Local Permutation Entropy Analysis

• Local permutation entropy is a method that computes the normalized Shannon entropy of ordinal-pattern distributions within localized signal windows.
• It employs the Bandt–Pompe framework to map time series into ordinal sequences, ensuring analyses are amplitude agnostic and robust to noise.
• Windowed applications balance parameters such as window size and embedding dimension, providing detailed temporal insights despite finite-sample constraints.

Searching arXiv for recent and foundational work on permutation entropy and local/windowed variants relevant to Local Permutation Entropy. Local permutation entropy is a localized form of permutation entropy in which the Shannon entropy of ordinal-pattern frequencies is computed on restricted neighborhoods of a signal rather than on an entire record. In the literature, the exact term is often not introduced as a standalone formal object; closely related constructions appear instead as windowed permutation entropy, permutation entropy profiles, sliding-window PE, multi-segment PE, and node-centered ordinal analyses on graphs. The common basis is the Bandt–Pompe representation, in which a real-valued signal is converted into an ordinal symbolic sequence and entropy is evaluated on the empirical distribution of those local order patterns (Mateos et al., 2017, Kay et al., 2023, Fabila-Carrasco et al., 2021).

1. Ordinal-pattern foundations

For a real-valued discrete-time series $\{X_t\}_{t\in\mathbb N}$, the Bandt–Pompe construction uses an embedding dimension $d \ge 2$ and delay $\tau \ge 1$ to form delay vectors

$$Y_t^{d,\tau} = [X_{t-(d-1)\tau}\ \cdots\ X_{t-\tau}\ X_t].$$

An equivalent notation used in signal-analysis work is

$$A_{n,\tau}^k(\mathbf{x}) := (x_k, x_{k+\tau}, x_{k+2\tau}, \ldots, x_{k+(n-1)\tau}),$$

where $n$ is the embedding dimension, $\tau$ is the delay, and $k$ is the starting index. Each embedded vector is mapped to an ordinal pattern by replacing values with their ranks after sorting. The ordinal alphabet is the set of all permutations of length $d$ or $n$, with cardinality $d \ge 2$0 or $d \ge 2$1 (Mateos et al., 2017, Kay et al., 2023).

Permutation entropy is then the Shannon entropy of the empirical distribution of these ordinal patterns. In the notation of the complexity–entropy-plane literature,

$d \ge 2$2

where $d \ge 2$3 is the empirical frequency of pattern $d \ge 2$4. Because the logarithm is taken in base $d \ge 2$5, the quantity is normalized to $d \ge 2$6. In the signal-analysis notation,

$d \ge 2$7

so again $d \ge 2$8 (Mateos et al., 2017, Kay et al., 2023).

The definition is ordinal rather than metric: it depends on the relative order of values, not on their amplitudes. In high-precision physical measurements, exact ties are treated as unlikely; when ties occur because of limited precision, one convention is to break them by order of appearance in the time series. The same stable-ranking convention also appears in software-oriented presentations of Bandt–Pompe methods (Kay et al., 2023, Pessa et al., 2021).

2. Windowed and localized constructions

The most direct operational form of local permutation entropy is a windowed PE profile. Given a signal $d \ge 2$9, a window length $\tau \ge 1$0, and an overlap proportion $\tau \ge 1$1, window start indices are defined by

$\tau \ge 1$2

and the $\tau \ge 1$3th window is

$\tau \ge 1$4

For fixed $\tau \ge 1$5, the windowed entropy profile is

$\tau \ge 1$6

This $\tau \ge 1$7-permutation entropy profile is the clearest published analogue of local permutation entropy in one-dimensional signal analysis (Kay et al., 2023).

A natural operational definition, consistent with that framework, is

$\tau \ge 1$8

Expanded in terms of local pattern probabilities $\tau \ge 1$9, this becomes

$$Y_t^{d,\tau} = [X_{t-(d-1)\tau}\ \cdots\ X_{t-\tau}\ X_t].$$0

In plain terms, one computes normalized PE separately on each local window and arranges the results in temporal order (Kay et al., 2023).

