The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> VII. PERMUTATION ENTROPY A. DESCRIPTION OF PERMUTATION ENTROPY <s> We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations. <s> BIB001 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> VII. PERMUTATION ENTROPY A. DESCRIPTION OF PERMUTATION ENTROPY <s> Timely detection of unusual and/or unexpected events in natural and man-made systems has deep scientific and practical relevance. We show that the recently proposed conceptually simple and easily calculated measure of permutation entropy can be effectively used to detect qualitative and quantitative dynamical changes. We illustrate our results on two model systems as well as on clinically characterized brain wave data from epileptic patients. <s> BIB002 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> VII. PERMUTATION ENTROPY A. DESCRIPTION OF PERMUTATION ENTROPY <s> Abstract Permutation entropy quantifies the diversity of possible orderings of the values a random or deterministic system can take, as Shannon entropy quantifies the diversity of values. We show that the metric and permutation entropy rates—measures of new disorder per new observed value—are equal for ergodic finite-alphabet information sources (discrete-time stationary stochastic processes). With this result, we then prove that the same holds for deterministic dynamical systems defined by ergodic maps on n -dimensional intervals. This result generalizes a previous one for piecewise monotone interval maps on the real line [C. Bandt, G. Keller, B. Pompe, Entropy of interval maps via permutations, Nonlinearity 15 (2002) 1595–1602.] at the expense of requiring ergodicity and using a definition of permutation entropy rate differing modestly in the order of two limits. The case of non-ergodic finite-alphabet sources is also studied and an inequality developed. Finally, the equality of permutation and metric entropy rates is extended to ergodic non-discrete information sources when entropy is replaced by differential entropy in the usual way. <s> BIB003 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> VII. PERMUTATION ENTROPY A. DESCRIPTION OF PERMUTATION ENTROPY <s> This paper investigates the usage of permutation entropy for working status characterization of rotary machines. As a statistical measure, the permutation entropy describes complexity of a time series or signal measured on a physical system through phase space reconstruction, and takes into account non-linear behavior of the time series, as often seen in vibration signals of rotary machines. Thus it can be served as a viable tool for detecting dynamic changes of the machine working status. The effect of embedded dimension and time delay on calculation of the permutation entropy value has been studied, and the validity of the permutation entropy for detecting dynamic change of a physical system is studied through a well known non-linear system, the Logistic Map. Comparison with other complexity measures using a numerically formulated signal has also been investigated. Experimental results on bearing vibration analysis have then confirmed that the permutation entropy provides an effective measure for monitoring the working status of rolling bearings. <s> BIB004 </s> The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review <s> VII. PERMUTATION ENTROPY A. DESCRIPTION OF PERMUTATION ENTROPY <s> Bilal Fadlallah,1,* Badong Chen,2,† Andreas Keil,3 and Jose Principe1,* 1Computational NeuroEngineering Laboratory, Department of Electrical and Computer Engineering, University of Florida, Gainesville, Florida 32611, USA 2Institute of Artificial Intelligence and Robotics, Xi’an Jiaotong University, Xi’an 710049, China 3NIMH Center for the Study of Emotion and Attention, Department of Psychology, University of Florida, Gainesville, Florida 32611, USA (Received 6 March 2012; revised manuscript received 5 December 2012; published 20 February 2013) <s> BIB005
As a statistical measure, permutation entropy describes complexity of a time series or signal measured on a physical system through phase space reconstruction, and takes into account non-linear behavior of the time series, as often seen in vibration signals of rotary machines. Thus, PE can be served as a viable tool for dynamic changes detection of the machine working status. Permutation entropy (PE) was proposed by Bandt and Pompe BIB001 to measure the irregularity of time series. Different from ApEn, SampEn and FE, PE only utilizes the order of the amplitude of time series BIB004 . Therefore, PE has a corresponding higher calculation efficiency BIB002 and is robust under non-linear distortion of time series BIB004 . PE has been widely applied into the fault diagnosis of rotating machinery due to its sensitivity to the dynamical change BIB004 , BIB003 . The concept of PE is defined as follows. where p(π) denotes the relative frequency for each permutation π. num{i|i ≤ T − n, (x i+1 , · · · , x i+n ) has type π} indicates the number of permutation π under order m. The calculation process of the PE is shown in Figure 12 . PE is an appropriate complexity measure for chaotic time series, especially in the presence of dynamical and observational noise. PE has four main advantages as follows. First, PE has a high calculation efficiency, which can be used to compute huge data set BIB004 ; Second, PE has a good performance of complexity estimation; Third, PE has good robust ability to noise BIB001 ; Last, PE does not require any model assumption and is suitable for the analysis of nonlinear processes. The main shortcoming of PE lies in its inability to classify well defined patterns of a particular design BIB005 .