Improved Multivariate Hierarchical Multiscale Dispersion Entropy: A New Method for Industrial Rotating Machinery Fault Diagnosis

This paper proposes an Improved Multivariate Multiscale Dispersion Entropy(IMMDE) combined with Hierarchical Entropy(HE) for vibration signal feature extraction. The traditional coarse-grained calculation is missing the relationship between neighboring sample points in the shift operation, which may lead to missing fault information. Secondly, as the scale increases, the original sequence is gradually shortened, may lead to instability and inaccuracy in entropy estimation when dealing with short-term sequences. The improved coarse-grained calculation method overcomes its limitations to improve the stability, and using the HE method to extract deep fault frequency information from the high and low frequency components of the multivariate signal. Then, the extracted features are dimensioned using the Max-Relevance Min-Redundancy (mRMR) to create a new set of fault features to improve diagnosis efficiency. Finally, the Support Vector Machine(SVM) determines the degree and type of fault. Experiments were conducted with three examples, the results show that IMHMDE can effectively extract the feature information according to mechanical faults’ characteristics and improve the efficiency of fault diagnosis.

fault diagnosis. 23 Faults in rotating machinery are primarily caused by 24 complex internal structures and changing the external 25 environment. Vibration analysis, Acoustic Emission (AE), 26 The associate editor coordinating the review of this manuscript and approving it for publication was Guillermo Valencia-Palomo . temperature trend analysis, and wear debris analysis all are 27 currently the most commonly used methods for analyzing 28 rotating machinery faults. The vibration monitoring signals 29 contain critical information about abnormalities in the inter- 30 nal structure of the machinery [2]. Vibration analysis is very 31 effective at determining the type of rotating machinery faults 32 and the extent of the damage. It can detect the widest range 33 of mechanical faults, such as mechanical looseness, bearing 34 faults, and rotor misalignment [3]. In the actual operation 35 process, the fault pulse information is easily masked [4] by 36 various noises and high-energy low-frequency components 37 because of the weakness of the early fault signal of rotating 38 machinery. There are primarily two methods to deal with 39 such problems. The first is to enhance the effective signal 40 by filtering out the noise, and the second is to select the 41 Dispersion Entropy (HDE) combined with the improved 97 Laplace fraction to determine the health of bearings, and 98 the results demonstrated that the method could accurately 99 identify different fault types of rolling bearings as well as 100 their severity. Song [26] proposed a fruit fly search algorithm 101 to improve the VMD combined with the HDE method for 102 detecting faults in diesel engine injectors. The entropy meth-103 ods proposed above all rely on single direction or single-104 channel vibration signal information to identify different 105 types of fault signals. However, rotating machinery in indus-106 trial production typically operates in a constantly changing 107 environment, signals collected from different directions also 108 contain important fault information [27]. As the impulses 109 generated by faulty components are propagated over long dis-110 tances, they are inevitably weakened, resulting in the loss of 111 fault information. Thus, the sensor signals must be collected 112 from various directions to explore the potential information 113 of various types of faults to improve fault detection accuracy. 114 Based on the multiscale entropy theory [28], it is more 115 practical to extract fault information by analyzing vibration 116 signals from multiple channels simultaneously. Theoretical 117 methods that have been proposed such as Multivariate Multi-118 scale Sample Entropy (MMSE) [29] and Multivariate Mul-119 tiscale Dispersion Entropy (MMDE) [30]. Because of its 120 high computational efficiency and robust noise immunity, 121 MMDE has been widely used in the fault diagnosis of rotating 122 machinery. In this paper, considering the shortage of the 123 traditional coarse-grained calculation that cannot consider 124 all-time series simultaneously, we improve it and propose 125 combining the Improved Multivariate Multiscale Dispersion 126 Entropy (IMMDE) and He. We called it Improved Multivari-127 ate Hierarchical Multiscale Dispersion Entropy (IMHMDE) 128 and used it to analyze and study the vibration signals to 129 improve the accuracy of fault damage identification. It allows 130 the extraction of fault information from the vibration signals 131 of multiple sensors simultaneously. Still, it also considers 132 the low-frequency and high-frequency components of the 133 signals, effectively avoiding the loss of fault information. 134 It can extract the fault information from the vibration sig-135 nals of multiple sensors simultaneously while the low and 136 high-frequency components of the signals are considered, 137 effectively avoiding the loss of fault information. Finally, the 138 extracted IMHMDE features are reduced in dimensionality 139 using [31]. The newly constructed feature set is divided into 140 the training and testing sets before being fed into SVM for 141 fault identification.

