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Citation: Song, M.; Wang, J.; Zhao,
H.; Wang, X. Fault Diagnosis Method
Based on AUPLMD and RTSMWPE
for a Reciprocating Compressor
Valve. Entropy 2022,24, 1480.
https://doi.org/10.3390/e24101480
Academic Editors: Claude Delpha
and Demba Diallo
Received: 8 September 2022
Accepted: 11 October 2022
Published: 17 October 2022
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4.0/).
entropy
Article
Fault Diagnosis Method Based on AUPLMD and RTSMWPE for
a Reciprocating Compressor Valve
Meiping Song 1, * , Jindong Wang 1, Haiyang Zhao 1and Xulei Wang 2
1Mechanical Science and Engineering Institute, Northeast Petroleum University, Daqing 163318, China
2PetroChina Daqing Refining and Chemical Company, Daqing 163318, China
*Correspondence: xpsong@nepu.edu.cn; Tel.: +86-0459-6504736
Abstract:
In order to effectively extract the key feature information hidden in the original vibration
signal, this paper proposes a fault feature extraction method combining adaptive uniform phase local
mean decomposition (AUPLMD) and refined time-shift multiscale weighted permutation entropy
(RTSMWPE). The proposed method focuses on two aspects: solving the serious modal aliasing
problem of local mean decomposition (LMD) and the dependence of permutation entropy on the
length of the original time series. First, by adding a sine wave with a uniform phase as a masking
signal, adaptively selecting the amplitude of the added sine wave, the optimal decomposition result
is screened by the orthogonality and the signal is reconstructed based on the kurtosis value to remove
the signal noise. Secondly, in the RTSMWPE method, the fault feature extraction is realized by
considering the signal amplitude information and replacing the traditional coarse-grained multi-
scale method with a time-shifted multi-scale method. Finally, the proposed method is applied to
the analysis of the experimental data of the reciprocating compressor valve; the analysis results
demonstrate the effectiveness of the proposed method.
Keywords:
adaptive uniform phase local mean decomposition; refined time-shift multiscale weighted
permutation entropy; reciprocating compressor valve; feature extraction; fault diagnosis
1. Introduction
Reciprocating compressors are the key equipment for compressing and transporting
gas in petroleum, chemical, and other fields [
1
]. Once an accident occurs, it will cause
huge economic losses and casualties. As a result of the complex structure, there are
various reasons for failure, of which more than 60% will occur on the valve [
2
]. As the
failure of mechanical equipment components is usually accompanied by the change in
vibration signal, it is one of the more suitable methods to collect the vibration signal of
the equipment and to make corresponding diagnosis and analysis. However, due to the
complicated operation process of the reciprocating compressor, the extracted vibration
signal is non-stationary and the extracted characteristic parameters are fuzzy, which brings
great difficulty to the fault diagnosis of the reciprocating compressor [
3
]. In recent years,
many fault diagnosis methods such as wavelet transform (WT), Wigner–Ville distribution
(WVD), short-time Fourier transform (STFT), and other signal processing methods have
been widely used and have achieved good results [
4
]. However, the above methods cannot
effectively take into account the global and local characteristics of non-stationary signals in
the time and frequency domains.
British scholar Jonathan S. Smith proposed an adaptive analysis method for non-
stationary signals in 2005, namely local mean decomposition (LMD). LMD can adaptively
decompose complex signals into the sum of several product functions (PF) and a residual
component, where each PF component is obtained by multiplying an envelope signal and a
pure FM signal in order to obtain the complete time–frequency distribution of the original
signal [
5
,
6
]. Cheng et al. conducted a comparative study between LMD and empirical
Entropy 2022,24, 1480. https://doi.org/10.3390/e24101480 https://www.mdpi.com/journal/entropy
Entropy 2022,24, 1480 2 of 19
mode decomposition (EMD) methods, and the results show that LMD is superior to the
EMD method in terms of the end effect and iteration number, and the decomposition
accuracy is more ideal, and the instantaneous frequency of PF components is determined
by a pure FM signal [
7
–
9
]. The problem of unexplained negative frequencies when the EMD
is decomposed does not arise. However, similar to EMD, the LMD algorithm still suffers
from some degree of modal aliasing and end effect issues. Inspired by EEMD, Yang et al.
proposed an ensemble local mean decomposition (ELMD) method and pointed out that
ELMD is more effective than EEMD in mechanical fault diagnosis. However, the ELMD
method still suffers from the two intractable problems mentioned above [
10
–
12
]. Wang et al.
applied the complementary integrated local mean decomposition CELMD method to the
composite fault diagnosis of gearboxes, and effectively extracted the composite fault fea-
tures, but the PF component was greatly affected by the amplitude of white noise. Lu et al.
proposed an adaptive complementary ensemble local mean decomposition (ACELMD)
method to decompose the marine underwater acoustic signal, reducing the modal aliasing
phenomenon. Wang et al. proposed a complete ELMD with adaptive noise (CELMDAN),
which further reduced residual noise, alleviated mode aliasing, and verified that CELM-
DAN outperformed CEEMDAN. Anh Ngoc-Lan Huynh et al. proposed the robust local
mean decomposition (RLMD), which optimized the moving average algorithm and its
screening process by adding adaptive amplitude white noise, and it achieved good results.
