Scientia et Technica Año XXVI, Vol. 26, No. 04, diciembre de 2021. Universidad Tecnológica de Pereira. ISSN 0122-1701 y ISSN-e: 2344-7214
443
Abstract Entropy measurements are an accessible tool to
perform irregularity and uncertainty measurements present in
time series. In signal processing, the Multiscale Permutation
Entropy is recently presented as a methodology of
characterization capable of measuring randomness and non-
linear dynamics existing in non-stationary time series, such as
mechanical vibration signals. In this article, the Multiscale
Permutation Entropy is combined with diverse feature selection
techniques and multiple classifiers based on machine learning
aiming to detect different operative states in an internal
combustion engine. The best combination is selected from the
evaluation of parameters like precision and computational time.
Finally, the proposed methodology is established as an effective
tool to diagnose failures in bearing systems with a high precision
rate and a reduced calculation time.
Index Terms Dynamics, Entropy, Machine, Multiescale,
Permutation, Vibration.
Resumen Las mediciones de entropía son una herramienta
accesible para realizar mediciones de irregularidades e
incertidumbres presentes en series de tiempo. En el
procesamiento de señales, la Entropía de Permutación
Multiescalar se presenta recientemente como una metodología de
caracterización capaz de medir la aleatoriedad y la dinámica no
lineal existente en series de tiempo no estacionarias, como las
señales de vibración mecánica. En este artículo, la entropía de
permutación multiescalar se combina con diversas técnicas de
selección de características y múltiples clasificadores basados en
el aprendizaje automático con el objetivo de detectar diferentes
estados operativos en un motor de combustión interna. La mejor
combinación se selecciona a partir de la evaluación de
This manuscript was sent on October 30, 2020 and accepted on November
20, 2021.
J. C. Mejía is with the Mechanical Engineering Faculty, Universidad
Tecnológica de Pereira, Pereira, Carrera 27 #10-02, PO 660003 Colombia (e-
mail: j.mejia1@utp.edu.co).
J. D. Echeverry-Correa, is associate profesor in the Electrical Engineering
Program in Universidad Tecnológica de Pereira, Pereira, Carrera 27 #10-02,
PO 660003 Colombia (e-mail: jde@utp.edu.co).
H. F. Quintero is with the Mechanical Engineering Faculty, Universidad
Tecnológica de Pereira, Pereira, Carrera 27 #10-02, PO 660003 Colombia (e-
mail: hquinte@utp.edu.co).
parámetros como precisión y tiempo computacional. Finalmente,
la metodología propuesta se establece como una herramienta
eficaz para diagnosticar fallas en sistemas de rodamientos con
una alta tasa de precisión y un tiempo de cálculo reducido.
Palabras clavesDinámica, Entropía, Máquina, Multiescalar,
Permutación, Vibración.
I. INTRODUCTION
EARING systems play an important role in rotary machines
and in the modern manufacturing industry.
Different methodologies have been developed for the
detection and diagnosis of faults in its main components [1].
