs) exhibits a clear maximum at an intermediate sampling period $t_M$ (1105.3927). Oversampling ($\tau \ll t_M$) increases redundancy and order, leading to low entropy and complexity; undersampling ($\tau \gg t_M$) erases correlations, resulting in high entropy but also low complexity.
• The maximum of complexity at $t_M$ robustly indicates the optimal sampling for reconstructing attractor dynamics. This $t_M$ aligns with Takens’ delay time recommendations and is compatible with the (majority power) Nyquist–Shannon criterion.

5. Ordinal Analysis Extensions: Multivariate, Complexity-Entropy Planes, and Networks

Multivariate and Bivariate Ordinal Patterns

The methodology generalizes to multivariate signals, either by assigning ordinal patterns to each variable independently and analyzing their joint statistics, or by constructing bivariate ordinal measures to estimate coupling, directionality, and evolving network properties (e.g., phase synchronization estimation via ordinal vectors) (Lehnertz, 2023).

Complexity–Entropy and Fisher–Shannon Planes

Plotting entropy against complexity (or Fisher information) creates diagnostic planes that visually and quantitatively discriminate periodic, random, and chaotic regimes (Weck et al., 2014, Rosso et al., 2015, Zhu et al., 2016). In these planes:

• White noise tends toward maximal entropy, minimal complexity.
• Low-dimensional chaos resides in regions of moderate entropy and higher complexity.
• Statistical complexity and entropy together, or in combination with Fisher information, provide sensitive detection of subtle dynamical transitions.

Ordinal-Pattern Transition Networks

Rather than considering only the distribution of ordinal patterns, transitions between them form a Markov process whose transition probabilities encode further dynamical structure ("ordinal networks") (Flores et al., 12 Jul 2025). These transition matrices can be vectorized for community detection or similarity analysis, enabling robust unsupervised classification of complex dynamics (e.g., in the clustering of chaotic vs. tumbling falling paper trajectories).

6. Robustness, Implementation Considerations, and Extensions

Structural properties of the Bandt–Pompe approach confer significant advantages:

• Noise Robustness: Ordinal patterns are invariant under monotonic transformations and relatively insensitive to amplitude noise.
• Handling of Tied Values: Several methods (chronological/rank extended alphabets, imputation, Bayesian schemes) have been developed to handle ties in discrete or quantized datasets (Traversaro et al., 2017).
• Real-Time and High-Dimensional Settings: The core algorithm is computationally efficient, does not require amplitude quantization thresholds, and is suited for online or high-dimensional applications. In the context of multivariate (e.g., scalp EEG or LFP arrays) or streaming data, distributed implementations can be constructed.

Advanced implementations combine Bandt–Pompe symbolization with other measures—e.g., Lempel–Ziv algorithmic complexity (Lempel–Ziv permutation complexity)—to capture both local (statistical) and sequential (algorithmic) aspects (Zozor et al., 2013, Mateos et al., 2017).

7. Application Highlights and Open Challenges

Applications include:

• Turbulent flows and plasma physics (e.g., benchmarking stochasticity in MHD turbulence vs. solar wind) (Weck et al., 2014).
• Biomedical signal analysis (EEG, HRV, stress response, atrial fibrillation determinism) (Rosso et al., 2016, Aronis et al., 2018).
• Traffic modeling and transport systems (Aquino et al., 2015).
• Dynamical regime detection and transition localization in semiclassical and mixed quantum–classical systems (Gonzalez et al., 21 Nov 2024).
• Clustering of experimental and simulated trajectory data (e.g., falling paper motion) via ordinal-pattern transition similarity networks (Flores et al., 12 Jul 2025).
• Neural encoding and functional network inference in brain dynamics (Lehnertz, 2023, Carlos et al., 22 Jul 2025).

Open challenges encompass:

• Deeper understanding of ordinal probability redundancy and its effect on feature selection for machine learning (Leyva et al., 2022).
• Optimal embedding parameter selection and statistical confidence assessment.
• Multiscale, multivariate, and nonstationary generalizations.
• Reliable surrogate data techniques and confound mitigation in neurophysiological contexts.
• Integration with advanced model identification and prediction/classification algorithms (Leyva et al., 2022).

Summary Table: Core Steps in Bandt–Pompe Symbolization

Step	Formula / Procedure	Typical Parameter Range
Build lagged vectors	$(x_{s-(d-1)T},...,x_{s-T},x_s)$	$d=3...7$, $T=1$
Assign ordinal pattern	Map vector to permutation $\pi$ that sorts the values in ascending order	$d!$ possible permutations
Estimate BP-PDF	$$p(\pi_i) = \text{fraction of segments with pattern }\pi_i$$	$p(\pi_i) \geq 0, \sum p(\pi_i)=1$
Compute entropy (normalized)	$H[P] = -\sum p(\pi_i)\ln p(\pi_i)/\ln d!$	$0 \leq H[P] \leq 1$
Compute statistical complexity	$C[P] = Q_J[P,P_e]\, H[P]$, with $Q_J$ from Jensen–Shannon divergence	$0 \leq C[P] \leq 1$

In conclusion, Bandt–Pompe symbolization provides a mathematically principled and empirically robust foundation for ordinal time series analysis. It translates raw time series into a symbolic domain capturing causal, dynamic information, supporting advanced characterization, detection, and classification tasks. The method continues to underpin state-of-the-art diagnostics in complex system dynamics, with ongoing developments targeting multivariate, network, and real-time analytic extensions.