ExSEnt: Extrema-Segmented Entropy

• ExSEnt is a framework that segments time series at local extrema to isolate and quantify temporal spacing and amplitude variations, offering a decomposition of overall complexity.
• It computes three complementary entropy measures—temporal, amplitude, and joint entropy—using Sample Entropy to assess predictability in both individual and combined feature sequences.
• The method provides actionable insights in nonlinear dynamics and physiological applications by pinpointing whether irregularity arises from timing, magnitude, or their coupling.

Searching arXiv for ExSEnt and closely related entropy/extrema papers. Extrema-Segmented Entropy (ExSEnt) is a feature-decomposed framework for quantifying time-series complexity that separates temporal from amplitude contributions by partitioning a signal into monotonic segments and then computing entropy on segment-derived feature sequences. Introduced as a method that detects sign changes in the first-order increments, extracts interval duration and net amplitude change for each segment, and evaluates sample entropy on the resulting duration, amplitude, and paired duration–amplitude sequences, ExSEnt is designed to distinguish whether irregularity is driven primarily by timing variability, magnitude variability, or their coupling (Kamali et al., 29 Aug 2025). Its stated motivation is that standard unidimensional measures, including SampEn, permutation entropy, and multiscale entropy, can indicate that a signal is more or less complex without identifying the source of that complexity.

1. Conceptual basis and scope

ExSEnt is formulated around the claim that local extrema define a natural segmentation of a time series into behaviorally meaningful monotonic pieces. Rather than treating the original series as a single homogeneous object, the method converts it into two feature sequences derived from consecutive extrema: a sequence of segment durations and a sequence of signed net amplitude changes. Complexity is then assessed separately on these two sequences and on their joint pairing, yielding three complementary entropy measures: temporal entropy, amplitude entropy, and joint entropy (Kamali et al., 29 Aug 2025).

This construction is explicitly presented as a response to a limitation of standard entropy summaries. SampEn, permutation entropy, multiscale entropy, and related variants may detect changes in irregularity, but they do not by themselves determine whether those changes originate in the spacing of salient events, in the size of oscillatory excursions, or in the relation between the two. ExSEnt therefore replaces a single aggregate descriptor with a decomposition into feature-specific components.

The intended interpretive gain is not merely descriptive. The framework is proposed as a way to attribute complexity to distinct signal features in nonlinear synthetic systems and in physiological recordings. This suggests a broader methodological position: the segmentation itself is part of the representation, not only a preprocessing step. A plausible implication is that ExSEnt belongs to a class of entropy methods in which the definition of the events being analyzed is inseparable from the definition of the entropy measure.

2. Extrema-based segmentation and feature construction

For a time series $X=\{x_1,x_2,\ldots,x_N\}$, ExSEnt defines the first-order amplitude increment as

$\Delta x_k = x(t_k) - x(t_{k-1})\,.$

A segment is a continuous set of data points over which the sign of all $\Delta x_k$ values remains constant. A local extremum $\mathcal{E}_i$ is identified where the sign of $\Delta x_k$ differs from the sign of its following increment $\Delta x_{k+1}$ (Kamali et al., 29 Aug 2025).

To improve robustness to noise, the method introduces a threshold based on the interquartile range of the increment sequence,

$$\theta=\lambda \times \text{IQR}(\{\Delta x_k\})\,.$$

A new segment begins only if

$$|\Delta x_{k+1}|>\theta \;\text{ and }\; \Delta x_k\,\Delta x_{k+1}<0\,.$$

The paper explicitly notes that a small $\lambda$ captures finer fluctuations while a larger one suppresses noise-induced changes; in the validation experiments, $\lambda=0.01$ was used (Kamali et al., 29 Aug 2025).