Generally, these diagnoses are made from the capture and
processing of vibration signals, since they contain relevant
information about the state of the machine [2]. However, these
signals have a large number of non-stationary and non-linear
characteristics, since their capture inevitably takes place with
friction and impacts. To overcome this problem, a series of
techniques have been developed for processing and classifying
these signals. A widely used approach is based on the analysis
of temporal and spectral characteristics of vibration signals [3]
[4]. However, analysis in time, frequency and time-frequency
domains are seriously affected by the signal length and
sampling frequency of the capture [5]. To solve this problem
another approach has been presented in recent years, which is
based on entropy of different natures, such as Simple Entropy
(ApEn) [6], Approximate Entropy (SampEn) [7], Multiscale
Entropy (MSE) [8], Permutation Entropy (PE) [9][31] and
Multiscale Permutation Entropy (MPE) [10]. All of the above
mentioned have been used successfully for the
characterization of signals of different nature. For instance, in
[6] the ApEn is used for diagnosis and clinical monitoring in
the area of physiology. In [11] PE is used for the classification
of patients from EEG signals. Finally, in [1] [8][32] is used the
MPE and MSE for the identification and diagnosis of faults in
bearing systems based on vibration signals. It should be noted
that the MPE is an evolution of the MSE and the PE since it
Use of Multiscale Permutation Entropy Feature
Selection and Supervised Classifiers for Bearing
Failures Diagnosis
Uso de la Entropía con Permutación Multiescalar, técnicas de Selección de
Características y Clasificadores Supervisados para el Diagnóstico de Fallas en
Rodamientos
J. C. Mejía-Hernández ; J. D. Echeverry-Correa ; H. F. Quintero-Riaza
DOI: https://doi.org/10.22517/23447214.24579
Artículo de investigación científica y tecnológica
B
Scientia et Technica Año XXVI, Vol. 26, No. 04, diciembre de 2021. Universidad Tecnológica de Pereira
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gives a much more complete measure of the nonlinear
dynamic parameters of a system. MPE includes a combination
of different scales and time delays, which identifies
particularities that are not perceptible within other entropies
[10][33]. All the previous methods of characterization allow to
obtain a great quantity of information of the system that is
being analyzed. When classifying, many of these
characteristics can be redundant or irrelevant, which can create
redundancy or over-training of the classifier, for which
characteristics selection techniques are implemented,
improving the quality and efficiency of the model. In the
bearing fault diagnosis application, Variance based on
Relevance Analysis (VRA) [1], Laplacian Score (LS) [12] and
Relief (REL) [5] are normally used. After the selection of
features, a classification process is performed with machine
learning algorithms such as Multiple Vector Support Machines
(SVMM) [1], [13], [14] and [15], Hidden Markov Models
(HMM) [16] and Artificial Neural Networks (ANN) [17], [18]
and [19]. However, these classifiers have a high degree of
complexity, computation time and initial parameters that must
be optimized. Few works have attempted to exploit the
potential of less complex conventional classifiers, such as
Nearest K-Neighbors (KNN) [20], [21], Decision Trees
(TREE) [22] or Naive Bayes (BAYES) [21]. This paper
proposes a methodology for the diagnosis of bearing failures
based on characteristics of the MPE. To make it more efficient
and effective, it is combined with some feature selection
techniques and multiple supervised classifiers. The verification
of the advantage of the chosen parameters is done by
comparison with different ones used in the state of the art.
The rest of the article is organized as follows: In section
II.A is presented the mathematical formulation of the MPE
and in section II.B is detailed the different feature selection
techniques used. In section II.C the classifier used in the state
of the art are exposed. Then, a novel methodology for the
diagnosis of bearing failures based on MPE, feature selection
techniques and supervised classifier is located in the section
III. Finally, results are presented in section IV and the
conclusions of this approach in section V.
II. METHODOLOGY
The proposed methodology combines a characterization
method, a feature selection technique and a supervised
classifier as show in Fig. 1. Each part of the methodology is
described in the following sections. The proposed
methodology begins with the characterization of vibration
signals, then using an automatic classifier combined with a
characteristic selection technique, the state of the system is
estimated.
Fig 1. Methodology for vibration classification
A. Multiscale Permutation Entropy and Acronyms
Multiscale Permutation Entropy (MPE) is used in this paper
as a signal characterization method, since it is a measure that
allows to detect dynamic changes in the time series. It is based
on the comparison in neighboring values without taking into
account the size of the values and, therefore, has a calculation
simple and fast [23]. The above, allows to position the MPE as
a particularly useful and robust tool in the presence of
dynamic noise [1]. In order to describe the MPE measure, it is
important to review the entropy proposed by Shannon [14],
described as follows: Considering a time series
T
t
t
x
1=
in a
space of representation of characteristics, where T is the length
of the time series, the entropy is represented as in (1):
(1)
Where x
i
R and p(x
i
) is the marginal probability. The time
series can be represented with a delay of time and dimension
given by (2):
( )
( )
1
,
...,,
++
=
miii
m
i
xxxX
(2)
Where i = 1, 2,...,T (m−1)τ, m is the dimension and τ the
delay. To perform the computation of MPE, the signal must be
truncated in N = T (m 1)τ subvectors. Then, for each
subvector is calculated the entropy mapped in a space of m!
different symbols denoted as
!
