Multiscale Permutation Entropy (MPE) is a presented nonlinear dynamic technology for measuring the randomness and detecting the nonlinear dynamic change of time sequences and can be used effectively to extract the nonlinear dynamic wear fault feature of gear tooth surface from vibration signals of gear set. To solve the subjectivity drawback of threshold parameter selection process in MPE method, a joint calculation method based on the Mutual Information (MI) and improved False Nearest Neighbor (FNN) principle for calculating threshold parameters for MPE method was presented in this article. Then, the influence of threshold parameters on the identification accuracy of fault features with the MPE was studied by analyzing simulation data. Through the simulation analysis, the effectiveness of the proposed MPE method is validated. Finally, the wear failure test of spur gear was carried out, and the proposed method was applied to analyze the experimental data of fault signal. Meanwhile, the vibration characteristics of the fault signal are acquired. The analysis results show that the proposed method can effectively realize the fault diagnosis of gear box and has higher fault identification accuracy than the existing methods

Lihua Wang

Longlong Li

Xiayi Liu

Xintao Zhou

Yahui Cui

English

Vibroengineering PROCEDIA

 24 VIBROENGINEERING PROCEDIA. OCTOBER 2019, VOLUME 28  
Fault identification technology for gear tooth surface 
wear based on MPE method by MI and improved FNN 
algorithm 
Xintao Zhou1, Yahui Cui2, Xiayi Liu3, Longlong Li4, Lihua Wang5 
School of Machinery and Precision Instrument Engineering, Xi’an University of Technology, Xi’an, China 
1, 2Corresponding authors 
E-mail: 1xtzhou2014sc@stu.xaut.edu.cn, 2cyhxut@xaut.edu.cn, 3liuxy@stu.xaut.edu.cn, 
4LonglongLi@stu.xaut.edu.cn, 51170211016@stu.xaut.edu.cn 
Received 9 September 2019; accepted 23 September 2019 
DOI https://doi.org/10.21595/vp.2019.21001 
Copyright © 2019 Xintao Zhou, et al. This is an open access article distributed under the Creative Commons Attribution License, which 
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 
Abstract. Multiscale Permutation Entropy (MPE) is a presented nonlinear dynamic technology 
for measuring the randomness and detecting the nonlinear dynamic change of time sequences and 
can be used effectively to extract the nonlinear dynamic wear fault feature of gear tooth surface 
from vibration signals of gear set. To solve the subjectivity drawback of threshold parameter 
selection process in MPE method, a joint calculation method based on the Mutual Information 
(MI) and improved False Nearest Neighbor (FNN) principle for calculating threshold parameters 
for MPE method was presented in this article. Then, the influence of threshold parameters on the 
identification accuracy of fault features with the MPE was studied by analyzing simulation data. 
Through the simulation analysis, the effectiveness of the proposed MPE method is validated. 
Finally, the wear failure test of spur gear was carried out, and the proposed method was applied 
to analyze the experimental data of fault signal. Meanwhile, the vibration characteristics of the 
fault signal are acquired. The analysis results show that the proposed method can effectively 
realize the fault diagnosis of gear box and has higher fault identification accuracy than the existing 
methods. 
Keywords: multiscale permutation entropy, mutual information, improved false nearest neighbor, 
delay time, embedded dimension, scale factor and fault feature. 
1. Introduction 
Gear trains always work under tough circumstance such as varying loading and heavy 
background noise, and the fault vibration signal about the gear wear is originally weak, prone to 
be contaminated by background noise [1, 2]. Recently, due to the computational simplicity, 
anti-noise and calculating robustness, entropy provides a new way to extract fault feature about 
mechanical equipment and has been widely applied in various applications [3, 4]. Due to the 
complexity and variability of practical problems, in order to make the value of entropy method to 
solve these problems efficiently. For illustration, the approximate entropy method, fuzzy entropy 
method, sample entropy method and multiscale permutation entropy (MPE) method [5, 6]. The 
MPE method expands from single scale to multi-scale phase space, so as to ensure the integrity of 
local information and overall information of the vibration characteristic [7, 8]. Additionally, this 
method can not only effectively solve the disturbance of noise to fault characteristics, but also 