From each pair of consecutive extrema, ExSEnt extracts two quantities. The duration of the $\Delta x_k = x(t_k) - x(t_{k-1})\,.$0 segment is

$\Delta x_k = x(t_k) - x(t_{k-1})\,.$1

and the net amplitude change is

$\Delta x_k = x(t_k) - x(t_{k-1})\,.$2

with the sign of the change preserved. These define two derived sequences, $\Delta x_k = x(t_k) - x(t_{k-1})\,.$3 and $\Delta x_k = x(t_k) - x(t_{k-1})\,.$4, which are then treated as new time series for entropy analysis.

The segmentation step reduces the effective sample size relative to the raw series because the entropy is computed on the number of extracted segments $\Delta x_k = x(t_k) - x(t_{k-1})\,.$5, not on $\Delta x_k = x(t_k) - x(t_{k-1})\,.$6 original samples. This is one reason the authors selected Sample Entropy. For convergence analysis on the logistic map, they report that the number of extracted segments grows approximately linearly with the original series length, with $\Delta x_k = x(t_k) - x(t_{k-1})\,.$7 (Kamali et al., 29 Aug 2025).

3. Entropy formalism and interpretation of the three ExSEnt metrics

ExSEnt uses Sample Entropy as the base estimator because it works well for small datasets. The formulation given is

$\Delta x_k = x(t_k) - x(t_{k-1})\,.$8

where $\Delta x_k = x(t_k) - x(t_{k-1})\,.$9 is the embedding dimension, $\Delta x_k$0 is the tolerance threshold, and $\Delta x_k$1 is the total number of data points. Template matching uses the Chebyshev distance,

$\Delta x_k$2

The counts of matched template pairs are

$\Delta x_k$3

and

$\Delta x_k$4

The paper reiterates the standard interpretation: lower SampEn indicates more predictability, whereas higher SampEn indicates more complexity or randomness (Kamali et al., 29 Aug 2025).

ExSEnt applies SampEn separately to the duration and amplitude sequences,

$\Delta x_k$5

$\Delta x_k$6

and to their paired sequence,

$\Delta x_k$7

For the joint entropy, the paired segment values are written as $\Delta x_k$8, and $\Delta x_k$9-length templates are constructed as

$\mathcal{E}_i$0

with

$\mathcal{E}_i$1

and similarly for $\mathcal{E}_i$2-length templates. Before pairing, the duration and amplitude sequences are normalized to zero mean and unit variance because they have different units and scales.

The three ExSEnt quantities have distinct meanings. $\mathcal{E}_i$3 measures temporal irregularity in the spacing between extrema. $\mathcal{E}_i$4 measures irregularity in the sizes of excursions between extrema. $\mathcal{E}_i$5 measures the complexity of the paired duration–amplitude evolution. The paper states that if

$\mathcal{E}_i$6

there is coherence or redundancy between duration and amplitude variations, making the paired sequence more predictable than the components alone. If

$\mathcal{E}_i$7

the two features contribute more independently to the signal’s complexity. This decomposition is the central interpretive novelty of ExSEnt.

The reported implementation uses $\mathcal{E}_i$8 unless otherwise noted, while the tolerance $\mathcal{E}_i$9 is set as the usual fraction of the standard deviation, typically $\Delta x_k$0, consistent with common SampEn practice. The authors emphasize that the same $\Delta x_k$1 should be used for joint and individual entropies to keep comparisons meaningful (Kamali et al., 29 Aug 2025).

4. Behavior on canonical nonlinear systems

The principal synthetic validation is carried out on the logistic map, the Rössler system, and the Rulkov map. Across these systems, ExSEnt is reported to track complexity changes across control-parameter sweeps and to detect transitions between periodic and chaotic regimes (Kamali et al., 29 Aug 2025).

For the logistic map,

$\Delta x_k$2

with $\Delta x_k$3, the joint entropy $\Delta x_k$4 largely follows the same pattern as SampEn, but the component entropies differ substantially. The paper reports that $\Delta x_k$5 over most of the parameter range except some semi-periodic windows. At $\Delta x_k$6, near-zero amplitude and joint entropies are observed while $\Delta x_k$7 remains $\Delta x_k$8 for $\Delta x_k$9; increasing the embedding dimension to $\Delta x_{k+1}$0 reduces $\Delta x_{k+1}$1 to zero as well, consistent with a period-3 structure.