The complexity–entropy-plane work did not explicitly introduce local or sliding-window permutation entropy, but it supplies the same ingredients in global form. Applying the Bandt–Pompe mapping, empirical frequency estimation, and normalized entropy formula inside each successive interval yields a time-resolved entropy trace without altering the underlying definition. That localizability is an inference from the methodology rather than a separately named construction in that paper (Mateos et al., 2017).

3. Parameters, estimation, and finite-sample constraints

Local permutation entropy is controlled by the same parameters as global PE—embedding dimension, delay, and ordinal frequency estimation—but finite-sample effects are more severe because every estimate is made inside a shorter window. A larger embedding dimension provides a richer ordinal alphabet, yet $$Y_t^{d,\tau} = [X_{t-(d-1)\tau}\ \cdots\ X_{t-\tau}\ X_t].$$1 or $$Y_t^{d,\tau} = [X_{t-(d-1)\tau}\ \cdots\ X_{t-\tau}\ X_t].$$2 grows rapidly. The complexity–entropy-plane literature explicitly warns that when $$Y_t^{d,\tau} = [X_{t-(d-1)\tau}\ \cdots\ X_{t-\tau}\ X_t].$$3 becomes very large, “the sequences to be analyzed must be drastically large to insure that both the permutation complexity and permutation entropy have a meaning.” In windowed analysis, the full record length is replaced by the window length, so only $$Y_t^{d,\tau} = [X_{t-(d-1)\tau}\ \cdots\ X_{t-\tau}\ X_t].$$4 or $$Y_t^{d,\tau} = [X_{t-(d-1)\tau}\ \cdots\ X_{t-\tau}\ X_t].$$5 ordinal patterns are available locally (Mateos et al., 2017, Kay et al., 2023).

The delay $$Y_t^{d,\tau} = [X_{t-(d-1)\tau}\ \cdots\ X_{t-\tau}\ X_t].$$6 sets the temporal scale of each ordinal motif. Increasing $$Y_t^{d,\tau} = [X_{t-(d-1)\tau}\ \cdots\ X_{t-\tau}\ X_t].$$7 makes each pattern span a longer physical interval, but it also reduces the number of admissible embeddings inside a fixed window. The signal-analysis literature therefore emphasizes a direct trade-off: if the window $$Y_t^{d,\tau} = [X_{t-(d-1)\tau}\ \cdots\ X_{t-\tau}\ X_t].$$8 is too small, the pattern distribution is not very informative; if $$Y_t^{d,\tau} = [X_{t-(d-1)\tau}\ \cdots\ X_{t-\tau}\ X_t].$$9 is too large, locality is reduced and computation increases. Likewise, if the overlap $$A_{n,\tau}^k(\mathbf{x}) := (x_k, x_{k+\tau}, x_{k+2\tau}, \ldots, x_{k+(n-1)\tau}),$$0 is too small, the PE sequence is coarse and may vary greatly from term to term, whereas larger $$A_{n,\tau}^k(\mathbf{x}) := (x_k, x_{k+\tau}, x_{k+2\tau}, \ldots, x_{k+(n-1)\tau}),$$1 yields finer temporal sampling at higher computational cost (Kay et al., 2023).

Experimental studies that use global PE often rely on much longer records than are available in local applications. In one complexity–entropy study, realizations had length $$A_{n,\tau}^k(\mathbf{x}) := (x_k, x_{k+\tau}, x_{k+2\tau}, \ldots, x_{k+(n-1)\tau}),$$2, and the main experiments used $$A_{n,\tau}^k(\mathbf{x}) := (x_k, x_{k+\tau}, x_{k+2\tau}, \ldots, x_{k+(n-1)\tau}),$$3, with additional tests at $$A_{n,\tau}^k(\mathbf{x}) := (x_k, x_{k+\tau}, x_{k+2\tau}, \ldots, x_{k+(n-1)\tau}),$$4 and $$A_{n,\tau}^k(\mathbf{x}) := (x_k, x_{k+\tau}, x_{k+2\tau}, \ldots, x_{k+(n-1)\tau}),$$5. The reported qualitative arrangements in the complexity–entropy plane were similar for $$A_{n,\tau}^k(\mathbf{x}) := (x_k, x_{k+\tau}, x_{k+2\tau}, \ldots, x_{k+(n-1)\tau}),$$6, suggesting robustness to moderate dimension changes, but the same study also notes that some quantization effects tend to disappear only when $$A_{n,\tau}^k(\mathbf{x}) := (x_k, x_{k+\tau}, x_{k+2\tau}, \ldots, x_{k+(n-1)\tau}),$$7, immediately followed by the caution that high $$A_{n,\tau}^k(\mathbf{x}) := (x_k, x_{k+\tau}, x_{k+2\tau}, \ldots, x_{k+(n-1)\tau}),$$8 requires much longer data because $$A_{n,\tau}^k(\mathbf{x}) := (x_k, x_{k+\tau}, x_{k+2\tau}, \ldots, x_{k+(n-1)\tau}),$$9 becomes huge (Mateos et al., 2017).

