The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> V. SAMPLE ENTROPY A. DESCRIPTION OF SAMPLE 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> BIB001 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> V. SAMPLE ENTROPY A. DESCRIPTION OF SAMPLE ENTROPY <s> Abnormal heart rate characteristics of reduced variability and transient decelerations are present early in the course of neonatal sepsis. To investigate the dynamics, we calculated sample entropy, a similar but less biased measure than the popular approximate entropy. Both calculate the probability that epochs of window length m that are similar within a tolerance r remain similar at the next point. We studied 89 consecutive admissions to a tertiary care neonatal intensive care unit, among whom there were 21 episodes of sepsis, and we performed numerical simulations. We addressed the fundamental issues of optimal selection of m and r and the impact of missing data. The major findings are that entropy falls before clinical signs of neonatal sepsis and that missing points are well tolerated. The major mechanism, surprisingly, is unrelated to the regularity of the data: entropy estimates inevitably fall in any record with spikes. We propose more informed selection of parameters and reexamination of studies where approximate entropy was interpreted solely as a regularity measure. <s> BIB002 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> V. SAMPLE ENTROPY A. DESCRIPTION OF SAMPLE ENTROPY <s> Multiscale entropy (MSE) was recently developed to evaluate the complexity of time series over different time scales. Although the MSE algorithm has been successfully applied in a number of different fields, it encounters a problem in that the statistical reliability of the sample entropy (SampEn) of a coarse-grained series is reduced as a time scale factor is increased. Therefore, in this paper, the concept of a composite multiscale entropy (CMSE) is introduced to overcome this difficulty. Simulation results on both white noise and 1/f noise show that the CMSE provides higher entropy reliablity than the MSE approach for large time scale factors. On real data analysis, both the MSE and CMSE are applied to extract features from fault bearing vibration signals. Experimental results demonstrate that the proposed CMSE-based feature extractor provides higher separability than the MSE-based feature extractor. <s> BIB003
As discussed above, ApEn has certain disadvantages in the complexity estimation of time series. To address this problem, Richman and Moorman BIB001 proposed sample entropy (SampEn). Unlike ApEn, SampEn can measure the irregularity of time series independent of the embedding dimension m and similarity coefficient r. Therefore, SampEn is relatively consistent and eliminating the bias of ApEn BIB001 . A larger SampEn value indicates the time series with higher complexity, while a samller SampEn value implies the time series with lower irregularity  . The definition of SampEn is given as follows. where B m (r) is the mean value of pattern mean count. B m (r) and B m+1 (r) can be expressed as: where r represents the tolerance of the time series; m represents the embedding dimension; N represents the length of time series. num{d[x(i), x(j)] < r} represents the count of the distance between x(i) and x(j) lower than the tolerance r. Noted that the i = j means the SampEn can not contain self-matches. In SampEn method, it is recommended to set m = 2 and r = (0.1 ∼ 0.25) * SD (SD represents the standard deviation) BIB002 , BIB003 . A flowchart of the SampEn method is illustrated in Figure 3 . SampEn method has four main advantages as follows. First, SampEn is an accurate value of negative average of logarithm of conditional probability, which doesn t rely on the length of data; Second, SampEn doesn t contain self-matches; Third, SampEn has better performance in the consistency; Last, SampEn is insensitive in data length. In addition, the main shortcomings of SampEn lies in its inconsistency for the sparse data and low calculation efficiency.