1
, m
i
m
i =
by (3):
( )
( ) ( )
,
:
,
ln,
,
m
i
i
m
i
ppmH
m
i
=
(3)
The probability
( )
,m
i
p
is calculated by (4):
( )
( )
( )
=
=
Nj
m
jutypeu
j
m
jutypeu
m
i
X
X
p
i
,
)(:
,
)(:
,
1
1
(4)
Where the judgment type denotes the map from pattern
Scientia et Technica Año XXVI, Vol. 26, No. 04, diciembre de 2021. Universidad Tecnológica de Pereira.
445
space to symbol space. Also, 1
A
(u) = 1 if u and 1
A
(u) = 0 if u
A. The MPE can take values between the ranges [0, ln(m!)]
and it is invariant under nonlinear monotonic transformations.
The values of m and τ vary from 1 to 8 and the values, which
are values used for calculating MSE [8] and PE [9]. In Fig 2 is
plotted the behavior of the MPE, when are varying m and τ for
a vibration signal.
Fig 2. MPE varying m and τ
When choosing the delay and dimension parameters, the
nature of the signals must be taken into account. If the
parameters are too small, the nonlinear dynamic of the
characteristics from signals will not be analyzed effectively. If
the parameters are too large, useful information will be deleted
in consequence, which will result in a wicked analysis.
B. Featured Selection Technique
For the comparison of the characteristic selection technique,
the ones most used in the literature were implemented.
Relief (REL). Unsupervised method to generate a ranking
based on the predictors give neighbors of the same class
or different class.
Laplacian Score (LS). Unsupervised method to generate
a ranking based on the input characteristics based on a
variability criterion [1].
Variance-based Relevance Analysis (VRA).
Unsupervised method, it generates a ranking with the
input features based on a variability criterion [12].
Non-negative matrix factorization (NF). Unsupervised
method to analyze the relevance of the characteristics
from a reduction in dimensionality by non-negative
factorization techniques [20].
Self-Weight Ranking (SW). Unsupervised method,
technique, it codes the feature relevance in terms of a self-
similarity measures [5].
Distance-Weight Ranking (DW). Supervised method,
this technique quantifies the relevance of the distance
between samples from different clusters by using
supervised information.
C. Supervised Classifiers
Characteristic selection techniques are combined with
different supervised classifiers to find the best combination
that suits the nature of the signal.
K-Nearest Neighbors (KNN). Supervised and non-
parametric classification method that estimates the
posterior probability that an element belongs to the class
from a set of information provided [24].
Decision Trees (TREE). Supervised classifier based on
prediction systems based on rules, which serve to
represent and categorize a series of conditions that occur
successively, for the resolution of a problem [22].
Naive bayes (BAYES). Supervised classifier for
multiclass learning. This classifier is based on estimation
of prediction and re-substitution [21].
Multiple Support Vector Machines (SVMM).
Supervised classifier based on a hyperplane or set of
hyperplanes in a space of very high (or even infinite)
dimensionality that can be used in problems of
classification or regression [1], [13], [15] and [14].
III. EXPERIMENTAL DATA
The validation of the proposed methodology is carried out
by evaluation performed when classifying bearing fault signals
obtained from the Case Western Reserve database [25]. In this
database, signals were collected for the normal bearings (Nor),
faults in the internal train (IR1), external train (IR2) and ball
(BE). Faults are also found in order of severity, 0.007 inches
in diameter to 0.040 inches in diameter and at variable engine
speeds of 1720 to 1797 RPM. Each experiment was repeated
three times and the data was collected at 12 kHz for 5 seconds.
Each signal was divided into 10 sub-signals in order to have
more samples per class and imitating the experimental
framework established in the literature [26]. A sub-signal of
each of the faults can be seen in Fig.3.
