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 claves—Dinámica, 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

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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

 

in a

space of representation of characteristics, where T is the length

of the time series, the entropy is represented as in (1):

)(ln)()(

xpxpXH



−=

(1)

Where x

∈ R and p(x

) is the marginal probability. The time

series can be represented with a delay of time and dimension

given by (2):

( )





...,,

−++

miii

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

 

, m

i =





by (3):

( )

( ) ( )











ln,

ppmH





−=

(3)

The probability

( )





is calculated by (4):

( )







jutypeu









)(:

(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

(u) = 1 if u ∈ and 1

(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.

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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

88.16

1.82

99.01

0.56

90.41±1.66

95.41

1.18

79.08

2.31

99.25

0.48

92.16±1.52

94.66

1.27

81.33±2.21

99.16±0.51

92.83±1.45

96.83

0.99

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

96.01

2.79

98.25

0.76

93.92

2.75

94.92

3.05

94.51

2.67

96.75

0.55

92.01

3.85

92.17

1.81

94.83±1.95

98.08±1.41

93.58±2.32

94.25

1.27

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

al.[

14]

PE+EMD

SVM

97.75

Yuwono

al.[

17]

WPT

HMM

95.8

Ben et al.[19]

TP+FR+EMD

ANN

Zhu et al.[28]

HE+SE+MSE

SVM+PSO

97.75

Han et

al.[

16]

SE+LDM

SVM

100

Zheng

al.[

15]

ANFIS

99.29

Liu et

al.[

29]

TP-FR

WPT+SVM+PSO

97.5

Tiwari et

al.[

MPE

ANFC

02.15

William et

al.[

20]

ANN

97.13

Ocak et

al.[

30]

LPM

HMM

99.6

Wei et

al.[

FR+WPT

Relief

Shao et

al.[

31]

DAE+CAE

LPP

Softmax

Zheng

al.[

GCMPE

SVM+PSO

98.81

Liang et

al.[

21]

TP+FR

NMF

KNN

92.86

Muru

al.[

18]

SSA

EMD

ANN

95.14

This work

MPE

REL

KNN

99.72

This work

MPE

REL

KNN

99.55

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447

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

Labeled

Nor

IR1

IR2

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