142
In summary, the main structure of this paper is as fol-143 lows: Section 2 introduces the relevant theory and analysis. 144 Section 3 explains the proposed fault diagnosis method and 145 steps and includes a flow chart. Section 4 demonstrates the 146 feasibility of analyzing vibration signals collected from rotat-147 ing machinery components through three case studies, and 148 Section 5 concludes with a summary of the findings and a 149 look ahead. The specific steps of the algorithm are shown 150 below.

151
µ k is the expected value, and σ 2 k denotes the variance. Y is c represents the class.

168
(2): The reconstruction is carried out as follows, according 169 to the theory of multi-dimensional embedding to reconstruct: (4): Finally, the definition of multivariate dispersion 189 entropy is derived from Shannon's entropy as follows: The order of the multi-channel signal has no effect on the  [33]. The pre-202 cise procedure is as follows:

203
(1): Given multi-channel data I = u k,b k=1,2,...,n b=1,2,...,L and 204 scale factor τ , the coarse-grained analysis I is as follows: (2): coarse-grained calculation of the entropy of the multi-207 variate time series {u τ k,b } with the same parameters. This leads 208 to the formula for MMDE: Multivariate signal I , embedding dimension m, class c, 212 delay d, p denotes the probability of obtaining a potential 213 dispersion pattern for a coarse-grained sequence. The coarse-214 grained calculation of MMDE is shown in the Fig. 1(a). 215 MMDE analyzes time series by extending them from a sin-216 gle scale to multiple scales. It is primarily calculated by 217 averaging non-overlapping signals to produce multiple series 218 and thus calculating multivariate entropy. This method still 219 excludes τ − 1 multivariate time series from the calcu-220 lation and does not account for the relationships between 221 coarse-grained time series, resulting in a lack of statistical 222 information [33]. The traditional method for performing multiscale analysis is 226 to compute the mean of adjacent elements using a scale factor. 227 As the scale factor increases, the value of multiscale disper-228 sion entropy decreases, resulting in a decrease in the number 229 of samples in the coarse-grained sequence and instability of 230 the entropy value. Thus, the coarse-grained calculation is 231 improved for the analysis of multi-channel sequences, and 232 the improved coarse-grained process is depicted in Fig. 1(b). 233 When the time series of the k-th channel is at τ = 2, the 234 calculation formula using equation (8): The mvDE of all coarse-grained time series at the same 238 scale is calculated and averaged to produce the final results. 239 This method can improve the estimation accuracy of entropy, 240 In this part, a three-channel white noise signal and a 246 three-channel 1/f noise signal are selected for analysis.

247
IMMDE, Refined Composite Multivariate Multiscale 248 Dispersion Entropy (RCMMDE) [32], and MMDE all 249 have the same parameters but use different coarse-grained 250 is too large, data with different amplitudes will be divided 261 too finely. This paper sets m = 3, τ = 20, d = 1 and first 262 discusses the influence of class c on the estimation results 263 of different entropy value. Fig. 2 shows the results after 264 testing, and the entropy estimation results of IMMDE and 265 RCMMDE are more stable than MMDE under the influence 266 of different c parameters. After the embedding dimension m 267 is determined, the more dispersion patterns exist, the more 268 information they contain. As c gets larger, more information 269 is captured, but the computation time also increases, there-270 fore, the paper sets c to 6 [19]. Additionally, the effectiveness 271 and precision of the diagnosis are impacted by the scale 272 selection. Too small scale features will lead to incomplete 273 extraction of fault information, and too large scale features 274 will result in feature redundancy and reduced computational 275 efficiency.

276
For the value of the embedding dimension m, the class c 277 is set to 6, so the value can only be 2 or 3. For the parameter 278 setting in reference [34], [35], this paper determines that the 279 value of each parameter m = 3, c = 6, d = 1, τ = 20.