For the Rössler system, with $\Delta x_{k+1}$2, $\Delta x_{k+1}$3, $\Delta x_{k+1}$4, $\Delta x_{k+1}$5, and $\Delta x_{k+1}$6, the bifurcation diagram color-coded by $\Delta x_{k+1}$7 highlights chaotic intervals and semi-periodic regimes. In this system, $\Delta x_{k+1}$8 is generally the dominant component and remains relatively high over much of the range, while $\Delta x_{k+1}$9 and $$\theta=\lambda \times \text{IQR}(\{\Delta x_k\})\,.$$0 can be very low in semi-periodic regions. The paper gives the example $$\theta=\lambda \times \text{IQR}(\{\Delta x_k\})\,.$$1, where $$\theta=\lambda \times \text{IQR}(\{\Delta x_k\})\,.$$2 and $$\theta=\lambda \times \text{IQR}(\{\Delta x_k\})\,.$$3; with $$\theta=\lambda \times \text{IQR}(\{\Delta x_k\})\,.$$4, these become $$\theta=\lambda \times \text{IQR}(\{\Delta x_k\})\,.$$5 and $$\theta=\lambda \times \text{IQR}(\{\Delta x_k\})\,.$$6. At $$\theta=\lambda \times \text{IQR}(\{\Delta x_k\})\,.$$7, $$\theta=\lambda \times \text{IQR}(\{\Delta x_k\})\,.$$8 is highest. SampEn is said not to mirror the joint ExSEnt as closely here; it more closely resembles an average of the three ExSEnt measures.

For the Rulkov map, with $$\theta=\lambda \times \text{IQR}(\{\Delta x_k\})\,.$$9, $$|\Delta x_{k+1}|>\theta \;\text{ and }\; \Delta x_k\,\Delta x_{k+1}<0\,.$$0, $$|\Delta x_{k+1}|>\theta \;\text{ and }\; \Delta x_k\,\Delta x_{k+1}<0\,.$$1, and $$|\Delta x_{k+1}|>\theta \;\text{ and }\; \Delta x_k\,\Delta x_{k+1}<0\,.$$2, the joint entropy identifies chaotic bursting intervals and semi-periodic regimes. The authors report step-like rises in $$|\Delta x_{k+1}|>\theta \;\text{ and }\; \Delta x_k\,\Delta x_{k+1}<0\,.$$3 and $$|\Delta x_{k+1}|>\theta \;\text{ and }\; \Delta x_k\,\Delta x_{k+1}<0\,.$$4 at $$|\Delta x_{k+1}|>\theta \;\text{ and }\; \Delta x_k\,\Delta x_{k+1}<0\,.$$5, while SampEn remains smooth. In the interval $$|\Delta x_{k+1}|>\theta \;\text{ and }\; \Delta x_k\,\Delta x_{k+1}<0\,.$$6, $$|\Delta x_{k+1}|>\theta \;\text{ and }\; \Delta x_k\,\Delta x_{k+1}<0\,.$$7, $$|\Delta x_{k+1}|>\theta \;\text{ and }\; \Delta x_k\,\Delta x_{k+1}<0\,.$$8, and SampEn are near zero, but $$|\Delta x_{k+1}|>\theta \;\text{ and }\; \Delta x_k\,\Delta x_{k+1}<0\,.$$9 remains $\lambda$0. At $\lambda$1, $\lambda$2 for $\lambda$3, but increasing to $\lambda$4 yields $\lambda$5.

Taken together, these examples support the paper’s claim that different feature sequences can have different effective embedding dimensions. This suggests that ExSEnt is not only a decomposition of entropy values, but also a way of exposing heterogeneity in the temporal organization of different signal features.

