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\DOI{10.5802/crmeca.292}
\datereceived{2024-12-23}
\daterevised{2025-02-27}
\dateaccepted{2025-03-01}
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\dateposted{2025-03-20}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Intelligent bearing faults diagnosis in non-stationary
conditions based on angular resampling and support vector machine}

\alttitle{Diagnostic intelligent des d\'{e}fauts de roulements en
conditions non stationnaires bas\'{e} sur le r\'{e}\'{e}chantillonnage
angulaire et les machines \`{a} vecteurs de support}

\author{\firstname{Fakhreddine} \lastname{Bouali}\CDRorcid{0000-0002-2996-1767}\IsCorresp}
\address{Laboratory of Applied Precision Mechanics, Institute of Optics
and Precision Mechanics, Ferhat Abbes University Setif 1, Setif 19137, Algeria}
\email[F. Bouali]{bouali.f@univ-setif.dz}
                              
\author{\firstname{Semchedine} \lastname{Fedala}\CDRorcid{0000-0002-3064-0880}}
\addressSameAs{1}{Laboratory of Applied Precision Mechanics, Institute of Optics
and Precision Mechanics, Ferhat Abbes University Setif 1, Setif 19137, Algeria}
                               
\author{\firstname{Hugo} \lastname{Andr\'e}\CDRorcid{0000-0003-1150-563X}}
\address{Universit\'{e} de Lyon, Universit\'{e} Jean Monnet de
Saint-Etienne, LASPI EA3059, 20 Avenue de Paris, 42334 Roanne, France}
                   
\author{\firstname{Ahmed}\nobreakauthor\lastname{Felkaoui}\CDRorcid{0009-0002-1000-389X}}
\addressSameAs{1}{Laboratory of Applied Precision Mechanics, Institute of Optics
and Precision Mechanics, Ferhat Abbes University Setif 1, Setif 19137, Algeria}

\begin{abstract} 
Over the past decades, intelligent bearing diagnostic methods have
become a research hotspot. These methods require the construction of a
Feature Vector (FV) generally composed of indicators calculated from
time sampled vibration signals. However, in non-stationary conditions,
these signals require the application of complex methods whose
calculation time is really important. For this reason, the use of
angular resampling techniques is recommended because they make it
possible to get rid of speed fluctuations and to employ simple
processing methods.

In this work, we propose to use angularly resampled acceleration
signals for intelligent bearing fault diagnosis under non-stationary
conditions. It is the question of comparing three types of FV: classic
angular indicators on angular signals \tralicstex{\textit{x}(\textit{Î¸})}{$x(\theta)$}, original order
spectrum indicators (peak amplitude), or the combination of the two
previous families of indicators. Then, the selection phase is performed
by the Minimum Redundancy Maximum Relevance (MRMR) algorithm to select
the most relevant features. Finally, the classification is carried out
by a cubic support vector machine (SVM) for the detection and
identification stages of various bearings fault conditions. The
effectiveness of the proposed method achieves a perfect classification
rate of 100\%.
\end{abstract} 


\begin{altabstract} 
Au cours des derni\`{e}res d\'{e}cennies, les m\'{e}thodes
intelligentes de diagnostic des roulements sont devenues un sujet de
recherche majeur. Ces m\'{e}thodes n\'{e}cessitent la construction
d'un vecteur forme (VF), g\'{e}n\'{e}ralement compos\'{e}
d'indicateurs calcul\'{e}s \`{a} partir de signaux de vibration
\'{e}chantillonn\'{e}s \`{a} pas de temps constant. Cependant, en
conditions non stationnaires, ces signaux exigent l'application de
m\'{e}thodes complexes, dont le temps de calcul est
particuli\`{e}rement \'{e}lev\'{e}. C'est pourquoi l'utilisation
des techniques de r\'{e}\'{e}chantillonnage angulaire est
recommand\'{e}e, car elles permettent de s'affranchir des
fluctuations de vitesse et d'employer des m\'{e}thodes de traitement
plus simples.

{Dans ce travail, nous proposons d'utiliser des signaux
d'acc\'{e}l\'{e}ration r\'{e}\'{e}chantillonn\'{e}s angulairement pour
le diagnostic intelligent des d\'{e}fauts de roulements en conditions
non stationnaires. Il s'agit de comparer trois types de vecteurs formes
: des indicateurs angulaires classiques sur les signaux angulaires
\tralicstex{\textit{x}(\textit{Î¸})}{$x(\theta)$}, des indicateurs spectraux d'ordre originaux (amplitude
de pic) ou la combinaison de ces deux familles d'indicateurs. Ensuite,
la phase de s\'{e}lection est r\'{e}alis\'{e}e \`{a} l'aide de
l'algorithme} {\textit{Minimum Redundancy Maximum Relevance}} {(MRMR)
afin de choisir les indicateurs les plus pertinents. Enfin, la
classification est effectu\'{e}e par la m\'{e}thode machine \`{a}
vecteurs de support cubique (SVM Cubique) pour les \'{e}tapes de
d\'{e}tection et d'identification des diff\'{e}rentes conditions de
d\'{e}faut des roulements. L'efficacit\'{e} de la m\'{e}thode
propos\'{e}e permet d'atteindre un taux de classification parfait de
100 \%.}
\end{altabstract} 

