Complexity and Entropy Analysis

Overview

Complexity measures quantify the irregularity, unpredictability, and multiscale structure of time series signals. NeuroKit2 provides comprehensive entropy, fractal dimension, and nonlinear dynamics measures for assessing physiological signal complexity.

Main Function

complexity()

Compute multiple complexity metrics simultaneously for exploratory analysis.

complexity_indices = nk.complexity(signal, sampling_rate=1000, show=False)

Returns:

• DataFrame with numerous complexity measures across categories:
  • Entropy indices
  • Fractal dimensions
  • Nonlinear dynamics measures
  • Information-theoretic metrics

Use case:

• Exploratory analysis to identify relevant measures
• Comprehensive signal characterization
• Comparative studies across signals

Parameter Optimization

Before computing complexity measures, optimal embedding parameters should be determined:

complexity_delay()

Determine optimal time delay (τ) for phase space reconstruction.

optimal_tau = nk.complexity_delay(signal, delay_max=100, method='fraser1986', show=False)

Methods:

• 'fraser1986': Mutual information first minimum
• 'theiler1990': Autocorrelation first zero crossing
• 'casdagli1991': Cao's method

Use for: Embedding delay in entropy, attractor reconstruction

complexity_dimension()

Determine optimal embedding dimension (m).

optimal_m = nk.complexity_dimension(signal, delay=None, dimension_max=20,
                                    method='afn', show=False)

Methods:

• 'afn': Average False Nearest Neighbors
• 'fnn': False Nearest Neighbors
• 'correlation': Correlation dimension saturation

Use for: Entropy calculations, phase space reconstruction

complexity_tolerance()

Determine optimal tolerance (r) for entropy measures.

optimal_r = nk.complexity_tolerance(signal, method='sd', show=False)

Methods:

• 'sd': Standard deviation-based (0.1-0.25 × SD typical)
• 'maxApEn': Maximize ApEn
• 'recurrence': Based on recurrence rate

Use for: Approximate entropy, sample entropy

complexity_k()

Determine optimal k parameter for Higuchi fractal dimension.

optimal_k = nk.complexity_k(signal, k_max=20, show=False)

Use for: Higuchi fractal dimension calculation

Entropy Measures

Entropy quantifies randomness, unpredictability, and information content.

entropy_shannon()

Shannon entropy - classical information-theoretic measure.

shannon_entropy = nk.entropy_shannon(signal)

Interpretation:

• Higher: more random, less predictable
• Lower: more regular, predictable
• Units: bits (information)

Use cases:

• General randomness assessment
• Information content
• Signal irregularity

entropy_approximate()

Approximate Entropy (ApEn) - regularity of patterns.

apen = nk.entropy_approximate(signal, delay=1, dimension=2, tolerance='sd')

Parameters:

• delay: Time delay (τ)
• dimension: Embedding dimension (m)
• tolerance: Similarity threshold (r)

Interpretation:

• Lower ApEn: more regular, self-similar patterns
• Higher ApEn: more complex, irregular
• Sensitive to signal length (≥100-300 points recommended)

Physiological applications:

• HRV: reduced ApEn in heart disease
• EEG: altered ApEn in neurological disorders

entropy_sample()

Sample Entropy (SampEn) - improved ApEn.

sampen = nk.entropy_sample(signal, delay=1, dimension=2, tolerance='sd')

Advantages over ApEn:

• Less dependent on signal length
• More consistent across recordings
• No self-matching bias

Interpretation:

• Same as ApEn but more reliable
• Preferred in most applications

Typical values:

• HRV: 0.5-2.5 (context-dependent)
• EEG: 0.3-1.5

entropy_multiscale()

Multiscale Entropy (MSE) - complexity across temporal scales.

mse = nk.entropy_multiscale(signal, scale=20, dimension=2, tolerance='sd',
                            method='MSEn', show=False)

Methods:

• 'MSEn': Multiscale Sample Entropy
• 'MSApEn': Multiscale Approximate Entropy
• 'CMSE': Composite Multiscale Entropy
• 'RCMSE': Refined Composite Multiscale Entropy

Interpretation:

• Entropy at different coarse-graining scales
• Healthy/complex systems: high entropy across multiple scales
• Diseased/simpler systems: reduced entropy, especially at larger scales

Use cases:

• Distinguish true complexity from randomness
• White noise: constant across scales
• Pink noise/complexity: structured variation across scales

entropy_fuzzy()

Fuzzy Entropy - uses fuzzy membership functions.

fuzzen = nk.entropy_fuzzy(signal, delay=1, dimension=2, tolerance='sd', r=0.2)

Advantages:

• More stable with noisy signals
• Fuzzy boundaries for pattern matching
• Better performance with short signals

entropy_permutation()

Permutation Entropy - based on ordinal patterns.

perment = nk.entropy_permutation(signal, delay=1, dimension=3)

Method:

• Encodes signal into ordinal patterns (permutations)
• Counts pattern frequencies
• Robust to noise and non-stationarity

Interpretation:

• Lower: more regular ordinal structure
• Higher: more random ordering

Use cases:

• EEG analysis
• Anesthesia depth monitoring
• Fast computation

entropy_spectral()

Spectral Entropy - based on power spectrum.

spec_ent = nk.entropy_spectral(signal, sampling_rate=1000, bands=None)

Method:

• Normalized Shannon entropy of power spectrum
• Quantifies frequency distribution regularity

Interpretation:

• 0: Single frequency (pure tone)
• 1: White noise (flat spectrum)

Use cases:

• EEG: spectral distribution changes with states
• Anesthesia monitoring

entropy_svd()

Singular Value Decomposition Entropy.

svd_ent = nk.entropy_svd(signal, delay=1, dimension=2)

Method:

• SVD on trajectory matrix
• Entropy of singular value distribution

Use cases:

• Attractor complexity
• Deterministic vs. stochastic dynamics

entropy_differential()

Differential Entropy - continuous analog of Shannon entropy.

diff_ent = nk.entropy_differential(signal)

Use for: Continuous probability distributions

Other Entropy Measures

Tsallis Entropy:

tsallis = nk.entropy_tsallis(signal, q=2)

• Generalized entropy with parameter q
• q=1 reduces to Shannon entropy

Rényi Entropy:

renyi = nk.entropy_renyi(signal, alpha=2)

• Generalized entropy with parameter α

Additional specialized entropies:

• entropy_attention(): Attention entropy
• entropy_grid(): Grid-based entropy
• entropy_increment(): Increment entropy
• entropy_slope(): Slope entropy
• entropy_dispersion(): Dispersion entropy
• entropy_symbolicdynamic(): Symbolic dynamics entropy
• entropy_range(): Range entropy
• entropy_phase(): Phase entropy
• entropy_quadratic(), entropy_cumulative_residual(), entropy_rate(): Specialized variants

Fractal Dimension Measures

Fractal dimensions characterize self-similarity and roughness.

fractal_katz()

Katz Fractal Dimension - waveform complexity.

kfd = nk.fractal_katz(signal)

Interpretation:

• 1: straight line

• 1: increasing roughness and complexity

• Typical range: 1.0-2.0

Advantages:

• Simple, fast computation
• No parameter tuning

fractal_higuchi()

Higuchi Fractal Dimension - self-similarity.

hfd = nk.fractal_higuchi(signal, k_max=10)

Method:

• Constructs k new time series from original
• Estimates dimension from length-scale relationship

Interpretation:

• Higher HFD: more complex, irregular
• Lower HFD: smoother, more regular

Use cases:

• EEG complexity
• HRV analysis
• Epilepsy detection

fractal_petrosian()

Petrosian Fractal Dimension - rapid estimation.

pfd = nk.fractal_petrosian(signal)

Advantages:

• Fast computation
• Direct calculation (no curve fitting)

fractal_sevcik()

Sevcik Fractal Dimension - normalized waveform complexity.

sfd = nk.fractal_sevcik(signal)

fractal_nld()

Normalized Length Density - curve length-based measure.

nld = nk.fractal_nld(signal)

fractal_psdslope()

Power Spectral Density Slope - frequency-domain fractal measure.

slope = nk.fractal_psdslope(signal, sampling_rate=1000)

Method:

• Linear fit to log-log power spectrum
• Slope β relates to fractal dimension

Interpretation:

• β ≈ 0: White noise (random)
• β ≈ -1: Pink noise (1/f, complex)
• β ≈ -2: Brown noise (Brownian motion)

fractal_hurst()

Hurst Exponent - long-range dependence.

hurst = nk.fractal_hurst(signal, show=False)

Interpretation:

• H < 0.5: Anti-persistent (mean-reverting)
• H = 0.5: Random walk (white noise)
• H > 0.5: Persistent (trending, long-memory)

Use cases:

• Assess long-term correlations
• Financial time series
• HRV analysis

fractal_correlation()

Correlation Dimension - attractor dimensionality.

corr_dim = nk.fractal_correlation(signal, delay=1, dimension=10, radius=64)

Method:

• Grassberger-Procaccia algorithm
• Estimates dimension of attractor in phase space

Interpretation:

• Low dimension: deterministic, low-dimensional chaos
• High dimension: high-dimensional chaos or noise

fractal_dfa()

Detrended Fluctuation Analysis - scaling exponent.

dfa_alpha = nk.fractal_dfa(signal, multifractal=False, q=2, show=False)

Interpretation:

• α < 0.5: Anti-correlated
• α = 0.5: Uncorrelated (white noise)
• α = 1.0: 1/f noise (pink noise, healthy complexity)
• α = 1.5: Brownian noise
• α > 1.0: Persistent long-range correlations

HRV applications:

• α1 (short-term, 4-11 beats): Reflects autonomic regulation
• α2 (long-term, >11 beats): Long-range correlations
• Reduced α1: Cardiac pathology

fractal_mfdfa()

Multifractal DFA - multiscale fractal properties.

mfdfa_results = nk.fractal_mfdfa(signal, q=None, show=False)

Method:

• Extends DFA to multiple q-orders
• Characterizes multifractal spectrum

Returns:

• Generalized Hurst exponents h(q)
• Multifractal spectrum f(α)
• Width indicates multifractality strength

Use cases:

• Detect multifractal structure
• HRV multifractality in health vs. disease
• EEG multiscale dynamics

fractal_tmf()

Multifractal Nonlinearity - deviation from monofractal.

tmf = nk.fractal_tmf(signal)

Interpretation:

• Quantifies departure from simple scaling
• Higher: more multifractal structure

fractal_density()

Density Fractal Dimension.

density_fd = nk.fractal_density(signal)

fractal_linelength()

Line Length - total variation measure.

linelength = nk.fractal_linelength(signal)

Use case:

• Simple complexity proxy
• EEG seizure detection

Nonlinear Dynamics

complexity_lyapunov()

Largest Lyapunov Exponent - chaos and divergence.

lyap = nk.complexity_lyapunov(signal, delay=None, dimension=None,
                              sampling_rate=1000, show=False)

Interpretation:

• λ < 0: Stable fixed point
• λ = 0: Periodic orbit
• λ > 0: Chaotic (nearby trajectories diverge exponentially)

Use cases:

• Detect chaos in physiological signals
• HRV: positive Lyapunov suggests nonlinear dynamics
• EEG: epilepsy detection (decreased λ before seizure)

complexity_lempelziv()

Lempel-Ziv Complexity - algorithmic complexity.

lz = nk.complexity_lempelziv(signal, symbolize='median')

Method:

• Counts number of distinct patterns
• Coarse-grained measure of randomness

Interpretation:

• Lower: repetitive, predictable patterns
• Higher: diverse, unpredictable patterns

Use cases:

• EEG: consciousness levels, anesthesia
• HRV: autonomic complexity

complexity_rqa()

Recurrence Quantification Analysis - phase space recurrences.

rqa_indices = nk.complexity_rqa(signal, delay=1, dimension=3, tolerance='sd')

Metrics:

• Recurrence Rate (RR): Percentage of recurrent states
• Determinism (DET): Percentage of recurrent points in lines
• Laminarity (LAM): Percentage in vertical structures (laminar states)
• Trapping Time (TT): Average vertical line length
• Longest diagonal/vertical: System predictability
• Entropy (ENTR): Shannon entropy of line length distribution

Interpretation:

• High DET: deterministic dynamics
• High LAM: system trapped in specific states
• Low RR: random, non-recurrent dynamics

Use cases:

• Detect transitions in system dynamics
• Physiological state changes
• Nonlinear time series analysis

complexity_hjorth()

Hjorth Parameters - time-domain complexity.

hjorth = nk.complexity_hjorth(signal)

Metrics:

• Activity: Variance of signal
• Mobility: Proportion of standard deviation of derivative to signal
• Complexity: Change in mobility with derivative

Use cases:

• EEG feature extraction
• Seizure detection
• Signal characterization

complexity_decorrelation()

Decorrelation Time - memory duration.

decorr_time = nk.complexity_decorrelation(signal, show=False)

Interpretation:

• Time lag where autocorrelation drops below threshold
• Shorter: rapid fluctuations, short memory
• Longer: slow fluctuations, long memory

complexity_relativeroughness()

Relative Roughness - smoothness measure.

roughness = nk.complexity_relativeroughness(signal)

Information Theory

fisher_information()

Fisher Information - measure of order.

fisher = nk.fisher_information(signal, delay=1, dimension=2)

Interpretation:

• High: ordered, structured
• Low: disordered, random

Use cases:

• Combine with Shannon entropy (Fisher-Shannon plane)
• Characterize system complexity

fishershannon_information()

Fisher-Shannon Information Product.

fs = nk.fishershannon_information(signal)

Method:

• Product of Fisher information and Shannon entropy
• Characterizes order-disorder balance

mutual_information()

Mutual Information - shared information between variables.

mi = nk.mutual_information(signal1, signal2, method='knn')

Methods:

• 'knn': k-nearest neighbors (nonparametric)
• 'kernel': Kernel density estimation
• 'binning': Histogram-based

Use cases:

• Coupling between signals
• Feature selection
• Nonlinear dependence

Practical Considerations

Signal Length Requirements

Parameter Selection

General guidelines:

• Use parameter optimization functions first
• Or use conventional defaults:
  • Delay (τ): 1 for HRV, autocorrelation first minimum for EEG
  • Dimension (m): 2-3 typical
  • Tolerance (r): 0.2 × SD common

Sensitivity:

• Results can be parameter-sensitive
• Report parameters used
• Consider sensitivity analysis

Normalization and Preprocessing

Standardization:

• Many measures sensitive to signal amplitude
• Z-score normalization often recommended
• Detrending may be necessary

Stationarity:

• Some measures assume stationarity
• Check with statistical tests (e.g., ADF test)
• Segment non-stationary signals

Interpretation

Context-dependent:

• No universal "good" or "bad" complexity
• Compare within-subject or between groups
• Consider physiological context

Complexity vs. randomness:

• Maximum entropy ≠ maximum complexity
• True complexity: structured variability
• White noise: high entropy but low complexity (MSE distinguishes)

Applications

Cardiovascular:

• HRV complexity: reduced in heart disease, aging
• DFA α1: prognostic marker post-MI

Neuroscience:

• EEG complexity: consciousness, anesthesia depth
• Entropy: Alzheimer's, epilepsy, sleep stages
• Permutation entropy: anesthesia monitoring

Psychology:

• Complexity loss in depression, anxiety
• Increased regularity under stress

Aging:

• "Complexity loss" with aging across systems
• Reduced multiscale complexity

Critical transitions:

• Complexity changes before state transitions
• Early warning signals (critical slowing down)

References

• Pincus, S. M. (1991). Approximate entropy as a measure of system complexity. Proceedings of the National Academy of Sciences, 88(6), 2297-2301.
• Richman, J. S., & Moorman, J. R. (2000). Physiological time-series analysis using approximate entropy and sample entropy. American Journal of Physiology-Heart and Circulatory Physiology, 278(6), H2039-H2049.
• Peng, C. K., et al. (1995). Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos, 5(1), 82-87.
• Costa, M., Goldberger, A. L., & Peng, C. K. (2005). Multiscale entropy analysis of biological signals. Physical review E, 71(2), 021906.
• Grassberger, P., & Procaccia, I. (1983). Measuring the strangeness of strange attractors. Physica D: Nonlinear Phenomena, 9(1-2), 189-208.