The white noise used in auxiliary noise decomposition is a broadband signal with complex
frequency components. In the process of mapping the signal to be decomposed to the cor-
responding frequency band of the white noise, modal aliasing appeared easily
[13–15]
. In
2018, Wang et al. reduced residual noise by adding the phase of the narrow-band sine wave,
and proposed uniform phase empirical mode decomposition (UPEMD) [
16
], then Zheng
et al. optimized the amplitude of the sine wave added by UPEMD [
17
]. The decomposition
effect was found to be better than CEEMDAN. Therefore, inspired by its predecessors,
this paper proposes adaptive uniform phase local mean decomposition (AUPLMD) with
an optimization parameter. A new noise addition strategy was adopted to improve the
computational efficiency and to alleviate the mode aliasing problem.
Entropy is a measure of system complexity. Depending on the type of failure and
the degree of damage to the reciprocating compressor valve, its entropy value will also be
different [
18
–
20
]. Permutation entropy (PE) is one of the most reliable and conceptually
simple tools, and has been widely used in many fields [
21
–
23
]. However, the main drawback
of this method is that it only considers the relative order structure of the time series, and
ignores the amplitude information [
24
]. Fadlallah et al. proposed weighted permutation
entropy (WPE) in 2013 to solve this problem [
25
–
27
]. WPE has a better anti-noise and anti-
interference ability, its algorithm is simple, and its operability is strong. It can effectively
amplify the small changes of the time series, and achieve good application results in
mechanical fault diagnosis, medicine, and other fields [
28
,
29
]. However, WPE is only
based on a single-scale measure. Therefore, a multiscale weighted permutation entropy
(MWPE), which takes into account the complex temporal fluctuations inherent in the
sequence, is proposed as an improved version of WPE [
30
–
32
]. However, it also exposes
some shortcomings of MWPE. Because of the introduction of a weighting factor, small
changes in the data series will lead to large fluctuations in MWPE. Moreover, the shortening
of the time series in the coarse-grained process leads to a loss of information, which may
lead to abrupt changes in MWPE and inaccurate estimates [
33
,
34
]. Therefore, this paper
proposes refined time-shifted multiscale weighted permutation entropy (RTSMWPE) to
measure the irregularity and complexity of the time series. In the RTSMWPE method, the
coarse-grained-based multi-scale method used in MWPE is replaced by a new time-shifted
multi-scale method, which has the following advantages. First, the method considers the
amplitude information of the original time series to construct a new time-shifted multi-
scale time series, which can effectively preserve the important structural information of
the original data. Second, it reduces the dependence on data length and outperforms other
similar methods when dealing with short time series.
Entropy 2022,24, 1480 3 of 19
In order to further reduce the interference of noise on feature extraction, a fault feature
extraction method based on AUPLMD and RTSMWPE is proposed to improve the accuracy
of fault diagnosis. In this paper, a masking signal with an adaptive sine wave amplitude
is added to the LMD algorithm to find the optimal decomposition result and reconstruct
the decomposition signal according to the value of kurtosis. Fault feature extraction is
realized by the RTSMWPE method. The AUPLMD-RTSMWPE method is applied to the
analysis of experimental data of the reciprocating compressor valves. By verifying the
simulated vibration signals and the measured vibration signals, the AUPLMD method
can effectively reduce the modal aliasing phenomenon. At the same time, compared
with AUPLMD-MWPE, LMD-RTSMWPE, and LMD-MWPE, the method presented in this
paper can extract early fault features accurately under a strong noise background, and has
better practicability.
The rest of this paper is organized as follows. In Section 2, the AUPLMD signal
decomposition method is introduced. In Section 3, the RTSMWPE feature extraction
method is proposed. Section 4verifies the effectiveness of the method in diagnosing the
fault of reciprocating compressor valve.
2. Adaptive Uniform Phase Local Mean Decomposition
2.1. Local Mean Decomposition
LMD is a novel time–frequency analysis method that adaptively decomposes complex
signals into PF components, in which multiple envelope signals are multiplied by pure
frequency modulated signals. The construction of the mean function and envelope estima-
tion in the traditional LMD method is obtained by smoothing the local mean line segment
and the local amplitude line segment using the moving average method. However, the
subjectivity of the moving average step size selection and the phase error may occur in
the multiple smoothing process, which are related to the LMD decomposition accuracy.
Therefore, a monotonic piecewise cubic Hermite interpolation is proposed to replace the
moving average method in order to solve the problems in envelope construction. For the
signal x(t), the LMD decomposition process is as follows.
