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Refined Multiscale Fuzzy Entropy based on Standard Deviation for Biomedical Signal Analysis

Hamed Azami, Alberto Fernandez, Javier Escudero

arXiv:1602.02847v2cs.CE

TL;DR

Biomedical complexity measures can misrepresent multiscale structure and become unstable at large scales or for short signals. The paper introduces refined composite fuzzy entropy using mean or standard deviation coarse-graining and evaluates it on synthetic and clinical signals. Standard-deviation-based features provide complementary information and higher classification accuracy than mean-based features within the reported analyses.

  • Problem

    Existing multiscale entropy methods can yield misleading complexity assessments, unstable large-scale results, or undefined values for short biological signals.

  • Method

    The paper introduces RCMFEσ and RCMFEμ, combining refined composite multiscale processing with fuzzy entropy and standard-deviation- or mean-based coarse-graining.

  • Results

    RCMFEσ-based features achieved higher classification accuracies than RCMFEμ-based features, while standard-deviation-based methods discriminated clinical groups better than variance- and mean-based methods.

  • Takeaways & Limitations

    RCMFEσ offers complementary information to mean-based coarse-graining and can distinguish dynamics that the alternative does not, and conversely.

  • Takeaways & Limitations

    For focal and non-focal EEG, band-pass filtering between 0.5 and 40 Hz removes information above 40 Hz, potentially affecting short-scale SampEn results.

Abstract

from arXiv · show

Multiscale entropy (MSE) has been a prevalent algorithm to quantify the complexity of fluctuations in the local mean value of biomedical time series. Recent developments in the field have tried to improve the MSE by reducing its variability in large scale factors. On the other hand, there has been recent interest in using other statistical moments than the mean, i.e. variance, in the coarse-graining step of the MSE. Building on these trends, here we introduce the so-called refined composite multiscale fuzzy entropy based on the standard deviation (RCMFEσ) to quantify the dynamical properties of spread over multiple time scales. We demonstrate the dependency of the RCMFEσ, in comparison with other multiscale approaches, on several straightforward signal processing concepts using a set of synthetic signals. We also investigate the complementarity of using the standard deviation instead of the mean in the coarse-graining process using magnetoencephalograms in Alzheimer disease and publicly available electroencephalograms recorded from focal and non-focal areas in epilepsy. Our results indicate that RCMFEσ offers complementary information to that revealed by classical coarse-graining approaches and that it has superior performance to distinguish different types of physiological activity.

1 Introduction

Biomedical time-series complexity is difficult to quantify because conventional entropy can confuse irregularity with meaningful multiscale structure. The paper builds on multiscale and moment-based approaches to propose refined composite fuzzy-entropy measures using mean and standard deviation.

  • MSE addresses multiscale structure by coarse-graining signals at scale factor τ and calculating entropy on the resulting sequences.
  • MSEσ extends coarse-graining by using variance instead of the mean to quantify volatility dynamics in heartbeat signals.
  • The paper proposes RCMFEμ and RCMFEσ, combining refined composite processing with fuzzy entropy and mean- or standard-deviation-based coarse-graining.Standard deviation is proposed because it has the same dimension as the original signal, unlike variance.
  • The proposed measures are intended to improve accuracy, robustness, and stability while characterizing dependencies on signal-processing concepts and clinical datasets.

2.1 Entropy approaches

The paper reviews sample entropy and fuzzy entropy as pattern-matching measures of time-series irregularity. Both compare matches at embedding dimensions m and m+1, but fuzzy entropy replaces a hard match with graded similarity.

  • 2.1.1 Sample entropy: Sample entropy constructs m-dimensional vectors using embedding dimension m and measures distances by the maximum difference between corresponding components.
  • 2.1.1 Sample entropy: A sample-entropy match is determined by whether vector distance falls within tolerance r, with B_m(r) counting matched m-dimensional vectors.
  • 2.1.1 Sample entropy: SampEn is computed as the negative natural logarithm of the ratio B_m+1(r)/B_m(r), comparing matches across successive embedding dimensions.
  • 2.1.2 Fuzzy entropy: Fuzzy entropy assigns each vector pair a similarity degree through a fuzzy function based on distance, power n, and tolerance r.
  • 2.1.2 Fuzzy entropy: FuzEn is the negative natural logarithm of the ratio of φ_m+1 to φ_m, computed analogously across embedding dimensions.