5. Physiological applications and feature-specific interpretation

The empirical applications in the ExSEnt paper are electromyography and Parkinson’s disease gait acceleration. In both cases, the method is used not simply to discriminate conditions, but to identify which feature of the signal accounts for the observed complexity change (Kamali et al., 29 Aug 2025).

For electromyography, the dataset consists of forearm EMG from 19 healthy subjects performing a finger-pinching task. The analysis compares a 2-second baseline, a 2-second movement window, and the full 7-second trial. The reported result is that movement reduces amplitude complexity substantially, with $\lambda$6 decreasing by $\lambda$7, whereas temporal complexity $\lambda$8 changes only slightly, by $\lambda$9. The authors further note a strong linear correlation between $\lambda=0.01$0 and $\lambda=0.01$1, suggesting that amplitude variations are the main driver of joint complexity in this task. The effect is reported as most prominent in the $\lambda=0.01$2 Hz band.

For Parkinson’s disease gait analysis, the signals are vertical ankle acceleration from the Daphnet Freezing of Gait dataset. The analysis separates the standard locomotor band $\lambda=0.01$3 and the freeze band $\lambda=0.01$4, and compares no-freezing gait windows with freezing-of-gait windows, all at least $\lambda=0.01$5 s long. In the $\lambda=0.01$6 Hz band, only $\lambda=0.01$7 differs significantly between the two conditions. In the $\lambda=0.01$8 Hz band, all three metrics—$\lambda=0.01$9, $\Delta x_k = x(t_k) - x(t_{k-1})\,.$00, and $\Delta x_k = x(t_k) - x(t_{k-1})\,.$01—differ significantly. The interpretation given is that slow locomotor-band changes are mainly amplitude-driven, whereas freezing-band changes involve both timing and amplitude irregularity.

The paper emphasizes that this band-specific decomposition is more informative than the traditional Freeze Index because it indicates which feature is altered rather than only reporting an overall power ratio. A plausible implication is that ExSEnt is particularly useful in settings where multiple physiological mechanisms can produce similar global changes in complexity but differ in whether they perturb timing, amplitude, or their coupling.

6. Relation to neighboring entropy frameworks

ExSEnt is part of a broader research landscape in which extrema, local shapes, or event-focused distributions are used to define entropy-like descriptors, but its formalism is distinct from several nearby approaches.

A close conceptual comparator is semantic entropy, introduced in “A Geometric Analysis of Time Series Leading to Information Encoding and a New Entropy Measure” (Majumdar et al., 2018). That framework represents a discrete time series as a string of 13 local 3-point geometric configurations determined by first and second differences and defines

$\Delta x_k = x(t_k) - x(t_{k-1})\,.$02

It also introduces information power and uses the ratio $\Delta x_k = x(t_k) - x(t_{k-1})\,.$03 as an indicator of synchronous behavior in epileptic EEG. Both ExSEnt and semantic entropy are geometry-based and neighborhood-sensitive, and both attach importance to peaks and troughs. The distinction is that semantic entropy encodes all 13 local configurations of 3-point neighborhoods, whereas ExSEnt explicitly segments the series at extrema and then computes SampEn on duration and amplitude feature sequences.

A second related line is the cumulative entropy construction proposed for extreme-event detection in market data (Drzazga-Szczȩśniak et al., 9 Mar 2025). There the central quantity is a joint or cumulative entropy over chronologically accumulated blocks,

$\Delta x_k = x(t_k) - x(t_{k-1})\,.$04

used to produce an entropic spectrum that increases as the reference point approaches an extreme event. This method is segmented and extrema-oriented in spirit, but its segmentation is chronological and cumulative rather than based on monotonic excursions between local extrema.