Extract the array of the local extreme points (maximum points and minimum points)
of the original signal x(t), and use the monotonic piecewise cubic Hermite to interpolate it to
construct the upper envelope function env
max
(t) and the lower envelope function env
min
(t).
Calculate the local mean function m
11
(t) and the envelope estimation function a
11
(t) using
the upper and lower envelopes:
m11 =envmax(t) + envmin (t)
2(1)
a11 =|envmax(t)−envmin (t)|
2(2)
Separate m
11
(t) from signal x(t) to obtain another signal
h11(t) = x(t)−m11(t)
, and
the frequency modulation signal s11(t) = h11 (t)/a11(t)is obtained.
Detect s
11
(t) to judge whether it is a pure FM signal. If
a12(t) =
1, then s
11
(t) is a pure
FM signal; if
a12(t)6=
1, s
11
(t) is used as the original signal to repeat the above iterative
process until s
1n
(t) becomes a pure FM signal, that is the envelope satisfies a
1(n+1)
= 1, and
the termination condition is lim
n→∞a1n(t) = 1.
The envelope estimate signal a
1q
(t) are multiplied together to obtain the envelope
signal of the first PF component.
a1(t) = a11(t)a12(t)· · · a1n(t) =
n
∏
q−1
a1q(t)(3)
The first PF component PF1(t) is defined as
PF1=a1(t)s1n(t)(4)
Entropy 2022,24, 1480 4 of 19
PF
1
(t) is separated from the original signal x(t) to obtain a new signal u
1
(t). The
above process is repeated ktimes with u
1
(t) as the original signal until u
k
(t) is a monotonic
function. The iterative equation is
uk(t) = uk−1(t)−PFk(t)(5)
Finally, the original signal x(t) is expressed as
x(t) =
k
∑
p=1
PFk(t) + uk(t)(6)
2.2. Uniform Phase Local Mean Decomposition
UPLMD homogenizes the distribution of extreme points by adding a narrow-band
cosine signal with a uniform phase change to the signal to be decomposed, and achieves the
purpose of suppressing modal aliasing. After removing the narrow-wave cosine signal in
the decomposition result, the integrated averaging can eliminate the residual of the auxiliary
signal, and increasing the phase number of the cosine wave can better suppress the modal
aliasing and spurious components. For the vibration signal x(t), the decomposition process
is as follows:
Calculate the number of loops n
imf
= log
2
(n) and the period T
w
according to the data
length n:
Tw=2m,m=1 : nim f (7)
The frequency f
w
= 1/T
w
of the cosine signal is obtained according to the period, and
then the number of phases n
p
and the amplitude
ε
are set according to the actual signal x(t).
The phase
θk
is evenly divided into equal parts in the range of [0, 2
π
], and the number
of phases np must be the integer power of 2, and different amplitudes should be selected
for different signals.
Let r
0
(t) = x(t), construct the narrow wave cosine signal w(t;
εm
;f
w
;
θk
), which is
as follows:
w(t;εm;fw;θk)=εm·cos(2π·fw·t+θk)(8)
where
εm
is the product of the amplitude
ε
and the standard deviation value of r
m−1
(t);
θk
is defined as θk= 2π(k−1)/np.
The operator L
i
(
·
) is defined as the i-th PF decomposed by LMD, and use the LMD to
decompose the signal after adding the narrow wave to obtain the first component:
cm,k(t) = L1(x(t) + w(t;εm;fw;θk)) k=1, 2, · · · ,np;m=1 : nim f (9)
After subtracting the narrow wave cosine signal w(t;
εm
;f
w
;
θk
) from c
m,k
(t), the PF
1
of
UPLMD is obtained through averaging, as follows
PF1= np
∑
k=1
cm,k(t)−w(t;εm;fw;θk)!np(10)
Separate the PF
1
component from the original signal x(t), and use the remaining part
r
i
(t) as a new signal, repeat the above steps until all PF
m
(m= 2,3,
. . .
, log
2
n) components
are decomposed, rlog2n(t)is a trend term.
2.3. Adaptive Uniform Phase Local Mean Decomposition
When UPLMD is used to process the collected vibration signal, its two important
parameters, the number of phases n
p
and the amplitude
ε
of narrow wave cosine signal,
need to be artificially selected in advance. Thus, it does not possess an adaptive capability.
The value of the phase number n
p
directly affects the decomposition ability of UPLMD.