2.2 Coarse-graining for multiscale entropy

Multiscale entropy begins by coarse-graining a signal into consecutive blocks at time scale τ. The resulting sequences can represent local means, variances, or standard deviations, with standard deviation retaining the signal’s dimensionality.

  • MSEμ/MFEμ coarse-grain a length-C signal by averaging consecutive samples within blocks determined by scale factor τ.
  • Variance can replace the mean in coarse-graining, providing a measure of signal volatility across time scales.
  • Standard deviation is proposed as an alternative spread measure because variance has a different dimension from the signal and amplifies deviations quadratically.
  • The standard-deviation coarse-graining is defined by the displayed formulation associated with the proposed spread-based approach.

2.3 Refined composite multiscale fuzzy entropy

Refined composite multiscale fuzzy entropy addresses limitations of conventional coarse-graining by using multiple shifted coarse-grained sequences. RCMFEσ and RCMFEμ thereby quantify spread- and mean-based dynamics with improved stability at challenging scales.

  • Conventional MSEμ coarse-graining is asymmetric to sample grouping and becomes more variable at high scales as coarse-grained sequences shorten.
  • RCMFEσ and RCMFEμ combine fuzzy entropy with refined composite processing to address these coarse-graining drawbacks.
  • RCMFEσ first constructs τ shifted time series at scale factor τ, whereas conventional MSE/MFE use only one coarse-grained sequence.
  • Fuzzy-entropy quantities are calculated separately for each shifted sequence and then averaged before computing RCMFEσ.
  • The approaches use embedding dimension m=2, FuzEn power n=2, and tolerance r=0.15 times the original signal’s standard deviation.

2.4 Evaluation signals

The evaluation uses synthetic signals to probe multiscale entropy behavior across noise, frequency, spectral content, periodicity, and nonlinear dynamics. It also tests discrimination of physiological activity using MEG recordings from Alzheimer disease and controls and intracranial EEG from focal and non-focal epilepsy regions.

  • 2.4.1 Noise and synthetic signals: Synthetic evaluations vary noise type, sinusoidal frequency, colored-noise spectral content, periodicity, and nonlinear dynamics to examine RCMFEσ and RCMFEμ.The signals include WGN, 1/f noise, a logarithmic chirp, AR(1), MIX, logistic-map, and Lorenz-system signals.
  • 2.4.1 Noise and synthetic signals: The logistic-map signal spans periodic behavior, progressively doubled periods, and chaos as α increases from 3.5 to 3.99.Periodic windows occur within the chaotic regime, including near α ≈ 3.8.
  • 2.4.1 Noise and synthetic signals: The Lorenz evaluation contrasts a chaotic segment at ρ = 28 with a torus-knot segment at ρ = 99.96 after standard-deviation normalization.Both segments use λ = 10 and β = 8/3, and the analyzed signal is the x coordinate.
  • 2.4.2 Clinical datasets: Clinical discrimination is tested with resting-state MEG from 36 probable Alzheimer disease patients and 26 age-matched controls.The recordings used a 148-channel whole-head magnetometer, with five minutes of eyes-closed resting-state activity per participant.
  • 2.4.2 Clinical datasets: Intracranial EEG comprises focal and non-focal recordings from five patients per set, with a subset of 50 signals per set used for evaluation.Each signal contains 10,240 samples over 20 seconds at 512 Hz, and signals were band-pass filtered from 0.5 to 40 Hz.

3 Results

Across synthetic and clinical signals, standard-deviation-based refined composite multiscale fuzzy entropy provided complementary information to mean-based approaches and often improved discrimination, while refined composite processing increased reliability but also computation time.