Another related approach is the complexity measure of extreme events based on normalized Shannon entropy and disequilibrium of local-extrema amplitude histograms (Das et al., 2024). That method extracts local maxima or minima, bins them into $\Delta x_k = x(t_k) - x(t_{k-1})\,.$05 amplitude classes, and computes

$\Delta x_k = x(t_k) - x(t_{k-1})\,.$06

with $\Delta x_k = x(t_k) - x(t_{k-1})\,.$07 the normalized Shannon entropy and $\Delta x_k = x(t_k) - x(t_{k-1})\,.$08 the normalized disequilibrium distance from the uniform distribution. The paper reports a general trend in which complexity increases during a transition to extreme events, reaches a maximum, and then decreases. The resemblance to ExSEnt lies in the explicit use of local extrema; the difference is that the histogram-based complexity measure discards segment durations, while ExSEnt treats duration and amplitude as separate entropy-bearing variables.

More distant analogies exist in stochastic thermodynamics and quantum information. “Path-Extrema Upper Bounds on Mean Entropy Production” organizes trajectories into classes determined by realized extrema envelopes $\Delta x_k = x(t_k) - x(t_{k-1})\,.$09 of the martingale $\Delta x_k = x(t_k) - x(t_{k-1})\,.$10 and derives the exact decomposition

$\Delta x_k = x(t_k) - x(t_{k-1})\,.$11

(Limkumnerd, 16 May 2026). “Local extrema of entropy functions under tensor products” studies when local minima or maxima of entropy-type matrix functionals are preserved under tensor products (Friedland et al., 2011). These works share the language of entropy and extrema, but they address different mathematical problems than ExSEnt.

7. Practical considerations, limitations, and open directions

The ExSEnt paper discusses several implementation details that constrain how the method should be interpreted in practice. The segmentation threshold is controlled by $\Delta x_k = x(t_k) - x(t_{k-1})\,.$12, and the authors explicitly state that smaller values capture finer fluctuations whereas larger values suppress noise-induced sign changes. In the reported validation experiments, $\Delta x_k = x(t_k) - x(t_{k-1})\,.$13 is used. For stochastic benchmarks they use $\Delta x_k = x(t_k) - x(t_{k-1})\,.$14, $\Delta x_k = x(t_k) - x(t_{k-1})\,.$15 Hz, 100 realizations, and fixed seeds (Kamali et al., 29 Aug 2025).

Convergence behavior is treated empirically rather than axiomatically. For the logistic map, the authors define a practical stability threshold when the IQR half-width is within $\Delta x_k = x(t_k) - x(t_{k-1})\,.$16 of the final value. They report stability at $\Delta x_k = x(t_k) - x(t_{k-1})\,.$17 for $\Delta x_k = x(t_k) - x(t_{k-1})\,.$18, $\Delta x_k = x(t_k) - x(t_{k-1})\,.$19 for $\Delta x_k = x(t_k) - x(t_{k-1})\,.$20, and $\Delta x_k = x(t_k) - x(t_{k-1})\,.$21 for $\Delta x_k = x(t_k) - x(t_{k-1})\,.$22. They also acknowledge that the optimal $\Delta x_k = x(t_k) - x(t_{k-1})\,.$23 depends on signal characteristics and should be assessed dataset by dataset.

Two methodological limitations are explicitly identified. First, when durations and amplitudes are analyzed separately, they may exhibit different effective embedding dimensions; the paper presents this as an open issue for a more detailed theory of feature-specific dimensionality. Second, the current method uses extrema-based segmentation, but other event definitions could be more appropriate in some applications. This suggests that ExSEnt is best viewed as a framework centered on a particular event construction rather than as a uniquely canonical decomposition.

Within the scope of the reported results, the main significance of ExSEnt is that it makes complexity attribution explicit. High $\Delta x_k = x(t_k) - x(t_{k-1})\,.$24 indicates irregular timing of salient events, high $\Delta x_k = x(t_k) - x(t_{k-1})\,.$25 indicates irregular amplitude excursions, and high $\Delta x_k = x(t_k) - x(t_{k-1})\,.$26 indicates weak coupling and richer combined variability; low values indicate greater regularity or stronger structure (Kamali et al., 29 Aug 2025). In that sense, ExSEnt complements standard entropy measures by reframing complexity analysis as a problem of feature-resolved event structure rather than a single global irregularity score.