The larger the n
p
, the higher the decomposition ability and the less noise residue, but
Entropy 2022,24, 1480 5 of 19
the calculation time will be longer. Usually, n
p
is in the range of 4~32 to balance the
decomposition ability and time cost, so the value of n
p
is 16. For the selection of the
amplitude value, different signals have different optimal amplitudes, so it is difficult to
choose empirically. Therefore, AUPLMD is adopted to automatically select the important
parameters of UPLMD. Through massive experiments and analysis, the amplitude
εm
is
selected in the range of 0.10~0.5, where 0.02 is the step size, and the orthogonality index of
the decomposition result is compared and the optimal amplitude is automatically selected
within the range. Orthogonality is an important index to measure modal aliasing. In theory,
the orthogonality index for single-component PF is 0, otherwise it will increase. Therefore,
the optimal PF criterion is adaptively selected based on the minimum orthogonality in
order to improve the decomposition ability and accuracy of AUPLMD. The flow chart
of AUPLMD is shown in Figure 1, and the specific steps of AUPLMD are introduced
as follows:
Step 1:
Set the number of phases as n
p
= 16, then adaptively select the optimal amplitude
εmo in the given range and construct the masking signal w(t;εmo;fw;θk).
Step 2: Decompose
x(t) + w(t;εmo;fw;θk)
by LMD to obtain the first component
cm,k(t)
, sub-
tracting the narrow wave cosine signal w(t;εmo;fw;θk) from cm,k(t), the first component
PF1jis obtained through averaging as PF1j=np
∑
k=1
cm,k(t)−w(t;εm;fw;θk)np.
Step 3:
Let j=j+ 1, implementing step (2) until
εm
takes different values in the range of
0.10~0.5 and it obtains a series of PF
1j
components. The orthogonality index is used
as the criterion to select the optimal PF
1
component. The smaller the value of OI,
the better the decomposition performance.
Step 4:
Separate the PF
1
component from the original signal x(t), and use the remaining
part r
i
(t) as a new signal. Then, repeat steps 1 to 3 until x(t) is finally decomposed
into the sum of the PFs and a trend item.
Entropy2022,24,xFORPEERREVIEW6of20
Figure1.TheflowchartofAUPLMD.
2.4.ComparisonAnalysis
2.4.1.SimulationSignalAnalysis
TodemonstratetheexcellentdecompositionperformanceoftheproposedAUPLMD
method,thedecompositionresultsofasimulationsignal(signalS
4
)constructedfrom
threetypicalmechanicalvibrationsignalsarecomparedusingLMD,CELMDAN,andthe
proposedAUPLMD.Thesimulationsignalisshownin(11).Specifically,S
1
isacosine
signalwithfrequencyf
1
,S
2
isafrequencymodulatedsignalwithacarrierfrequencyf
2
and
amodulationfrequencyf
m
,andS
3
isanimpulsesignal[35].S
4
wasamixedsignalconsist‐
ingofthesethreesignals,withasamplingfrequencyf=1000Hzandthenumberofsample
pointsN=2000.ThetimedomainwaveformandspectrumsofS
4
areshowninFigure2.
3214
3
10
3
22
11
2sin*
2cos2sin
2sin
SSSS
tfeS
tftfS
tfS
t
m
(11)
wheref
1
=10Hz,f
2
=400Hz,f
3
=800Hz,andf
m
=25Hz.
Figure 1. The flow chart of AUPLMD.
Entropy 2022,24, 1480 6 of 19
2.4. Comparison Analysis
2.4.1. Simulation Signal Analysis
To demonstrate the excellent decomposition performance of the proposed AUPLMD
method, the decomposition results of a simulation signal (signal S
4
) constructed from
three typical mechanical vibration signals are compared using LMD, CELMDAN, and the
proposed AUPLMD. The simulation signal is shown in (11). Specifically, S
1
is a cosine
signal with frequency f
1
,S
2
is a frequency modulated signal with a carrier frequency f
2
and a modulation frequency f
m
, and S
3
is an impulse signal [
35
]. S
4
was a mixed signal
consisting of these three signals, with a sampling frequency f= 1000 Hz and the number of
sample points N= 2000. The time domain waveform and spectrums of S
4
are shown in
Figure 2.
S1=sin(2πf1t)
S2=sin(2πf2t+cos(2πfmt))
S3=e−10t∗sin(2πf3t)
S4=S1+S2+S3
(11)
where f1= 10 Hz, f2= 400 Hz, f3= 800 Hz, and fm= 25 Hz.
Signal S
4
is decomposed by three methods, namely LMD, CELMDAN, and AUPLMD,
as shown in Figure 3. The decomposition results of LMD show serious modal aliasing.
Compared with the LMD method, the CELMDAN method still has modal aliasing. Com-
pared with the other two methods, the decomposition result of AUPLMD is the best, and its
performance is very encouraging, because its decomposition results show only slight mode
aliasing. PF
1
,PF
2
, and PF
6
in the AUPLMD decomposition result are highly consistent with
signal S
2
, signal S
3
, and signal S
1
, respectively (Figure 2), which indicates that AUPLMD
can effectively reduce modal aliasing.