  • Noise signals: Entropy profiles reflected signal structure: WGN generally decreased with scale, whereas 1/f noise remained approximately constant at larger scales.Fuzzy entropy reduced variability relative to sample entropy, and refined composite profiles had smaller standard deviations than their basic counterparts.
  • Sensitivity of multiscale methods to signal length: Refined composite methods improved reliability by combining τ coarse-grained signals, although RCMSEμ remained undefined at some short-signal scales.For fuzzy entropy, RCMFEμ avoided undefined values and had considerably smaller standard deviations than MFEμ, especially for short signals.
  • Clinical datasets: RCMFEσ and RCMFEμ showed complementary scale ranges, with smaller adjusted p values for RCMFEμ at scales 1–9 and for RCMFEσ at scales 10–30.This pattern indicates that either measure may distinguish dynamics that the other does not at particular scales.
  • Clinical datasets: RCMFEσ achieved higher classification accuracy than RCMFEμ and provided complementary information when mean-based measures could not distinguish dynamics.The average accuracies for focal versus non-focal EEG were 79.62% for RCMFEσ and 71.58% for RCMFEμ.
  • Clinical datasets: Standard-deviation-based methods discriminated Alzheimer disease and control groups better than variance- and mean-based multiscale algorithms.For AD MEG, RCMFEσ values were lower for patients than controls at every scale, whereas RCMFEμ differed only at scale factors 1–3.

4 Discussions

Across synthetic and clinical signals, refined composite and fuzzy-entropy variants generally improved stability, while standard-deviation coarse-graining captured dynamics complementary to mean-based measures. Clinical analyses showed scale-dependent group differences and higher classification accuracy for RCMFEσ-based features.

  • Noise signals: Fuzzy-entropy multiscale methods were more stable than SampEn-based methods, while refined composite techniques improved the stability of MSE and MFE methods.Standard-deviation-based methods also performed better than variance-based methods at shorter temporal scales for WGN and 1/f noise.
  • Synthetic signals: RCMFEσ and RCMFEμ were generally more stable than their basic counterparts, and refined composite processing reduced variability across synthetic-signal analyses.For the Lorenz system, RCMFEμ was more stable than RCMSEμ, which was more stable than MSEμ.
  • Synthetic signals: RCMFEμ and RCMFEσ revealed different dynamical properties because they extract mean and spread features, respectively, across multiple temporal scales.The logistic-map results specifically showed that mean- and standard-deviation-based approaches produce different feature types.
  • Clinical datasets: MEG group differences were most significant around scales 7–14 for RCMFEμ and 3–9 for RCMFEσ, demonstrating complementary discrimination across scales.Entropy profiles also increased at scales 1–3 for RCMFEμ and 2–6 for RCMFEσ in the MEG dataset.
  • Clinical datasets: EEG interpretation was constrained because band-pass filtering from 0.5 to 40 Hz removed information above 40 Hz, affecting short-scale frequency content.The authors suggest this may explain very low SampEn values at short time scales.

5 Conclusions

The paper introduces refined composite multiscale fuzzy entropy based on mean and standard deviation to characterize distinct dynamics across multiple scales. Across synthetic and clinical evaluations, the proposed methods improved stability, alleviated undefined short-signal values, and RCMFEσ features achieved higher classification accuracies than RCMFEμ features.

  • 5 Conclusions: The study introduced RCMFEσ and RCMFEμ to extract dynamical properties of spread and mean, respectively, over multiple time scales.The methods were evaluated on noise, synthetic signals, and two clinical datasets.
  • 5 Conclusions: FuzEn-based methods were more stable than SampEn-based algorithms, refined composite processing improved stability, and the proposed methods alleviated undefined values for short signals.These conclusions were supported across the examined multiscale methods and signal types.
  • 5 Conclusions: RCMFEσ-based features produced higher classification accuracies than RCMFEμ-based features, while the two feature types were complementary across signal dynamics.When one could not distinguish particular dynamics, the other could do so.
  • 5 Conclusions: The authors expect the developments to support applications distinguishing different kinds of dynamics in physiologic and nonphysiologic studies.This is stated as a prospective application scope.

Appendix

The appendix provides access to the analysis code and states the distribution terms for the article.

  • Appendix: The analysis code includes implementations of SampEn, FuzEn, MSE, MFE, RCMSE, and RCMFE variants based on mean, variance, and standard deviation.The listed code covers the methods used in the study.
  • Appendix: The article is distributed under the Creative Commons Attribution 4.0 International License, subject to attribution and license-link requirements.The license permits reuse and distribution when the stated conditions are met.
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