competently detect weak fault signals. Simultaneously, the MPE method can solve the multi-scale 
coupling problem between multiple faults and accurately reflect the dynamic mutation capability 
of the fault system [9].  
The calculation of MPE method is closely interconnected to the algorithm parameters, and 
dissimilar embedding dimensions and delay time will have great influence on the calculation 
results of entropy value. In this paper, a new technique for gear fault identification is developed 
using MI and improved FNN techniques for MPE method. Firstly, according to the basic principle 
and advantage of MI method, the minimum parameter of delay time 𝜏 is implemented of fault 
FAULT IDENTIFICATION TECHNOLOGY FOR GEAR TOOTH SURFACE WEAR BASED ON MPE METHOD BY MI AND IMPROVED FNN ALGORITHM.  
XINTAO ZHOU, YAHUI CUI, XIAYI LIU, LONGLONG LI, LIHUA WANG 
 ISSN PRINT 2345-0533, ISSN ONLINE 2538-8479, KAUNAS, LITHUANIA 25 
vibration signal. Then, the optimal embedding dimension 𝑚 is calculated by the improved FNN 
method of the phase space. Finally, through simulation analysis and gear fault experimental 
research, and a joint calculation method based on the MI and improved PNN principle for 
calculating threshold parameters for MPE method was confirmed in this article. 
2. Review of the theoretical backgrounds 
2.1. Principle of MPE 
(1) Construct the phase space matrix of time sequences is shown in Eq. (1): 
𝑌 =
⎣
⎢
⎢
⎢
⎢
⎡𝑌ଵ𝑌ଶ
⋮
𝑌௝
⋮
𝑌௄⎦
⎥
⎥
⎥
⎥
⎤
=
⎣
⎢
⎢
⎢
⎢
⎡𝑥(1) 𝑥(1 + 𝜏) ⋯ 𝑥(1 + (𝑚 − 1)𝜏)𝑥(2) 𝑥(2 + 𝜏) ⋯ 𝑥(2 + (𝑚 − 1)𝜏)
⋮ ⋮ ⋮ ⋮
𝑥(𝑗) 𝑥(𝑗 + 𝜏) ⋯ 𝑥(𝑗 + (𝑚 − 1)𝜏)
⋮ ⋮ ⋮ ⋮
𝑥(𝐾) 𝑥(𝐾 + 𝜏) ⋯ 𝑥(𝐾 + (𝑚 − 1)𝜏)⎦
⎥
⎥
⎥
⎥
⎤
, (1)
where, 𝑚 is the embedding dimension; 𝜏 is the delay time; 𝐾  is the number of reconstructed 
components. 
(2) Ascending arrangement of elements in each reconstructed component is shown in Eq. (2): 
𝑥൫𝑗 + ൫𝑗(ଵ) − 1൯𝜏൯ ≤ 𝑥൫𝑗 + ൫𝑗(ଶ) − 1൯𝜏൯ ≤ ⋯ ≤ 𝑥൫𝑗 + ൫𝑗(௠) − 1൯𝜏൯. (2)
(3) Calculation the entropy value and normalization is shown in Eq. (3): 
𝐻௉ = −
∑ 𝑃௚ln𝑃௚௄௚ୀଵ
ln(𝑚!) . (3)
(4) Coarse granulation time sequences and calculate the permutation entropy value is shown 
in Eq. (4): 
𝑀𝑃𝐸(𝑋, 𝑠, 𝑚, 𝜏) = 𝑃𝐸 ቆ1𝑠 ෍ 𝑥௜
௝௦
௜ୀ(௝ିଵ)௦ାଵ
, 𝑚, 𝜏ቇ，1 ≤ 𝑗 ≤ 𝑁𝑠 . (4)
2.2. The method of calculation crucial threshold parameters 
2.2.1. Principle of the MI method 
That is, the relationship between MI of time sequences and delay time is shown in Eq. (5): 
𝐼(𝜏) = 𝐼(𝑥௜, 𝑥௜ାఛ) = ෍ 𝑃(𝑥௜, 𝑥௜ାఛ)logଶ ቈ
𝑃(𝑥௜, 𝑥௜ାఛ)
𝑃(𝑥௜)𝑃(𝑥௜ାఛ)቉
ே
௜ୀଵ
, (5)
where, 𝑝(𝑥௜) is the probability distribution function of 𝑥௜; 𝑝(𝑥௜ାఛ) is the probability distribution 
function of 𝑥௜ାఛ; 𝑝(𝑥௜, 𝑥௜ାఛ) is the combined probability distribution function of 𝑥௜, 𝑥௜ାఛ. 
2.2.2. Principle of the Improved FNN Method 
The FNN points method is applied to calculate the relationship between 𝐸(𝑚) and 𝑚, as 
shown in Eqs. (6-7): 
FAULT IDENTIFICATION TECHNOLOGY FOR GEAR TOOTH SURFACE WEAR BASED ON MPE METHOD BY MI AND IMPROVED FNN ALGORITHM.  
XINTAO ZHOU, YAHUI CUI, XIAYI LIU, LONGLONG LI, LIHUA WANG 
26 VIBROENGINEERING PROCEDIA. OCTOBER 2019, VOLUME 28  
𝐸(𝑚) = 1𝑁 − 𝑚𝜏 ෍
ฮ𝑋௠ାଵ(𝑖) − 𝑋௠ାଵ௙ (𝑖)ฮ
ฮ𝑋௠(𝑖) − 𝑋௠௙ (𝑖)ฮ
ேି௠ఛ
௜ୀଵ
, (6)
𝐸∗(𝑚) = 1𝑁 − 𝑚𝜏 ෍ ฮ𝑋௠ାଵ(𝑖) − 𝑋௠ାଵ
௙ (𝑖)ฮ
ேି௠ఛ
௜ୀଵ
, (7)
where, 𝑚-dimensional time sequences 𝑋௠(𝑖), and the nearest neighbor point 𝑋௠௙ (𝑖); (𝑚 + 1)-
dimensional time sequences 𝑋௠ାଵ(𝑖), and the nearest neighbor point 𝑋௠ାଵ௙ (𝑖); ‖•‖ is the ∞-norm. 
3. Numerical simulation and verification 
In this section, the simulation model is shown in Eq. (8): 
𝑓ଵ(𝑡) = 𝑓ୱ୧୬(𝑡) + 𝜉ீି௡௢௜௦௘(𝑡) + 𝜁ீି௣௨௟௦௘(𝑡). (8)
And the parameter settings are shown in Table 1. 
3.1. The analysis of delay time  
This is illustrates in Fig. 1, and shows that: when the length of time sequences is short, the 
correlation between sequences is powerful. Conversely, the correlation of time sequences is faintly. 
3.2. Embedded dimension analysis of time sequences 
According to the basic theory of improved (FNN) point method, and four different lengths of 
the simulation signal are taken to calculate the embedding dimension are shown in Fig. 2. The 
analysis showed that: While the length of the time sequences is short, the correlation between the 
signals is high, but the effective characteristics are difficult to distinguish from the noise signals. 
On the contrary, the correlation between signals is weak, and the fault signal characteristics are 
not obvious. 
Table 1. Input parameters of the simulation signal 
Signal designation Parameter setting 
Initialize signal 𝑓௦௜௡(𝑡) 𝑓௦௜௡(𝑡) = sin(28𝜋𝑡) 
Random signal 
𝜉ீି௡௢௜௦௘(𝑡) 
Mean: 0; Variance: 
1 
Pulse signal 𝜉ீି௣௨௟௦௘(𝑡) Frequency of 0.1 Hz 
Sample length 𝑁 800/1200/1600/1800 
 
