The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> IV. APPROXIMATE ENTROPY A. DESCRIPTION OF APPROXIMATE ENTROPY <s> Abstract ::: Techniques to determine changing system complexity from data are evaluated. Convergence of a frequently used correlation dimension algorithm to a finite value does not necessarily imply an underlying deterministic model or chaos. Analysis of a recently developed family of formulas and statistics, approximate entropy (ApEn), suggests that ApEn can classify complex systems, given at least 1000 data values in diverse settings that include both deterministic chaotic and stochastic processes. The capability to discern changing complexity from such a relatively small amount of data holds promise for applications of ApEn in a variety of contexts. <s> BIB001 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> IV. APPROXIMATE ENTROPY A. DESCRIPTION OF APPROXIMATE ENTROPY <s> Approximate entropy (ApEn) is a recently developed statistic quantifying regularity and complexity, which appears to have potential application to a wide variety of relatively short (greater than 100 points) and noisy time‐series data. The development of ApEn was motivated by data length constraints commonly encountered, e.g., in heart rate, EEG, and endocrine hormone secretion data sets. We describe ApEn implementation and interpretation, indicating its utility to distinguish correlated stochastic processes, and composite deterministic/ stochastic models. We discuss the key technical idea that motivates ApEn, that one need not fully reconstruct an attractor to discriminate in a statistically valid manner—marginal probability distributions often suffice for this purpose. Finally, we discuss why algorithms to compute, e.g., correlation dimension and the Kolmogorov–Sinai (KS) entropy, often work well for true dynamical systems, yet sometimes operationally confound for general models, with the aid of visual ... <s> BIB002 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> IV. APPROXIMATE ENTROPY A. DESCRIPTION OF APPROXIMATE ENTROPY <s> Entropy, as it relates to dynamical systems, is the rate of information production. Methods for estimation of the entropy of a system represented by a time series are not, however, well suited to analysis of the short and noisy data sets encountered in cardiovascular and other biological studies. Pincus introduced approximate entropy (ApEn), a set of measures of system complexity closely related to entropy, which is easily applied to clinical cardiovascular and other time series. ApEn statistics, however, lead to inconsistent results. We have developed a new and related complexity measure, sample entropy (SampEn), and have compared ApEn and SampEn by using them to analyze sets of random numbers with known probabilistic character. We have also evaluated cross-ApEn and cross-SampEn, which use cardiovascular data sets to measure the similarity of two distinct time series. SampEn agreed with theory much more closely than ApEn over a broad range of conditions. The improved accuracy of SampEn statistics should make them useful in the study of experimental clinical cardiovascular and other biological time series. <s> BIB003 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> IV. APPROXIMATE ENTROPY A. DESCRIPTION OF APPROXIMATE ENTROPY <s> Characteristic patterns of variation over time, namely rhythms, represent a defining feature of complex systems, one that is synonymous with life. Despite the intrinsic dynamic, interdependent and nonlinear relationships of their parts, complex biological systems exhibit robust systemic stability. Applied to critical care, it is the systemic properties of the host response to a physiological insult that manifest as health or illness and determine outcome in our patients. Variability analysis provides a novel technology with which to evaluate the overall properties of a complex system. This review highlights the means by which we scientifically measure variation, including analyses of overall variation (time domain analysis, frequency distribution, spectral power), frequency contribution (spectral analysis), scale invariant (fractal) behaviour (detrended fluctuation and power law analysis) and regularity (approximate and multiscale entropy). Each technique is presented with a definition, interpretation, clinical application, advantages, limitations and summary of its calculation. The ubiquitous association between altered variability and illness is highlighted, followed by an analysis of how variability analysis may significantly improve prognostication of severity of illness and guide therapeutic intervention in critically ill patients. <s> BIB004 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> IV. APPROXIMATE ENTROPY A. DESCRIPTION OF APPROXIMATE ENTROPY <s> Abstract The effective fault diagnosis of rotating machinery is critical to ensure the continuous operation of equipment and is more economical than scheduled maintenance. Traditional signal processing-based and artificial intelligence-based methods, such as wavelet packet transform and support vector machine, have been proved effective in fault diagnosis of rotating machinery, which prevents unexpected machine breakdowns due to the failure of significant components. However, these methods have several disadvantages that make them unable to automatically and effectively extract valid fault features for the effective fault diagnosis of rotating machinery. A novel adaptive learning rate deep belief network combined with Nesterov momentum is developed in this study for rotating machinery fault diagnosis. Nesterov momentum is adopted to replace traditional momentum to enable declining in advance and to improve training performance. Then, an individual adaptive learning rate method is used to select a suitable step length for accelerating descent. To confirm the utility of the proposed deep learning network architecture, two examinations are implemented on datasets from gearbox and locomotive bearing test rigs. Results indicate that the method achieves impressive performance in fault pattern recognition. Comparisons with existing methods are also conducted to demonstrate that the proposed method is more accurate and robust. <s> BIB005
Approximate entropy (ApEn) was first proposed by Pincus BIB001 , BIB002 to quantify the irregularity and unpredictability of time series. ApEn evaluates the probability of occurrence of a new pattern through observing the embedding dimension m and similarity coefficient r. Since the similarity criterion is equivalent to the standard deviation of time series, ApEn is a scale invariant indicator BIB002 . A larger ApEn value means a higher probability of a new pattern occurring of time series, and a smaller ApEn value indicates the time series has lower irregularity. The concept of ApEn is defined as follows. where φ m (r) represents the mean value of logarithm pattern mean count. φ m (r) and φ m+1 (r) can be calculated using Eq. BIB005 . where r represents the tolerance of the time series, m represents the pattern length, N represents the length of time series, and num{d[x(i), x(j)] < r} represents the count of the distance between x(i) and x(j) lower than the tolerance r. Here, the distance is defined as the maximum absolute difference of their corresponding scalar components. In mathematics, ApEn is an approximate value of the negative average natural logarithm of the conditional probability. To achieve an accurate complexity estimation performance, the parameters of ApEn are recommend as follows: pattern length m = 2 and similarity coefficient r = 0.2 * SD (SD represents the standard deviation) BIB001 , BIB002 . A flowchart of the ApEn method is shown in Figure 2 . ApEn has three main advantages as follows. Frist, ApEn has some ability to resist interference and noise, especially the transient anti-interference ability BIB001 ; Second, ApEn has a stable estimation with relatively short data; Third, ApEn is suitable for random signal, certain signal and their combinations BIB003 . In addition, ApEn has five main shortcomings as follows. First, ApEn is biased statistic and heavily dependent on the input signal length. Signals with short length lead to a lower value than expected. VOLUME 6, 2018 Second, ApEn lacks of consistent results for different values of m and r; Third, ApEn is susceptible to strong noises BIB004 ; Fourth, ApEn counts self-matches which is against the basic definition of entropy; Last, ApEn has a low calculation efficiency.