Complexity, entropy, fractals, and RQA

Checked 2026-10-01 against NeuroKit2 0.2.13 stable runtime/source, the official Complexity API, and the NeuroKit2 complexity comparison paper.

Return convention changed from older examples

Most stable 0.2.13 complexity functions return:

value, info = function(signal, ...)

Examples:

sampen, sampen_info = nk.entropy_sample(
    signal, delay=1, dimension=2, tolerance="sd"
)
dfa, dfa_info = nk.fractal_dfa(signal)
hfd, hfd_info = nk.fractal_higuchi(signal, k_max=10)
lyapunov, lyapunov_info = nk.complexity_lyapunov(signal)
fi, fi_info = nk.fisher_information(signal)

Do not treat the tuple as a scalar. The current public name is fisher_information(); information_fisher() is not exported.

Exceptions exist: for example, mutual_information() returns a float. Check the stable signature and persist runtime type/schema.

complexity() is a selected panel

features, details = nk.complexity(
    signal,
    which="makowski2022",
    delay=1,
    dimension=2,
    tolerance="sd",
)

The default does not compute “all complexity measures.” The pinned 0.2.13 probe returned a one-row DataFrame with 15 columns:

AttEn, BubbEn, CWPEn, Hjorth, LL,
MFDFA_Asymmetry, MFDFA_Delta, MFDFA_Fluctuation,
MFDFA_Increment, MFDFA_Max, MFDFA_Mean,
MFDFA_Peak, MFDFA_Width, MSPEn, SVDEn

The accompanying dict had method-specific details. This panel reflects a published empirical comparison and implementation choices; it is not a universal optimum for every signal, endpoint, or population.

Parameter selection

Phase-space/entropy estimates depend on:

• delay (tau);
• embedding dimension (m);
• tolerance/radius (r);
• scale/coarse-graining;
• symbolization/binning;
• detrending/integration/order;
• sampling rate and bandwidth; and
• usable length and stationarity.

Stable utilities also return metadata:

delay, delay_info = nk.complexity_delay(
    signal, delay_max=100, method="fraser1986", show=False
)
dimension, dimension_info = nk.complexity_dimension(
    signal, delay=delay, dimension_max=10, method="afnn", show=False
)
tolerance, tolerance_info = nk.complexity_tolerance(
    signal,
    method="maxApEn",
    delay=delay,
    dimension=dimension,
    show=False,
)

Optimization can return no solution or raise when search bounds are inadequate. Do not silently replace failure with an arbitrary default. Predefine the algorithm/search range, report failures, and test sensitivity.

tolerance="sd" commonly maps to a fraction of standard deviation, but amplitude normalization, outliers, and signal length alter it. One conventional parameter set is not method validation.

Major stable families

Entropy

Available functions include approximate, sample, fuzzy, permutation, spectral, multiscale, dispersion, symbolic-dynamic, SVD, Shannon, Rényi, Tsallis, and other variants.

Pinned probes confirmed (value, info) for:

• entropy_approximate();
• entropy_sample();
• entropy_multiscale();
• entropy_permutation(); and
• entropy_spectral().

Some values are corrected/normalized by default (for example corrected permutation entropy). Record every parameter and logarithm base. Entropy values from different algorithms/normalizations are not interchangeable.

Fractals

Stable functions include Katz, Higuchi, Petrosian, Sevcik, NLD, PSD slope, Hurst, correlation dimension, DFA/MFDFA, density, line length, and tMF.

fractal_dfa() returns (float, info) for monofractal mode and can return a DataFrame-like multifractal summary. Report scales, overlap, integration, detrending order, q values, and fit diagnostics. Do not interpret alpha values without checking which regime and preprocessing generated them.

Lyapunov and RQA

lle, lle_info = nk.complexity_lyapunov(
    signal,
    delay=1,
    dimension=2,
    method="rosenstein1993",
    separation="auto",
)
rqa, rqa_info = nk.complexity_rqa(
    signal,
    dimension=3,
    delay=1,
    tolerance="sd",
    method="python",
)

The pinned RQA DataFrame had fields such as RecurrenceRate, Determinism, Laminarity, TrappingTime, line-length/entropy, divergence, and vertical/white-line statistics. rqa_info included full recurrence and distance matrices, which scale quadratically in signal length. Bound input length and memory.

A positive estimated Lyapunov exponent does not by itself prove deterministic chaos. RQA results depend strongly on embedding, tolerance, norm, Theiler window, line thresholds, nonstationarity, and sample size.

Signal preparation

1. Preserve raw signal and physical unit.
2. Apply modality-specific artifact/missing-data policy first.
3. Define the analysis window and usable length.
4. Decide detrending, filtering, resampling, and standardization before viewing group effects.
5. Check stationarity or segment according to the estimand.
6. Compute prespecified measures and diagnostics.
7. Compare with surrogates/nulls and parameter sensitivity.

Do not apply blanket z-scoring: amplitude-sensitive measures may change, while scale invariant measures may not. Report both rationale and implementation.

Length and comparability

There is no universal minimum sample count across complexity measures. Required length grows with embedding dimension, delay, scale count, tolerance, and estimator. Multiscale entropy loses points at each coarse-graining scale; RQA and correlation dimension can be computationally and statistically unstable on short data.

• Use equal-duration/beat-count windows for direct comparisons unless a validated correction is used.
• Quantify estimate reliability with simulation/resampling.
• Avoid comparing measures computed at different sample rates or bandwidths without explicit validation.
• Return missing/unsupported rather than a numerically convenient but invalid value.

Interpretation

High entropy can mean noise, not useful complexity. “Healthy complexity,” “complexity loss,” consciousness, disease, stress, and aging claims require a prespecified theory, validated acquisition/preprocessing, appropriate controls, and independent evidence.

Do not use these measures for diagnosis, anesthesia/consciousness monitoring, seizure detection, prognosis, or medical-device validation based on this toolbox alone.

Reproducible report

Record:

• NeuroKit2 version and function return schema;
• signal type/unit/rate/bandwidth/window/length;
• exclusions, interpolation, filtering, detrending, resampling, normalization;
• algorithm, delay, dimension, tolerance, scales/bins/q/order;
• optimization method/search space and failures;
• fit/convergence diagnostics and runtime warnings;
• surrogate/null and sensitivity results; and
• multiplicity control and participant-level statistical design.

Sources checked 2026-10-01

• Official Complexity API
• Stable v0.2.13 complexity source
• Makowski et al. (2022), empirical comparison using NeuroKit2
• Richman & Moorman (2000), sample entropy
• Peng et al. (1995), DFA
• Costa et al. (2005), multiscale entropy