Fuzzy Entropy and Its Application for Enhanced Subspace Filtering

Fuzzy entropy (FuzzyEn), which employs the fuzzy probability to characterize the similarity between vectors, is a robust nonlinear statistic to quantify the complexity or regularity of nonlinear time series. The aim of this study is to investigate the statistical properties of FuzzyEn and improve the subspace denoising technique using FuzzyEn. We first show the asymptotic normality of FuzzyEn and derive its variance for finite sample behavior. We then analyze the two pending and fundamental issues in subspace denoising, i.e., depending on the so-called “noise floor” and the unaltered noise existing in signal subspace, from the point of view of fuzzy logic. A FuzzyEn-assisted subspace iterative soft threshold (FESIST) denoising method, which can effectively overcome the deficiency in the existing subspace filtering (SSF) techniques, is presented. The effectiveness of the method is first demonstrated on two synthetic chaotic series and then tested on real biological signals. The results demonstrate the superiority of the proposed method over existing SSF techniques, as well as the empirical mode decomposition and wavelet decomposition approaches.

[1]  Hong-Bo Xie,et al.  Complexity analysis of the biomedical signal using fuzzy entropy measurement , 2011, Appl. Soft Comput..

[2]  Marlize Barkhuizen,et al.  Analysis of process data with singular spectrum methods , 2003 .

[3]  Wei Guo,et al.  Noise Smoothing for Nonlinear Time Series Using Wavelet Soft Threshold , 2007, IEEE Signal Processing Letters.

[4]  J. Kurths,et al.  Estimation of dynamical invariants without embedding by recurrence plots. , 2004, Chaos.

[5]  Thierry Chonavel,et al.  Fast adaptive eigenvalue decomposition: a maximum likelihood approach , 1997, 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[6]  Xin Chen,et al.  Cross-fuzzy entropy: A new method to test pattern synchrony of bivariate time series , 2010, Inf. Sci..

[7]  Zhizhong Wang,et al.  Identification determinism in time series based on symplectic geometry spectra , 2005 .

[8]  Settimo Termini,et al.  A Definition of a Nonprobabilistic Entropy in the Setting of Fuzzy Sets Theory , 1972, Inf. Control..

[9]  Slawomir J. Nasuto,et al.  Evaluation of Empirical Mode Decomposition for Event-Related Potential Analysis , 2011, EURASIP J. Adv. Signal Process..

[10]  C. Diks,et al.  Estimating invariants of noisy attractors. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Wangxin Yu,et al.  Characterization of Surface EMG Signal Based on Fuzzy Entropy , 2007, IEEE Transactions on Neural Systems and Rehabilitation Engineering.

[12]  Jing Hu,et al.  Denoising Nonlinear Time Series by Adaptive Filtering and Wavelet Shrinkage: A Comparison , 2010, IEEE Signal Processing Letters.

[13]  J. Richman,et al.  Sample Entropy Statistics and Testing for Order in Complex Physiological Signals , 2007 .

[14]  Wei-Yen Hsu,et al.  Assembling A Multi-Feature EEG Classifier for Left-Right Motor Imagery Data Using Wavelet-Based Fuzzy Approximate Entropy for Improved Accuracy , 2015, Int. J. Neural Syst..

[15]  Jürgen Kurths,et al.  Statistical Mechanics and Information-Theoretic Perspectives on Complexity in the Earth System , 2013, Entropy.

[16]  David Chelidze,et al.  Smooth local subspace projection for nonlinear noise reduction. , 2014, Chaos.

[17]  Paulo José de Freitas Filho,et al.  Automatic Generation of Type-1 and Interval Type-2 Membership Functions for Prediction of Time Series Data , 2016, IBERAMIA.

[18]  Alan Wee-Chung Liew,et al.  Symplectic geometry spectrum analysis of nonlinear time series , 2014, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[19]  Jiang Wang,et al.  Characterization of complexity in the electroencephalograph activity of Alzheimer's disease based on fuzzy entropy. , 2015, Chaos.

[20]  Saeid Nahavandi,et al.  A New Fuzzy-Based Combined Prediction Interval for Wind Power Forecasting , 2016, IEEE Transactions on Power Systems.

[21]  Hong-Bo Xie,et al.  Classification of ventricular tachycardia and fibrillation using fuzzy similarity-based approximate entropy , 2011, Expert Syst. Appl..

[22]  Milan Paluš,et al.  Detecting Oscillations Hidden in Noise: Common Cycles in Atmospheric, Geomagnetic and Solar Data , 2008 .