Fig 3. Vibration signals for different states
The signals are characterized with the MPE and then
dividing it into training and testing. A feature selection
technique is applied to the training group to obtain a ranking
of relevance and it is applied to the test group. Then, with the
reorganized training group, a classifier is trained and the test
group is evaluated. The evaluation process is explained in Fig.
4.
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446
Fig 4. Methodology proposed for the detection of faults in bearings
systems
IV. RESULTS AND DISCUSSION
This paper exposes a methodology for classify bearing fault
in vibration signal. The classification carried out through a
cross validation of (k = 5) and repeated by changing the
number of training characteristics. After the validation, the
best results were chosen through the shortest distance to the
ideal point [100% (Acc); 0 (Std); 0 (Ca)], where Acc is the
accuracy, Std is the standard deviation of the accuracy of the
cross validation and Ca is the number of characteristics. In
tables II and II is exposed the accuracy with a 95% confidence
interval for the combinations of classifiers and feature
selection techniques for 4 (Nor, IR1, IR2 and BE) and 10
(Nor, IR1, IR2 and BE, combined with different speeds)
classes are appreciated.
TABLE I.
ACCURACY OF THE CLASSIFICATION OF 4 CLASSES
Bayes
KNN
SVMM
Tree
-
72.08
±
2.53
97.5
±
0.88
89.75±1.71
92.91
±
1.45
VRA
96.75
±
1.01
99.5
±
0.39
99.08±0.54
98.16
±
0.76
REL
89.56
±
1.73
99.92
±
0.16
97.58±0.86
97.75
±
0.83
LS
88.16
±
1.82
99.01
±
0.56
90.41±1.66
95.41
±
1.18
NF
79.08
±
2.31
99.25
±
0.48
92.16±1.52
94.66
±
1.27
DW
81.33±2.21
99.16±0.51
92.83±1.45
96.83
±
0.99
SW
81.25
±
2.21
97.75
±
0.83
95.51
±
1.17
96.38
±
1.05
TABLE II.
ACCURACY OF THE CLASSIFICATION OF 10 CLASSES
Bayes
KNN
SVMM
Tree
-
88.42
±
2.27
97.91
±
1.18
86.52±1.81
97.17
±
2.26
VRA
96.41
±
1.67
99.68
±
0.81
99.59
±
0.79
98.25
±
1.32
REL
97.51
±
1.04
99.75
±
0.21
94.83
±
2.71
96.25
±
1.63
LS
96.01
±
2.79
98.25
±
0.76
93.92
±
2.75
94.92
±
3.05
NF
94.51
±
2.67
96.75
±
0.55
92.01
±
3.85
92.17
±
1.81
DW
94.83±1.95
98.08±1.41
93.58±2.32
94.25
±
1.27
SW
81.75
±
2.76
97.75
±
1.03
93.51
±
1.95
93.91
±
1.46
The best results are obtained with the KNN classifier,
regardless of the feature selection technique. Specifically, the
best accuracy was obtained with the KNN classifier combined
with the Relief feature selection technique. The quantity of
characteristics used for the classification were 9 and 10 for 4
and 10 classes respectively. It should be noted that the results
are obtained thanks to the characterization made with the
MPE, which achieves a high level of separability of the classes
that allows the classifiers to adapt and solve the proposed
application. Finally, a summary of the best classifications can
be seen in the table III.
TABLE III.
COMPARISON OF THE BEST CLASSIFICATION RESULTS.
Author
Number
Classes
Character.
Feat.
Sel.