280
In this section, synthetic signals are investigated to demon-281 strate the stability of the improved coarse-grained compu-282 tation for entropy estimation during feature extraction and 283 the effectiveness of identifying multi-channel signals with 284 different levels of complexity. A three-channel WGN signal, 285 a two-channel WGN signal and a one-channel 1/f noise, 286 a one-channel WGN and a two-channel 1/f noise, a three-287 channel 1/f noise. 288 Firstly, the paper extracts the features of three methods 289 separately, the results are shown in Fig. 3. According to 290 the results, because the uncertainty of the WGN noise time 291 series is higher than that of 1/f noise, the information of 292 the WGN noise signal is primarily located on the low scale, 293 so when using entropy calculation, the entropy value of 294 its entire sequence is larger, and the entropy value of the 295 signal containing more WGN noise channels is higher at 296 the low scale. Then, as the scale increases, the entropy 297 converges gradually, 1/f noise will have a higher signal 298 complexity than WGN noise due to its long-range corre-299 lation, so its entropy converges slowly. When the three 300 types of methods are compared, IMMDE and RCMMDE 301 are found to provide more consistent entropy estimations for 302 more complex multi-channel non-smooth signals. Also, the 303 improved coarse-grained calculation method is more effec-304 tive in differentiating different types of channel signals at low 305 scales.

306
Three-channel noise signals with a length of 1024 are 307 tested 30 times in order further to demonstrate the stability of 308 the improved multiscale calculation method. The coefficient 309 of variation is obtained by dividing the standard deviation 310 of the sequence by the average value. Fig. 4 shows the final 311 results, the improved coarse-grained calculation method used 312 in this research has a lower coefficient of variation than both 313 MMDE and RCMMDE at different scales. The difference is 314 obvious when processing the three-channel 1/f signal with the 315 highest complexity. frequency Q 0 (x) and low-frequency Q 1 (x).

325
(2) When j = 0 or j = 1, the operator matrix Q j (x) obtained 326 by decomposing the components of the first level is defined 327 as follows:  in equation (13).
The decomposition process when k = 3 is shown in Fig. 5. 337 (4) Repeat the preceding steps to perform hierarchical 338 decomposition of the multi-channel time series. The mul-339 tivariate hierarchical component of each node is obtained, 340 and its IMMDE value is calculated, allowing the IMHMDE 341 calculation formula to be obtained. In comparison to IMMDE, IMHMDE has an additional 345 parameter k that represents the number of layers. Because a 346 too large k will increase computation time and affect accuracy 347 VOLUME 10, 2022 and k is too small to extract the high-frequency and low-   The degree of similarity between variables is frequently 392 measured using mutual information methods. The following 393 formula can be used to calculate the degree of similarity 394 between two random variables, x, and y.
p(x), p(y) denotes their probabilities respectively and 397 p(x, y) is the joint probability density. The criterion for max-398 imum correlation is as follows: |S| 2 Min-redundancy eliminates features that are more depen-408 dent on one another and chooses features that are the most 409 dissimilar to one another. The method is created by combin-410 ing and optimizing these two criteria.      Firstly, the bearing fault signals of the above test data 466 are used for research to verify the accuracy of the proposed 467 method results. Fig. 11 depicts the waveforms of the various 468 FIGURE 11. Original bearing fault vibration signal waveform diagram; red represents fan end, and blue represents drive end.   features. Fig. 12 shows the results of the IMHMDE feature 479 selection. The reconstructed feature set is then fed into SVM 480 to compare the results of various fault identification methods. 481 As the number of sensitive features increases, Fig. 13  Due to the random nature of the test results, the scale is 496 set to 4 for 25 trials and the results are shown in Fig. 15. The 497 specific diagnostic results are recorded in Table 4. It can be 498 found that the average accuracy of IMHMDE reaches about 499 99.68%, and the accuracy of each fault identification is better 500 than that of other methods. Receiver Operator Characteristic 501 (ROC), Area Under the Curve (AUC) can evaluate the model. 502       Table 6. Comparing the experimental results, the single chan-518 nel signal also achieves good fault diagnosis results. IHMDE, 519 HMDE, IMDE is more effective than MMDE, which shows 520 that the improved coarse-grained calculation and hierarchical 521 entropy method can improve the fault diagnosis accuracy 522 truly. Considering that the mechanical rolling bearings usually 526 work in a changing environment, to test the signal processing 527 ability of this method under varying working conditions, the 528       Table 7 shows the specifics of the signal. Here, 400 test  samples and 400 training samples are randomly selected to 534 verify the bearing signals in these ten states. 535 Fig. 17 depicts the fault features selected for the variable 536 operating conditions. The number of features is set to 1 to 20 537 for 15 trials respectively, and the average diagnostic rate 538 results are shown in Fig. 18. It can be found that the accuracy 539 of the different methods starts to stabilize at a scale of 13,540 indicating that increasing the number of features will not 541 have a significant impact on the results of fault diagnosis. For 542 further analysis, the scales of different entropy are set to 13 in 543 this case. Fig. 19 shows the confusion matrix of IMHMDE 544 with a scale of 13, and only two samples are misclassified in 545 its classification results.