2 4 6 8 10 12 14 16 18 20 22
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
M
utu
al 
inf
orm
ati
on
 I （
τ ）
Delay time τ
 N=256
 N=512
 N=800
 N=1200
 N=1600
 N=1800
Fig. 1. The relationship between delay time  
and time sequences length 
3.3. Analysis of the relationship between permutation entropy and threshold parameters 
The relationship between the permutation entropy and the length of the time sequences is 
investigated based on the six groups of different lengths in Table 1. And the results are shown in 
Fig. 3. The entropy value decreases of the equivalent time sequences with the increase of 
embedding dimension. While the length of time sequences increases, the corresponding 
permutation entropy of value also increases slightly. 
The fluctuation relationship between the length of time sequences and scale factor is obtained 
in Fig. 5, and the scale factor and the embedded dimension of time sequences is achieved in Fig. 6. 
FAULT IDENTIFICATION TECHNOLOGY FOR GEAR TOOTH SURFACE WEAR BASED ON MPE METHOD BY MI AND IMPROVED FNN ALGORITHM.  
XINTAO ZHOU, YAHUI CUI, XIAYI LIU, LONGLONG LI, LIHUA WANG 
 ISSN PRINT 2345-0533, ISSN ONLINE 2538-8479, KAUNAS, LITHUANIA 27 
It can be obtained that: the permutation entropy value of time sequences is inversely proportional 
to the embedded dimension and scale factor. 
  