2.4.2. Valve Analog Signal Analysis
The advantages of the AUPLMD method are verified by the example of a normal
working state of valve. Vibration signals of the valve in a normal working state containing
two full-cycle data points are selected, as shown in Figure 4, and signal decomposition
is performed using the above methods, as shown in Figure 5. It can be seen that the
decomposition ability of LMD is poor, and the mode aliasing phenomenon of the PF
2
com-
ponent is the most serious. Although CELMDAN is superior to LMD, modal aliasing also
appears, such as the PF
4
component. However, the PF
1
–PF
4
components in CELMDAN are
decomposed into PF
1
–PF
6
components in AUPLMD, and the modal aliasing phenomenon
among each component is alleviated, which indicates that AUPLMD also has a good effect
on vibration signal decomposition under actual working conditions, and the mode aliasing
can be effectively reduced.
Entropy2022,24,xFORPEERREVIEW7of20
Figure2.Waveformsofthefoursimulatedsignals.
SignalS4isdecomposedbythreemethods,namelyLMD,CELMDAN,and
AUPLMD,asshowninFigure3.ThedecompositionresultsofLMDshowseriousmodal
aliasing.ComparedwiththeLMDmethod,theCELMDANmethodstillhasmodalalias‐
ing.Comparedwiththeothertwomethods,thedecompositionresultofAUPLMDisthe
best,anditsperformanceisveryencouraging,becauseitsdecompositionresultsshow
onlyslightmodealiasing.PF1,PF2,andPF6intheAUPLMDdecompositionresultare
highlyconsistentwithsignalS2,signalS3,andsignalS1,respectively(Figure2),which
indicatesthatAUPLMDcaneffectivelyreducemodalaliasing.
Figure3.DecompositionresultsofthesimulatedsignalS4usingLMD,CELMDAN,and
AUPLMD.
2.4.2.ValveAnalogSignalAnalysis
TheadvantagesoftheAUPLMDmethodareverifiedbytheexampleofanormal
workingstateofvalve.Vibrationsignalsofthevalveinanormalworkingstatecontaining
–1
0
1
–1
0
1
–1
0
1
0 500 1000 1500 2000
Sam
p
lin
g
p
oint
s
–2
0
2
–2
0
2LMD
–1
0
1
0 500 1000 1500 2000
–0.01
0
0.01
0.02
0.03
–2
0
2CELMDAN
–0.2
0
0.2
–0.2
0
0.2
–1
0
1
–0.1
0
0.1
–0.2
0
0.2
–0.05
0
0.05
0 500 1000 1500 2000
–0.01
0
0.01
0.02
–1
0
1
2AUPLMD
–1
–0.5
0
0.5
–0.2
–0.1
0
0.1
–0.2
0
0.2
–0.2
0
0.2
–1
0
1
–0.2
0
0.2
0 500 1000 1500 2000
–0.1
0
0.1
Figure 2. Waveforms of the four simulated signals.
Entropy 2022,24, 1480 7 of 19
Entropy2022,24,xFORPEERREVIEW7of20
Figure2.Waveformsofthefoursimulatedsignals.
SignalS4isdecomposedbythreemethods,namelyLMD,CELMDAN,and
AUPLMD,asshowninFigure3.ThedecompositionresultsofLMDshowseriousmodal
aliasing.ComparedwiththeLMDmethod,theCELMDANmethodstillhasmodalalias‐
ing.Comparedwiththeothertwomethods,thedecompositionresultofAUPLMDisthe
best,anditsperformanceisveryencouraging,becauseitsdecompositionresultsshow
onlyslightmodealiasing.PF1,PF2,andPF6intheAUPLMDdecompositionresultare
highlyconsistentwithsignalS2,signalS3,andsignalS1,respectively(Figure2),which
indicatesthatAUPLMDcaneffectivelyreducemodalaliasing.
Figure3.DecompositionresultsofthesimulatedsignalS4usingLMD,CELMDAN,and
AUPLMD.
2.4.2.ValveAnalogSignalAnalysis
TheadvantagesoftheAUPLMDmethodareverifiedbytheexampleofanormal
workingstateofvalve.Vibrationsignalsofthevalveinanormalworkingstatecontaining
–1
0
1
–1
0
1
–1
0
1
0 500 1000 1500 2000
Sam
p
lin
g
p
oint
s
–2
0
2
–2
0
2LMD
–1
0
1
0 500 1000 1500 2000
–0.01
0
0.01
0.02
0.03
–2
0
2CELMDAN
–0.2
0
0.2
–0.2
0
0.2
–1
0
1
–0.1
0
0.1
–0.2
0
0.2
–0.05
0
0.05
0 500 1000 1500 2000
–0.01
0
0.01
0.02
–1
0
1
2AUPLMD
–1
–0.5
0
0.5
–0.2
–0.1
0
0.1
–0.2
0
0.2
–0.2
0
0.2
–1
0
1
–0.2
0
0.2
0 500 1000 1500 2000
–0.1
0
0.1
Figure 3.