\shortrunauthors

\keywords{\kwd{Bearings}
\kwd{Fault diagnosis}
\kwd{Support vector machines}
\kwd{Time-varying rotating speed}
\kwd{Angular resampling}}

\altkeywords{\kwd{Roulements}
\kwd{Diagnostic des d\'{e}fauts} 
\kwd{Machines \`{a} vecteurs de support} 
\kwd{Vitesse de rotation variable}
\kwd{R\'{e}\'{e}chantillonnage angulaire}}

\thanks{Algerian Ministry of Higher Education and Scientific Research (PNE 2019--2020)}

\maketitle
{\vspace*{-5pt}}

\end{noXML}

\section{Introduction}

The complicated form of machines in modern industry makes their faults
diagnosis more complex. Bearings are the most commonly used elements in
these machines and their failure is known as the main causes of machine
breakdowns. Effective bearing faults diagnosis plays a very important
role in preventing faults and increasing machine disponibility and
reliability. Research has focused on the detection and diagnosis of
bearing faults, intending to increase the accuracy and speed of
calculation, reducing costs, performing real-time monitoring and
diagnosis, and improve the efficiency of repair and troubleshooting,
which has resulted in the use of machine learning algorithms (MLA) to
automatically diagnose faults in rotating machinery~\cite{1, 2}.
Previous research have proven the performance of many MLA in bearing
diagnosis such as the K-Nearest Neighbors (KNN)~\cite{3}, Gaussian
process~\cite{4}, Hidden Markov Model (HMM)~\cite{5,6}, and Bayesian
Neural Networks (BNN)~\cite{7}, These non-parametric methods are based
on the probability laws and require a large number of samples i.e.\ 
several features vectors (FV), leading to a problem of storage and
computational costs. Unlike most classification methods, the Support
Vector Machine (SVM) does not require a large number of training
samples to solve a learning problem~\cite{8}. Another popular
classification method called Artificial Neural Networks (ANN)
incorporates recursive algorithms that adjust system parameters based
on a risk function such as empirical risk minimization (ERM) during the
training process. Nevertheless, this method requires a relatively large
number of samples to perform the learning of the different classes.
However, the SVM uses a risk function known as structural risk
minimization (SRM) which allows the algorithm to take data sparseness
into account and minimize the upper bound error of an expected risk~\cite{9}.



The construction of the Features Vectors (FV) is a primary task. Each
FV is composed of indicators that must be able to define the different
operating modes, thus reflecting the exact meaning of the classes that
represent the different operating states~\cite{10, 11}. Before
calculating the features, it is necessary to pre-process the signals.
Various signal processing techniques suitable for bearing faults
diagnosis such as Fourier transform or envelope analysis have been
widely studied. The latter aims at demodulating HF band expected to
carry bearing fault signature, since the signal induced by a bearing
defect excites the high resonance frequencies. Therefore, automatic
determination of the optimal demodulation band is often
necessary~\cite{12}. For this, Spectral Kurtosis (SK) is a useful
technique designed to determine the optimal demodulation-band
automatically~\cite{13,14}. However, for varying speed conditions,
the previous methods are unable to detect the defects. So, different
time-frequency analysis techniques have been established to eliminate
the influence of speed variation~\cite{15}, Huang et~al.\ proposed
Empirical Mode Decomposition (EMD) to solve the problem of
non-stationary conditions~\cite{16}. Although it presents some
drawbacks such as the lack of mathematical formulation and the mode
mixing under singularities~\cite{17}. Researchers have proposed new
methods to adjust these problems. For instance, Smith proposed the
Local Mean Decomposition (LMD)~\cite{18}, Ziani et~al.\ proposed the EMD,
TKO and Shock Detector to diagnose gears faults in non-stationary
conditions~\cite{19}, and the Local Characteristic scale Decomposition
(LCD) was introduced by Cheng et~al.~\cite{20}. Other methods based on
the exploitation of the Amplitude and Frequency Modulation property
(AM-FM) of the Intrinsic Mode Function (IMF) have been proposed to
obtain a rigorous mathematical formulation and better robustness to
noise compared to EMD method like the Empirical Wavelet Transform
(EWT)~\cite{21}, the Variational Mode Decomposition (VMD)~\cite{22} and
the Adaptive Local Iterative Filtering (ALIF)~\cite{23}. Thus, all
these Adaptive Mode Decomposition (AMD) methods share the same
disadvantage, which makes them only suitable to analyze signals without
spectral overlap. Whereas, Order Analysis (OA) is considered to be the
most widely used method for detecting bearing faults under variable
speed conditions. This method transforms the vibration signal from the
time domain to the angular domain by resampling it at a constant angle
interval using interpolation~\cite{24,25}. These angular signals have
the particularity of ensuring an integer constant number of samples per
revolution and thus eliminate the effects of speed variations. Several
techniques are used in the literature for the angular resampling of
vibration signals~\cite{10,26,27}.