[23]  K Lehnertz,et al.  Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: dependence on recording region and brain state. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  S. Beheshti,et al.  A new information-theoretic approach to signal denoising and best basis selection , 2005, IEEE Transactions on Signal Processing.

[25]  Hong-Bo Xie,et al.  A comparative study of pattern synchronization detection between neural signals using different cross-entropy measures , 2010, Biological Cybernetics.

[26]  Julien Clinton Sprott,et al.  Extraction of dynamical equations from chaotic data , 1992 .

[27]  M. O. Sokunbi,et al.  Fuzzy approximate entropy analysis of resting state fMRI signal complexity across the adult life span. , 2015, Medical engineering & physics.

[28]  Laura Sardonini.  Invariants estimation in nonlinear time series , 2007 .

[29]  Haolin Li,et al.  A rolling element bearing fault diagnosis approach based on hierarchical fuzzy entropy and support vector machine , 2016 .

[30]  D. L. Donoho,et al.  Ideal spacial adaptation via wavelet shrinkage , 1994 .

[31]  Boualem Boashash,et al.  Wavelet Denoising Based on the MAP Estimation Using the BKF Prior With Application to Images and EEG Signals , 2013, IEEE Transactions on Signal Processing.

[32]  Elizabeth Bradley,et al.  Nonlinear time-series analysis revisited. , 2015, Chaos.

[33]  Junjie Chen,et al.  The detection of epileptic seizure signals based on fuzzy entropy , 2015, Journal of Neuroscience Methods.

[34]  Mees,et al.  Singular-value decomposition and embedding dimension. , 1987, Physical review. A, General physics.

[35]  Gene H. Golub,et al.  Matrix computations , 1983 .

[36]  Hongyu Li,et al.  Diagnosing planetary gear faults using the fuzzy entropy of LMD and ANFIS , 2016 .

[37]  Noel Cressie,et al.  Deterministic/Stochastic Wavelet Decomposition for Recovery of Signal From Noisy Data , 2000, Technometrics.

[38]  Bin Yu,et al.  Wavelet thresholding via MDL for natural images , 2000, IEEE Trans. Inf. Theory.

[39]  Socrates Dokos,et al.  Hybrid soft computing systems for electromyographic signals analysis: a review , 2014, BioMedical Engineering OnLine.

[40]  Hossein Hassani,et al.  Singular Spectrum Analysis Based on the Minimum Variance Estimator , 2010 .

[41]  Gerhard Keller,et al.  Rigorous statistical procedures for data from dynamical systems , 1986 .

[42]  Hong-Bo Xie,et al.  Measuring time series regularity using nonlinear similarity-based sample entropy , 2008 .

[43]  B. AfeArd.  CALCULATING THE SINGULAR VALUES AND PSEUDOINVERSE OF A MATRIX , 2022 .

[44]  Hong-Bo Xie,et al.  Assessment of Time Series Complexity Using Improved Approximate Entropy , 2011 .

[45]  G. Pfurtscheller,et al.  Brain-Computer Interfaces for Communication and Control. , 2011, Communications of the ACM.

[46]  Søren Holdt Jensen,et al.  FIR filter representations of reduced-rank noise reduction , 1998, IEEE Trans. Signal Process..

[47]  Lotfi A. Zadeh,et al.  Fuzzy probabilities , 1996, Inf. Process. Manag..

[48]  Steve McLaughlin,et al.  Development of EMD-Based Denoising Methods Inspired by Wavelet Thresholding , 2009, IEEE Transactions on Signal Processing.

[49]  Gene H. Golub,et al.  Singular value decomposition and least squares solutions , 1970, Milestones in Matrix Computation.

[50]  D. D. Kosambi.  Statistics in Function Space , 2016 .

[51]  Ling Li,et al.  Fuzzy entropy based nonnegative matrix factorization for muscle synergy extraction , 2016, 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[52]  Olivier Caspary,et al.  Denoising Depth EEG Signals During DBS Using Filtering and Subspace Decomposition , 2013, IEEE Transactions on Biomedical Engineering.

[53]  Martin Vetterli,et al.  Adaptive wavelet thresholding for image denoising and compression , 2000, IEEE Trans. Image Process..

[54]  Hong-Bo Xie,et al.  Fuzzy Approximate Entropy Analysis of Chaotic and Natural Complex Systems: Detecting Muscle Fatigue Using Electromyography Signals , 2010, Annals of Biomedical Engineering.

[55]  Ivan Dvořák,et al.  Singular-value decomposition in attractor reconstruction: pitfalls and precautions , 1992 .

[56]  Jan J. Gerbrands,et al.  On the relationships between SVD, KLT and PCA , 1981, Pattern Recognit..