Decomposition results of the simulated signal S
4
using LMD, CELMDAN, and AUPLMD.
Entropy2022,24,xFORPEERREVIEW8of20
twofull‐cycledatapointsareselected,asshowninFigure4,andsignaldecompositionis
performedusingtheabovemethods,asshowninFigure5.Itcanbeseenthatthedecom‐
positionabilityofLMDispoor,andthemodealiasingphenomenonofthePF2component
isthemostserious.AlthoughCELMDANissuperiortoLMD,modalaliasingalsoap‐
pears,suchasthePF4component.However,thePF1‐PF4componentsinCELMDANare
decomposedintoPF1‐PF6componentsinAUPLMD,andthemodalaliasingphenomenon
amongeachcomponentisalleviated,whichindicatesthatAUPLMDalsohasagoodeffect
onvibrationsignaldecompositionunderactualworkingconditions,andthemodealias‐
ingcanbeeffectivelyreduced.
Figure4.Vibrationsignalofareciprocatingcompressorvalveinanormalworkingstate.
Figure5.DecompositionresultsofthevalveanalogsignalusingLMD,CELMDAN,and
AUPLMD.
PF1
PF2
PF3
PF4
PF5
0 0.05 0.1 0.15 0.2 0.25
–0.02
0
0.02
PF6
0 0.05 0.1 0.15 0.2 0.25
–2
0
2
PF7
0 0.05 0.1 0.15 0.2 0.25
–4
–2
0
2
PF8
0 0.05 0.1 0.15 0.2 0.25
–5
0
5
10
PF9
0 0.05 0.1 0.15 0.2 0.25
–1.8
–1.6
–1.4
–1.2
PF10
0 0.05 0.1 0.15 0.2 0.25
–1
0
1
PF1
AUPLMD
0 0.05 0.1 0.15 0.2 0.25
–0.2
0
0.2
PF2
0 0.05 0.1 0.15 0.2 0.25
–0.1
0
0.1
PF3
0 0.05 0.1 0.15 0.2 0.25
–0.05
0
0.05
PF4
0 0.05 0.1 0.15 0.2 0.25
–0.02
0
0.02
0.04
PF5
0 0.05 0.1 0.15 0.2 0.25
–0.01
0
0.01
PF6
0 0.05 0.1 0.15 0.2 0.25
–0.01
0
0.01
PF7
0 0.05 0.1 0.15 0.2 0.25
–0.04
–0.02
0
0.02
PF8
Figure 4. Vibration signal of a reciprocating compressor valve in a normal working state.
Entropy 2022,24, 1480 8 of 19
Entropy2022,24,xFORPEERREVIEW8of20
twofull‐cycledatapointsareselected,asshowninFigure4,andsignaldecompositionis
performedusingtheabovemethods,asshowninFigure5.Itcanbeseenthatthedecom‐
positionabilityofLMDispoor,andthemodealiasingphenomenonofthePF2component
isthemostserious.AlthoughCELMDANissuperiortoLMD,modalaliasingalsoap‐
pears,suchasthePF4component.However,thePF1‐PF4componentsinCELMDANare
decomposedintoPF1‐PF6componentsinAUPLMD,andthemodalaliasingphenomenon
amongeachcomponentisalleviated,whichindicatesthatAUPLMDalsohasagoodeffect
onvibrationsignaldecompositionunderactualworkingconditions,andthemodealias‐
ingcanbeeffectivelyreduced.
Figure4.Vibrationsignalofareciprocatingcompressorvalveinanormalworkingstate.
Figure5.DecompositionresultsofthevalveanalogsignalusingLMD,CELMDAN,and
AUPLMD.
PF1
PF2
PF3
PF4
PF5
0 0.05 0.1 0.15 0.2 0.25
–0.02
0
0.02
PF6
0 0.05 0.1 0.15 0.2 0.25
–2
0
2
PF7
0 0.05 0.1 0.15 0.2 0.25
–4
–2
0
2
PF8
0 0.05 0.1 0.15 0.2 0.25
–5
0
5
10
PF9
0 0.05 0.1 0.15 0.2 0.25
–1.8
–1.6
–1.4
–1.2
PF10
0 0.05 0.1 0.15 0.2 0.25
–1
0
1
PF1
AUPLMD
0 0.05 0.1 0.15 0.2 0.25
–0.2
0
0.2
PF2
0 0.05 0.1 0.15 0.2 0.25
–0.1
0
0.1
PF3
0 0.05 0.1 0.15 0.2 0.25
–0.05
0
0.05
PF4
0 0.05 0.1 0.15 0.2 0.25
–0.02
0
0.02
0.04
PF5
0 0.05 0.1 0.15 0.2 0.25
–0.01
0
0.01
PF6
0 0.05 0.1 0.15 0.2 0.25
–0.01
0
0.01
PF7
0 0.05 0.1 0.15 0.2 0.25
–0.04
–0.02
0
0.02
PF8
Figure 5.