Automatic parameter-selection work on standard PE reinforces the same constraints. It concludes that the success of any automatic method depends on the category of the studied system: a frequency-domain approach based on the least median of squares and the Fourier spectrum is accurate for periodic systems, nonlinear difference equations, and ECG/EEG data, whereas the mutual information function computed using adaptive partitions is most accurate for chaotic differential equations. For the permutation dimension $n$0, both False Nearest Neighbors and MPE provide accurate values for most systems, with $n$1 being suitable in most cases (Myers et al., 2019). This suggests that local analyses ordinarily benefit from fixing $n$2 and $n$3 from representative data before windowing, because windowing sharply reduces sample support.

4. Interpretation and common pitfalls

Permutation entropy measures the spread of the ordinal-pattern distribution. High PE means that the permutation distribution is close to uniform; low PE means that only a subset of patterns dominate. Because PE depends on relative order rather than absolute amplitudes, it is amplitude agnostic in the global sense emphasized in signal-analysis work: signals at different scales can be compared so long as the ordering structure is preserved (Kay et al., 2023).

That interpretation is incomplete if local PE is treated as a direct proxy for dynamical mechanism. The complexity–entropy-plane literature makes a central distinction between the statistical component of a signal, captured by permutation entropy, and the temporal organization of the ordinal sequence, captured by permutation Lempel–Ziv complexity. On that view, PE records uncertainty in pattern frequencies but not the temporal organization of those patterns beyond frequencies. This is why entropy alone may not decisively distinguish chaos from randomness, even though forbidden patterns can reduce PE in some chaotic systems. The same literature explicitly cautions that not all chaotic maps have forbidden patterns and some noisy sequences may also exhibit them (Mateos et al., 2017).

Several concrete misconceptions follow. A local rise in PE indicates a more diverse local ordinal-pattern distribution, but it does not by itself determine whether the local regime has become more random, more structured, more persistent, or more anti-persistent. For fractional Gaussian noise, permutation entropy for $n$4 and $n$5 is reported as “more or less identical,” whereas permutation Lempel–Ziv complexity distinguishes them. A plausible implication is that local PE should be interpreted more cautiously when the question concerns the sign of dependence rather than the amount of ordinal uncertainty (Mateos et al., 2017).

A further caution comes from null-model analysis. Standard normalized permutation entropy is equivalent to a comparison with the uniform distribution over permutations, which is appropriate for i.i.d. continuous noise. For random walks, however, the ordinal-pattern distribution is not uniform even under the null. The random-walk literature therefore implies that a low entropy in a short window can reflect ordinary random-walk ordering bias rather than local predictability or determinism. This is especially relevant in price-like or integrated time series, where local departures from uniformity need not indicate anomalous dynamics (Deford et al., 2017).

5. Empirical uses in signal analysis and anomaly detection

Local or segment-wise PE has been used most explicitly as a time-evolving complexity descriptor in signal analysis. In radio-frequency classification, permutation entropy was evaluated not only on the full signal but also on coarse segmentations into 2 and 4 windows, producing 7 segments in total. For each segment, multi-scale permutation entropy was computed for

$n$6

This yielded a $n$7 matrix per segment, flattened to length 40, and a 280-dimensional feature vector across all 7 segments. The resulting MSPE-based CNN achieved about $n$8–$n$9 average classification accuracy, compared with $\tau$0 for raw waveform input to a small CNN and $\tau$1 for spectrogram input (Kay et al., 2023).