546
Here, 25 trials are conducted for six multivariate entropy 547 models to verify the stability of IMHMDE, and the results are 548 shown in Fig. 20, and detailed information on the diagnostic 549 results of the six methods is recorded in       Similarly, we conduct some contrast experiments to detect 561 the advantages of the multivariate approach compared to 562 unvariate approaches. Without loss of generality, 15 trials are 563 performed for each approach. From Table 9, it can be found 564 that some of the univariate signals are diagnosed better than 565 the multivariate method under variable working conditions, 566 while the results of IMHMDE are similar to those of the 567 univariate method, which indicates that the method is better 568 at feature extraction. The bearing compound fault data of Xi'an Jiaotong Uni-571 versity is selected to verify the fault diagnosis results. The 572 test device is shown in Fig. 22. The platform comprises an 573 AC motor, motor speed controller, shaft, supporting bear-574 ing, hydraulic loading system, and tested bearings. The test 575 bearing type is LDK UER204 rolling bearing, the sampling 576  frequency is 25.6 kHz/min, and the number of balls is 8. parts of the test bearing cover the outer ring, inner ring, cage 583 and rolling element, and the types of failure are outer ring 584 wear, outer ring cracking, inner ring wear, cage fracture, etc. 585 VOLUME 10, 2022        scales are shown in Table 12. It can be found that the average 593 accuracy of IMHMDE at various scales is significantly higher 594 than that of other methods. At a scale equal to 4, IMHMDE 595 achieves an accuracy of around 96 %, and the final result 596 is stable at about 99% with the increase of the scale. Since 597 IMHMDE contains more fault information when extracting 598 features, it can effectively locate different faults when identi-599 fying such more complex fault types.

600
The Confusion matrix results of fault diagnosis are shown 601 in Fig. 25. It can be found that only one sample was misclas-602 sified. The feature scale of IMHMDE is then set to 17 and 603   the feature set is fed into the SVM for 15 trials.  Case 1 and case 2 fully prove the excellent performance 609 of this method in various fault diagnoses of bearings in dif-610 ferent environments. Similarly, the components of rotating 611 machinery that often fail include gears, the gearbox contains 612 more components, and the environment is more complex. 613 Therefore, the vibration signal data of the gearbox are used 614 to verify the method.

615
The primary test environment is shown in Fig. 27. In this 616 paper, the signal of the four sensors under the condition of 617 20Hz-0V is analyzed. The fault type is shown in Table 15, and 618 the waveform of the vibration signal collected by the sensor 619 is shown in Fig. 28.

620
The IMHMDE is used to extract high-dimensional fault 621 features from the original signal of the gearbox. In order to 622 verify the identification performance of this method for the 623 gear fault of a gearbox, the corresponding number of features 624 is selected. 20 features are selected in Fig. 29. Samples are 625 selected to input into the SVM classifier for recognition. 626 Table 16 records the results after 15 trials. It is found that the 627 average accuracy and scale are positively correlated. It can 628 VOLUME 10, 2022 suitable for different methods is not the same. The method 630 proposed in this paper has higher accuracy than other models, 631 and the recognition accuracy at a lower scale is also higher 632 than that of other models, so the efficiency of fault diagnosis 633 will also be improved. method proposed in this paper is 634 significantly better than the other models.

799
His research interests include aero engine, gas 800 turbine coupling structure dynamics and vibration 801 control, vibration analysis, testing and control of 802 rotating machinery, structural dynamics of new 803 composite materials, intelligent material, structure 804 vibration utilization and control, active, and pas-805 sive vibration control in other special fields. 806 807 VOLUME 10, 2022