Decomposition results of the valve analog signal using LMD, CELMDAN, and AUPLMD.
3. Refined Time-Shifted Multiscale Weighted Permutation Entropy
3.1. Multiscale Weighted Permutation Entropy
PE is an effective method to measure the complexity of the nonlinear time series
proposed by Bandt and Pompe. For a given time series, X= {x(n), n= 1, 2,
. . .
,N},
with the length being N, the phase space reconstruction of the time series is obtained
XK={x(i),x(i+d), ..., x(i+(m−1)d)}
,
i=
1, 2, ...,
K
, where
K=N−(m−1)d
,mis the
embedding dimension, and dis the time delay. Arrange each vector in the reconstructed
time series
XK
in ascending order and obtain
x(i+(j1(i)−1)d)≤x(i+(j2(i)−1)d)≤ · · ·
≤x(i+(jm(i)−1)d)
, let
πi=(j1(i),j2(i),· · · ,jm(i))
,
i=
1, 2,
· · ·
,
K
represents the index
of the column in the reconstruction component where the element is located, and
πi
is
a certain arrangement of (1,2,
···
,m). Obviously, m elements have at most m! different
arrangements, denoted as
Π
. Count all of the permutations of
πi
, find out the frequency of
occurrence of each permutation πr(0<r≤m!), namely:
p(πr)=
K
∑
j=1
1u:type(u)=πrXj
K
∑
j=1
1u:type(u)∈∏Xj
(12)
where 1
u(v) = 1, v∈u
0, v/∈u
,
∏={πr}m!
r=1
.Considering the amplitude information of the
time series, there is an amplitude difference between the sequences with the same arrange-
Entropy 2022,24, 1480 9 of 19
ment, so when calculating the frequency of each arrangement with the corresponding
weight, the frequency of each arrangement pattern is defined as follows:
p(ωπr)=
K
∑
j=1
1u:type(u)=πrXjωr
K
∑
j=1
1u:type(u)∈∏Xjωr
(13)
where
ωr
is the weight of the reconstructed component X
j
, and it is represented by the
variance of X
j
:
ωr=1
m
m
∑
q=1x(j+(q−1)d)−Xj2
,
Xj
is the mean of the reconstructed
component Xj:Xj=1
m
m
∑
q=1
x(j+(q−1)d).
WPE is an improved algorithm of permutation entropy, which considers the amplitude
information contained in the time series, and the expression is defined as follows:
WPE(x,m,d)=−
m!
∑
r=1
p(ωπr)ln(p(ωπr)) (14)
The calculation process of MWPE mainly includes two steps:
First, coarse-grained processing is performed on the original time series X. According
to the scale factor
τ
, divide Xinto N/
τ
non-overlapping segments, each segment contains
τ
data points, calculate the arithmetic mean of
τ
data points in each segment to represent
the value of this segment, and the coarse-grained time series is defined as follows:
yτ
k,j=1
τ
jτ+k−1
∑
i=(j−1)τ+k
x(i), 1 ≤j≤N
τ, 1 ≤k≤τ(15)
Second, calculate the MWPE value of the coarse-grained time series
yτ
k,j
in the scale
factor
τ
. The entropy value of the coarse-grained series when
τ
= 1 is the value ob-
tained by the WPE method, the constraint
N/τ>> m
! must be satisfied in order to gain
reliable statistics.
MWPE(x,m,d,τ)=−
m!
∑
r=1
pτ,k(ωπr)lnpτ,k(ωπr)(16)
3.2. Refined Time-Shift Multiscale Weighted Permutation Entropy
The insufficient coarse-graining process of MWPE makes the obtained coarse-grained
time series greatly reduce the information richness of the original time series. In order
to solve the excessive dependence of the coarse-grained time series on the length of the
original time series in the MWPE algorithm, this paper proposes the RTSMWPE algorithm,
and the detailed steps of RTSMWPE can be described as follows:
First, for a given original time series {x(i), i= 1,2, . . . , N}, define Xβ
kby
Xβ
k=xβ,xβ+k,xβ+2k,· · · ,xβ+nβk(17)
where k(k=
τ
) and
β
(1
≤β≤
k) are positive integers, representing the starting point and
interval time of the time series, respectively. n
β
is a rounded integer, indicating the number
of upper boundaries, nβ= (N−β)/k. For convenience, kis still called the scale factor.
Second, for a given scale factor k,
Xβ
k
can obtain knew time series consisting of ktime
shift from the
β
th (
β
= 1,2,
. . .
,k) data point, and thus the
pβ
k(ωπr)
of each
Xβ
k
can be
obtained according to WPE’s computation. Then, define p(ωπr)=1
k∑k
1pβ
k(ωπr).
Entropy 2022,24, 1480 10 of 19
Last, the RTSMWPE of X= {x(i), i= 1,2, . . . , N} can be defined as
RTSMWPE(X,m,d,τ)=−
m!