Classifier
Number
Features
Acc.%
Zhang
et
al.[
14]
3
PE+EMD
-
SVM
12
97.75
Yuwono
et
al.[
17]
3
WPT
-
HMM
12
95.8
Ben et al.[19]
7
TP+FR+EMD
-
ANN
10
93
Zhu et al.[28]
10
HE+SE+MSE
-
SVM+PSO
9
97.75
Han et
al.[
16]
14
SE+LDM
-
SVM
-
100
Zheng
et
al.[
15]
7
EF
-
ANFIS
4
99.29
Liu et
al.[
29]
4
TP-FR
-
WPT+SVM+PSO
81
97.5
Tiwari et
al.[
2]
4
MPE
-
ANFC
16
02.15
William et
al.[
20]
4
ZC
-
ANN
10
97.13
Ocak et
al.[
30]
3
LPM
-
HMM
30
99.6
Wei et
al.[
5]
6
FR+WPT
Relief
AP
18
96
Shao et
al.[
31]
16
DAE+CAE
LPP
Softmax
19
96
Zheng
et
al.[
1]
6
GCMPE
LS
SVM+PSO
2
98.81
Liang et
al.[
21]
4
TP+FR
NMF
KNN
3
92.86
Muru
et
al.[
18]
4
SSA
EMD
ANN
10
95.14
This work
4
MPE
REL
KNN
9
99.72
This work
10
MPE
REL
KNN
10
99.55
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The proposed experimental methodology achieved a mean
accuracy of 99.72% for classification of 4 different bearing
system failures. Table IV shows the confusion matrix for this
classifier.
TABLE IV.
CONFUCTION MATRIX OF THE BEST RANKING.
Classified
Nor
IR1
IR2
BE
Labeled
Nor
28
0
0
0
IR1
0
70
1
0
IR2
0
0
70
0
BE
0
0
0
71
The proposed methodology is capable of classifying the
failures with a high success rate, to the point that only one
sample avoided obtaining 100% accuracy. The effectiveness
of the methodology can be seen with also achieved a mean
accuracy of 99.55% for the classification of 10 different
bearing system failures.
V. CONCLUSIONS
This article presents a methodology for the diagnosis of
bearing failures based on the Multiscalar Permutation Entropy
(MPE) technique. The MPE proves to be a highly effective
characterization methodology to find information that allows
to separate classes. Specifically, in the mechanical vibration
signals that have a high non-stationary, the MPE manages to
find characteristics that would not be detected by other
methodologies. The MPE measures the non-linear dynamics
existing in non-stationary time series and when combined with
Relieff as a feature selection technique, a robust tool for
classification applications is obtained. For the classification a
method of K-Neighbors Nearest (KNN) was used, which
manages to adapt to the nature of the characteristics. The
results confirm a classification accuracy of more than 99:9%
with a computation time of 16:37 seconds, which exceeds the
results established in the literature.
ACKNOWLEDGMENT
The authors thank the Master in Electrical and Mechanical
Engineering of the Technological University of Pereira for
their support throughout the investigation. In addition, we
would like to thank COLCIENCIAS for supporting the project
entitled: Desarrollo de un sistema de monitoreo para el
análisis energético y de condición de emisiones en motores de
combustión interna diésel con base en técnicas no
destructivas with code 1110-776-57801, through which the
research described in this article was developed.
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Juan C. Mejia
Electronic Engineer of the
University of Quindío. Professor at the
Technological University of Pereira. He has
extensive experience as a researcher in fields of
instrumentation, digital systems programming,
signal processing and machine learning
algorithms. He has multiple bibliographic
productions in indexed journals and a wide variety of participation in
scientific events. He is part of the research group Manufacturing
Processes and Machine Design attached to the Technological University
of Pereira.
ORCID: https://orcid.org/0000-0001-7798-2688
Héctor Fabio Quintero Riaza
received Ph.D.
degree in Mechanical Engineering from Cataluña
Polytechnic University, Barcelona, Spain, in 2006.
Now he works at Technological University of
Pereira. His current research interests include
analysis and synthesis of mechanisms, mechanical
vibrations, and fault diagnosis. He is part of the
research group Manufacturing Processes and Machine Design attached
to the Technological University of Pereira.
ORCID: https://orcid.org/0000-0002-4275-739X
Julian D. Echeverry-Correa
received his PhD in
Electronic Systems from Universidad Politécnica
de Madrid, Spain, in 2015. Since 2007 he is
professor at the Program of Electrical Engineering
at Universidad Tecnológica de Pereira, currently as
an Associate Professor. His main research areas
include signal processing, data analysis and pattern
recognition.
ORCID: https://orcid.org/0000-0002-4275-739X