  
Fig. 2. The relationship between embedding dimension and length of signal 
3 4 5 6 7 8
0.5
0.6
0.7
0.8
0.9
1.0
Pe
rm
ut
at
io
n 
en
tr
op
y
Embedded dimension m
 N=256
 N=512
 N=800
 N=1200
 N=1600
 N=1800
 
Fig. 3. The relationship between the length and time 
of embedded dimension of time sequences 
2 3 4 5 6 7 8 9 10 11 12
0.600
0.625
0.650
0.675
0.700
0.725
0.750
0.775
0.800
0.825
0.850
0.875
Pe
rm
ut
at
io
n 
en
tro
py
Delay time τ
 N=256
 N=512
 N=800
 N=1200
 N=1600
 N=1800
 
Fig. 4. The relationship between the length and 
delay time sequences 
 
2 4 6 8 10 12
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Pe
rm
ut
at
io
n 
en
tr
op
y
Scale factor s
 N=256
 N=512
 N=800
 N=1200
 N=1600
 N=1800
 
Fig. 5. The relationship between scale factor and 
sample length of time sequences 
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Pe
rm
ut
at
io
n 
en
tr
op
y
 Scale factor s
 m=3
 m=4
 m=5
 m=6
 m=7
 m=8
 
Fig. 6. The relationship between scale factor and 
embedded dimension 
4. Experimental verification and analysis 
The layout of the laboratory table is shown in Fig. 7. Then, the four experiment points and 
parameters as follows: test point I, test point II, test point III and test point IV; input speed is 
1500 r/min; the input current of the magnetic powder loader is 0.1 A; Modulus of gear 𝑚 = 2 mm, 
Number of teeth 𝑧 = 55, Tooth width 𝑏 = 20 mm. 1. Motor input shaft, 2. Measured gear box, 
3. Torque transducer, 4. Magnetic powder loader. 
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Embedded dimension m
，
E1
(m
)
E2
(m
)
N=800
 
 
E1(m)
E2(m)
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Embedded dimension m
，
E1
(m
)
E2
(m
)
N=1200
 
 
E1(m)
E2(m)
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Embedded dimension m
，
E1
(m
)
E2
(m
)
N=1600
 
 
E1(m)
E2(m)
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Embedded dimension m
，
E1
(m
)
E2
(m
)
N=1800
 
 
E1(m)
E2(m)
FAULT IDENTIFICATION TECHNOLOGY FOR GEAR TOOTH SURFACE WEAR BASED ON MPE METHOD BY MI AND IMPROVED FNN ALGORITHM.  
XINTAO ZHOU, YAHUI CUI, XIAYI LIU, LONGLONG LI, LIHUA WANG 
28 VIBROENGINEERING PROCEDIA. OCTOBER 2019, VOLUME 28  
4.1. Calculate the delay time of the time sequences 
Through simulation analysis, the results are obtained as shown in Fig. 8. As can be seen from 
the figure, the analysis shows that the delay time values of the time sequence at the four measuring 
points on the gearbox are all 2. 
 