∑
r=1
p(ωπr)·ln(p(ωπr)) (18)
Compared with MWPE, RTSMWPE avoids the phenomenon of amplitude “neutraliza-
tion” in the coarse-grained process, and the calculation process of RTSMNDE is shown in
Figure 6.
Entropy2022,24,xFORPEERREVIEW10of20
bytheWPEmethod,theconstraint
!m
N
mustbesatisfiedinordertogainreliable
statistics.
!
1
,,
ln,,,MWPE
m
r
r
k
r
k
ppdmx
(16)
3.2.RefinedTime‐ShiftMultiscaleWeightedPermutationEntropy
Theinsufficientcoarse‐grainingprocessofMWPEmakestheobtainedcoarse‐
grainedtimeseriesgreatlyreducetheinformationrichnessoftheoriginaltimeseries.In
ordertosolvetheexcessivedependenceofthecoarse‐grainedtimeseriesonthelengthof
theoriginaltimeseriesintheMWPEalgorithm,thispaperproposestheRTSMWPEalgo‐
rithm,andthedetailedstepsofRTSMWPEcanbedescribedasfollows:
First,foragivenoriginaltimeseries{x(i),i=1,2,…,N},define
k
X
by
knkkk
xxxxX
,,,,
2
(17)
wherek(k=τ)andβ(1≤β≤k)arepositiveintegers,representingthestartingpoint
andintervaltimeofthetimeseries,respectively.n
β
isaroundedinteger,indicatingthe
numberofupperboundaries,n
β
=(N−β)/k.Forconvenience,kisstillcalledthescalefactor.
Second,foragivenscalefactork,
k
X
canobtainknewtimeseriesconsistingofk
timeshiftfromtheβth(β=1,2,…,k)datapoint,andthusthe
r
k
p
ofeach
k
X
can
beobtainedaccordingtoWPE’scomputation.Then,define
k
r
k
r
p
k
p
1
1
.
Last,theRTSMWPEofX={x(i),i=1,2,…,N}canbedefinedas
!
1
ln,,,
m
r
rr
ppdmXRTSMWPE
(18)
ComparedwithMWPE,RTSMWPEavoidsthephenomenonofamplitude“neutral‐
ization”inthecoarse‐grainedprocess,andthecalculationprocessofRTSMNDEisshown
inFigure6.
Figure6.CalculationprocessofRTSMWPE.
Figure 6. Calculation process of RTSMWPE.
3.3. Comparison Analysis
In this section, a widely used simulation signal with white Gaussian noises (WGN)
is used to analyze the proposed RTSMWPE method and it is compared with multiscale
permutation entropy (MPE), multiscale weighted permutation entropy (MWPE), compos-
ite multiscale weighted permutation entropy (CMWPE), time-shift multiscale weighted
permutation entropy (TSMWPE), and refined composite multiscale weighted permutation
entropy (RCMWPE) to verify its effectiveness. The lengths of the time series of white noise
are 2048, 4096, 6144, 8192, and 10,240. The entropy results are shown in Figure 7a–f. The
calculation of RTSMWPE is related with the following parameters: embedding dimension
m and time delay d. According to the authors of [
36
], choose m= 5, d= 1, and scale factor
τ= 20.
It can be seen from Figure 7a,b that when the scale factor
τ
= 1, the entropy value is
the largest, close to 1. As the scale factor increases, the entropy value gradually decreases,
while under the same scale factor, the entropy value increases with the length of the time
series. However, compared with MPE, the MWPE curve exhibits small fluctuations, which
is because the MWPE takes into account the amplitude information of the time series.
At the same time, under the same scale factor and data points, the entropy obtained by
MWPE is lower than that of MPE. Figure 7c,d shows that with the increase in the scale
factor, the change trend of the entropy curves of CMWPE and TSMWPE is the same as the
previous two methods, but the CMWPE and TSMWPE curves are more stable. Figure 7e,f
indicates that the entropy curves of RCMWPE and RTSMWPE remain stable as the scale
factor increases. However, when the time series length is N= 2048, the entropy value of
RCMWPE decreases slowly with the increase in the scale factor, while the entropy value of
RTSMWPE remains basically unchanged, and there are small fluctuations. From the above
Entropy 2022,24, 1480 11 of 19
analysis, it can be concluded that the RTSMWPE method proposed in this paper has strong
independence on the length of the time series.
Entropy2022,24,xFORPEERREVIEW11of20
3.3.ComparisonAnalysis
Inthissection,awidelyusedsimulationsignalwithwhiteGaussiannoises(WGN)
isusedtoanalyzetheproposedRTSMWPEmethodanditiscomparedwithmultiscale
permutationentropy(MPE),multiscaleweightedpermutationentropy(MWPE),compo‐
sitemultiscaleweightedpermutationentropy(CMWPE),time‐shiftmultiscaleweighted