Sliding-window PE and WPE have also been used as forensic anomaly detectors in paleoclimate records. In deep polar ice-core isotope data, PE and WPE were computed in rolling 2400-point windows after temporal regularization to $\tau$2 year per point, so each window summarized about 120 years of data. Using $\tau$3 and $\tau$4, the local entropy traces detected abrupt complexity changes later linked to an older laboratory instrument, interpolation artifacts, a GUI freeze during analysis, and a CRDS-CFA malfunction. A remeasurement of the 4.5–6.5 kybp segment removed the large square-wave anomaly from the PE/WPE traces, showing that localized ordinal-entropy profiles can identify sections requiring targeted re-analysis (Garland et al., 2018).

In local-mixing detection, the core diagnostic is not a single windowed PE trace but a family of PE traces indexed by embedding stride $\tau$5. With $\tau$6, window length $\tau$7, and $\tau$8, the method evaluates whether the usual ordering of PE across $\tau$9 is reversed at small scales. A reversal metric

$k$0

quantifies this effect, and the average $k$1 is then tracked as a function of non-overlapping bin size $k$2. The heuristic is to choose the first bin size where $k$3 goes to zero, or the first minimum if it never reaches zero. On Antarctic ice-core data, this procedure recovered a broad minimum at $k$4, close to the 17-point binning chosen by laboratory experts; on Mauna Loa methane, the first minimum occurred at $k$5, corresponding to one hour (Neuder et al., 2020).

Physical measurement applications further emphasize tie handling and weighted variants. In full-scale fire-test temperature records, several permutation-based entropies were compared, including PE, WPE, modified PE, two-length PE, and weighted variants. The weighted entropies were reported to provide “clearer and more accurate” results, and all principal non-modified methods identified thermocouple $k$6 as the most disordered location. The same study also found modified entropies less suitable for those data because $k$7 had fewer equalities, underscoring that local entropy estimation in tie-prone measurements depends strongly on the interaction between sensor resolution, sampling interval, and ordinal encoding rule (Mitroi-Symeonidis et al., 2019).

6. Generalizations, software, and pointwise viewpoints

Local permutation entropy extends naturally beyond one-dimensional time series. In graph-signal analysis, the graph permutation entropy framework constructs a local embedding at each node,

$k$8

where the coordinates summarize the signal over increasingly distant graph neighborhoods. One ordinal pattern is obtained per node, and the published graph-level entropy is the Shannon entropy of the global histogram of these nodewise patterns. The method is therefore local in pattern formation and global in aggregation. A plausible implication is that a truly local graph permutation entropy would retain the nodewise patterns and replace the global histogram by a histogram computed in a neighborhood around each node (Fabila-Carrasco et al., 2021).

Software support for localized PE is available primarily through repeated application of global Bandt–Pompe routines. The Python package ordpy implements permutation_entropy, ordinal_sequence, ordinal_distribution, tsallis_entropy, renyi_entropy, missing_patterns, and related methods for one-dimensional and two-dimensional data. It does not provide a dedicated local-permutation-entropy function, but it explicitly reproduces a transient logistic-map experiment in which normalized permutation entropy is calculated within a sliding window of 1024 observations, and its core functions support moving-window or patchwise computation on time series and images (Pessa et al., 2021).

A more abstract extension appears in permutation-symbolic local information dynamics. Symbolic local transfer entropies replace raw continuous values by ordinal symbols and evaluate pointwise log-ratio information measures at each spacetime location. Although that framework targets local information transfer rather than local entropy, it shows how permutation-symbolic descriptions support pointwise information quantities in continuous-state spatiotemporal systems. A plausible implication is that a pointwise local permutation-entropy analogue can be read as the surprise of an observed ordinal symbol, rather than as a window average (Nakajima et al., 2014).

This pointwise interpretation is also consistent with finite-state duality results on values and orderings. In that setting, block permutation entropy is the Shannon entropy of permutation probabilities $k$9, and the value-to-ordering map $d$0 precisely characterizes how many value blocks collapse onto the same ordinal pattern. A plausible pointwise counterpart of block permutation entropy is therefore the local self-information $d$1 of the realized ordinal pattern. The published theory does not name this quantity “local permutation entropy,” but it provides the exact block-probability structure from which such a definition follows (Haruna et al., 2011).