Fig. 7. The test bench 
2 4 6 8 10 12 14 16 18 20
1
2
3
4
5
6
M
ut
ua
l i
nf
or
m
at
io
n  
I（
τ）
Delay time τ
 Measurement point  I
 Measurement point  II
 Measurement point  III
 Measurement point  IV
 
Fig. 8. The MI curve at each measuring point 
4.2. Calculate the embedding dimension of time sequences 
According to the calculation principle of the improved FNN point method, the results shown 
in Fig. 9 is acquired. It is shows that: while the value of the embedding dimension 𝑚 is 5, it is the 
most reasonable of the time series at the four measuring points on the gearbox. 
  
  
Fig. 9. The improved FNN proportion at each measuring point 
1 2 3 4 5 6 7 8 9 10 11 12
0.675
0.700
0.725
0.750
0.775
0.800
0.825
0.850
0.875
0.900
0.925
0.950
Pe
rm
ut
at
io
n 
en
tr
op
y
 Scale factor  s
 Measurement point  I
 Measurement point  II
 Measurement point  III
 Measurement point  IV
 
Fig. 10. The MPE curve of each measuring point  
in normal state 
1 2 3 4 5 6 7 8 9 10 11 12
0.76
0.78
0.80
0.82
0.84
0.86
0.88
0.90
0.92
0.94
Pe
rm
ut
at
io
n 
en
tr
op
y
Scale factor  s
 Measurement point I
 Measurement point II
 Measurement point III
 Measurement point IV
 
Fig. 11. The MPE curve of each measuring point  
in malfunction state 
0 5 10 15 200.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Embedded dimension  m
，
E1
(m
)
E2
(m
)
Measurement point  I
 
 
E1(m)
E2(m)
0 5 10 15 200.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Embedding dimension  m
E1
(m
),E
2(m
)
Measurement point  II
 
 
E1(m)
E2(m)
0 5 10 15 200.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Embedded dimension  m
E1
(m
),E
2(m
)
Measurement point  III
 
 
E1(m)
E2(m)
0 5 10 15 200.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Embedding Dimension  m
E1
(m
),E
2(m
)
Measurement point IV
 
 
E1(m)
E2(m)
FAULT IDENTIFICATION TECHNOLOGY FOR GEAR TOOTH SURFACE WEAR BASED ON MPE METHOD BY MI AND IMPROVED FNN ALGORITHM.  
XINTAO ZHOU, YAHUI CUI, XIAYI LIU, LONGLONG LI, LIHUA WANG 
 ISSN PRINT 2345-0533, ISSN ONLINE 2538-8479, KAUNAS, LITHUANIA 29 
4.3. Multiscale permutation entropy method for fault identification 
According to the research results of the above relevant chapters, the analysis parameters were 
set by 𝑚 = 5, 𝜏 = 2, 𝑠 = 12. In the normal state (No fault), calculate the MPE value of the time 
sequences of the four measuring points on the gearbox is shown in Fig. 10, and the fault state is 
shown in Fig. 11. Shows that, according to the vibration variation of the signal, it can be 
adjudicated that the gear has gear tooth wear fault in the gearbox. 
5. Conclusions 
Conclusions are drawn as follows: The joint calculation method based on the MI and improved 
FNN principle for calculating threshold parameters for MPE method was presented. Compared 
with the trial algorithm given in related literature, the joint calculation method can improve the 
recognition of dynamic mutation of fault sequence by MPE method. Then, the fault identification 
simulation model for wear failure of gear tooth surface by MPE method was simulated, and the 
effectiveness of the proposed MPE method was validated. Finally, the wear failure test of spur 
gear was carried out, and the vibration characteristics of the fault signal are obtained by the method 
of MPE. Furthermore, With the experiment, it is efficient to identification the vibration signal for 
gear tooth surface wear fault. 
Acknowledgements 
This work was supported partially by the National Natural Science Foundations of China 
(No. 51175419) and Shaanxi Key Laboratory of Machinery Manufacturing Equipment 
Construction Project, which are highly appreciated by the authors. 
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Fault identification technology for gear tooth surface wear based on MPE method by MI and improved FNN algorithm

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Fault identification technology for gear tooth surface wear based on MPE method by MI and improved FNN algorithm

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