The Multiscale Entropy Algorithm and Its Variants: A Review <s> Multiscale Fuzzy Sample Entropy <s> Sampe Entropy (SampEn), a measure quantifying regularity and complexity, is believed to be an effective analyzing method of diverse settings that include both deterministic chaotic and stochastic processes, particularly operative in the analysis of physiological signals that involve relatively small amount of data. However, the similarity definition of vectors is based on Heaviside function, of which the boundary is discontinuous and hard, may cause some problems in the validity and accuracy of SampEn. Sigmoid function is a smoothed and continuous version of Heaviside function. To overcome the problems SampEn encountered, a modified SampEn (mSampEn) based on nonlinear Sigmoid function was proposed. The performance of mSampEn was tested on the independent identically distributed (i.i.d.) uniform random numbers, the MIX stochastic model, the Rossler map, and the Hennon map. The results showed that mSampEn was superior to SampEn in several aspects, including giving entropy definition in case of small parameters, better relative consistency, robust to noise, and more independence on record length when characterizing time series generated from either deterministic or stochastic system with different regularities. <s> BIB001 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Multiscale Fuzzy Sample Entropy <s> A modified sample entropy (mSampEn), based on the nonlinear continuous and convex function, has been proposed and proven to be superior to the standard sample entropy (SampEn) in several aspects. In this Letter, we empirically investigate the ability of the mSampEn statistic combined with surrogate data method to detect determinism. The effects of the datasets length and noise on the proposed method to differentiate between deterministic and stochastic dynamics are tested on several benchmark time series. The noise performance of the mSampEn statistic is also compared with the singular value decomposition (SVD) and symplectic geometry spectrum (SGS) based methods. The results indicate that the mSampEn statistic is a robust index for detecting determinism in short and noisy time series. <s> BIB002 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Multiscale Fuzzy Sample Entropy <s> Feature extraction from vibration signals has been investigated extensively over the past decades as a key issue in machine condition monitoring and fault diagnosis. Most existing methods, however, assume a linear model of the underlying dynamics. In this study, the feasibility of devoting nonlinear dynamic parameters to characterizing bearing vibrations is studied. Firstly, fuzzy sample entropy (FSampEn) is formulated by defining a fuzzy membership function with clear physical meaning. Secondly, inspired by the multiscale sample entropy (multiscale SampEn) which is originally proposed to quantify the complexity of physiological time series, we placed approximate entropy (ApEn), fuzzy approximate entropy (FApEn) and the proposed FSampEn into the same multiscale framework. This led to the developments of multiscale ApEn, multiscale FApEn and multiscale FSampEn. Finally, all four multiscale entropies along with their single-scale counterparts were employed to extract discriminating features from bearing vibration signals, and their classification performance was evaluated using support vector machines (SVMs). Experimental results demonstrated that all four multiscale entropies outperformed single-scale ones, whilst multiscale FSampEn was superior to other multiscale methods, especially when analyzed signals were contaminated by heavy noise. Comparisons with statistical features in time domain also support the use of multiscale FSampEn. <s> BIB003 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Multiscale Fuzzy Sample Entropy <s> Abstract A study of ventricular fibrillation (VF) and ventricular tachycardia (VT) was undertaken using modified sample entropy analysis. Sample entropy (SampEn) is believed to provide the quantitative information about the complexity of experimental data that is corrupted by noise, short in data length. However, the similarity definition of vectors is based on a Heaviside function, of which the boundary is discontinuous and hard, which may cause the problems in the validity and accuracy of SampEn. To deal with the encountered problems, a modified sample entropy ( m SampEn) based on fuzzy membership function was proposed. Its performances on characterizing VF and VT signals, as well as several simulated time series, demonstrate that m SampEn can more efficiently measure the complexity of time series. It is shown that, as a criteria for discriminating between VF and VT, m SampEn provides a significantly ( p <s> BIB004 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Multiscale Fuzzy Sample Entropy <s> Brain activities commonly recorded using the electroencephalogram (EEG) are contaminated with ocular artifacts. These activities can be suppressed using a robust independent component analysis (ICA) tool, but its efficiency relies on manual intervention to accurately identify the independent artifactual components. In this paper, we present a new unsupervised, robust, and computationally fast statistical algorithm that uses modified multiscale sample entropy (mMSE) and Kurtosis to automatically identify the independent eye blink artifactual components, and subsequently denoise these components using biorthogonal wavelet decomposition. A 95% two-sided confidence interval of the mean is used to determine the threshold for Kurtosis and mMSE to identify the blink related components in the ICA decomposed data. The algorithm preserves the persistent neural activity in the independent components and removes only the artifactual activity. Results have shown improved performance in the reconstructed EEG signals using the proposed unsupervised algorithm in terms of mutual information, correlation coefficient, and spectral coherence in comparison with conventional zeroing-ICA and wavelet enhanced ICA artifact removal techniques. The algorithm achieves an average sensitivity of 90% and an average specificity of 98%, with average execution time for the datasets ( $N = 7$ ) of 0.06 s ( ${\rm SD} = 0.021$ ) compared to the conventional wICA requiring 0.1078 s ( ${\rm SD} = 0.004$ ). The proposed algorithm neither requires manual identification for artifactual components nor additional electrooculographic channel. The algorithm was tested for 12 channels, but might be useful for dense EEG systems. <s> BIB005
Xie et al. proposed a modification in the sample entropy algorithm BIB001 . In order to measure the match degree of two vectors, they suggested to use the nonlinear Sigmoid function instead of the Heaviside function (the Heaviside function presents a discontinuous and hard boundary, whereas the Sigmoid function is smooth and continuous). Moreover, in the modified sample entropy algorithm, the vector sequences are generalized by removing the local baseline. Xie et al. reported that the modified sample entropy leads to better performances in term of relative consistency, freedom of parameter selection, robustness to noise and independence on the data length when characterizing time series with different regularities BIB001 . This new algorithm has been used in different kinds of studies (see, e.g., BIB005 ). Moreover, Kong and Xie applied this modified sample entropy algorithm to classify ventricular tachycardia and fibrillation and reported good results BIB004 . In 2010, Xie et al. proposed to use the modified sample entropy (using a Gaussian function instead of the nonlinear Sigmoid function) in order to determine whether a time series arises from a stochastic or a deterministic process BIB002 . They reported that the modified sample entropy is robust for detecting determinism in short and noisy signals BIB002 . In 2009, Chen et al. also proposed the use of the concept of fuzzy sets [30] . They thus developed the FuzzyEn measure, which relies on fuzzy membership functions instead of the Heaviside function. This leads to make a fuzzy boundary [30] . Chen et al. reported that FuzzyEn corresponds to a more accurate entropy definition than sample entropy; it owns stronger relative consistency and less dependence on data length [30] . Zhang et al. used the same approach in a multiscale framework for fault feature extraction: they used a Gaussian type fuzzy membership function in the similarity definition of vectors  . Xiong et al. went further in the work of Chen et al. [30] and proposed a new fuzzy membership function leading to a new fuzzy sample entropy and its multiscale version BIB003 . They reported that, when used to extract discriminating features of bearing vibration signals, their multiscale fuzzy sample entropy outperforms other multiscale entropy methods, especially when the signals contain heavy noise BIB003 .