In this study, we propose to demonstrate the effectiveness of angularly
resampled acceleration signals in the intelligent diagnosis of bearing
faults operating only under non-stationary conditions. It is a question
to show how to extract relevant indicators to build a feature vector
from the angular signals. Thus, we compare three FVs types: the first
is composed only from the classic angular indicators on angular signals
\textit{x}(${\theta}$), the second is composed only from the order
spectrum indicators (peak amplitude), and the last is the combination
of both previous families of indicators. Then, to select the most
relevant features, the Minimum Redundancy Maximum Relevance (MRMR)
algorithm is applied. Finally, the classification is carried out by a
cubic support vector machine (Cubic SVM) for the stages of bearing
fault detection and identification.

The layout of this paper is as follows: Section~\ref{sec2} introduces the
principle of angular resampling. In Section~\ref{sec3} we outline the
classification procedure by SVM. Section~\ref{sec4} is devoted to the
description of the database. Section~\ref{sec5} describes the proposed
algorithm, the obtained results and discussions. Finally, Section~\ref{sec6}
concludes this paper.

\section{Angular resampling}\label{sec2}

Generally, temporal vibration signals are the most used in the field of
rotating machinery monitoring despite their sensitivity to
non-stationary conditions~\cite{25}. However, angular resampling
eliminates the effects of speed variations since it ensures a constant
number of points per period. There are several techniques for obtaining
angular signals~\cite{26,27}. The angular resampling technique is an
interesting alternative to direct angular sampling because it does not
require expensive instrumentation and also offers the possibility to
exploit both temporal and angular signals. This technique can rely on
the simultaneous recording of vibration and encoder signals. Then, the
angular acceleration signal is obtained through the interpolation of
the vibration signal at each angular position delivered by the
encoder~\cite{10}.

Figure~\ref{fig1} presents the angular resampling method. The technique
consists in determining the angular positions $\Delta \theta (i)$\ 
$i=1,2,\ldots , n$ according to the rising edges appearance time
$\Delta t(i)\  i=1,2,\ldots , n$  of the encoder pulse. Then, the
resampling of the temporal signal in the angular domain is performed by
cubic ``spline'' interpolation~\cite{28}. Algorithm~\algOneREF{alg1} 
summarizes the
steps of the angular resampling used in this study using zero crossing
method.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Angular resampling.}
\end{figure}

\beqimg\begin{eqimg*}
\begin{algorithm}
\begin{inftab}
\begin{tabular}{@{}lll}
\xtbody
 \textbf{Input}  &amp; \multicolumn{2}{l}{$x(t_{i}), F_{s}$, \textit{Resolution, TTL signal}} \\\cline{1-1}
 {\textbf{Procedure}}  &amp; \textit{Step 1} &amp; Determine the low and high states of the encoder signal \\
 &amp; \textit{Step 2} &amp; Find the times at which each pulse starts $(t_{i})$ \\\cline{1-1}
 &amp; \textit{Step 3} &amp; Determine the time intervals between pulse $(\Delta t_{i})$ \\
 &amp; \textit{Step 4} &amp; Estimate: ``angle time'' function \\
 &amp; \textit{Step 5} &amp; Determine $\Delta \theta _{i}$ \\
 &amp; \textit{Step 6} &amp; Cubic interpolation of $x(t_{i})$ at $x(\theta _{i})$ \\\cline{1-1}
 \textbf{Output}  &amp; \multicolumn{2}{l}{$x(\theta _{i})$} 
\end{tabular}
\end{inftab}
\caption{Angular resampling using a zero-crossing}
\label{alg1}
\end{algorithm}
\eeqimg\end{eqimg*}


\section{Classification procedure by SVM method}\label{sec3}

SVM is a powerful classification tool that is widely used in various
machine-learning applications~\cite{29,30,31}, SVM's basic principle is
to separate two classes using an ideal hyperplane that maximizes the
margin between the two hyperplanes. Figure~\ref{fig2} summarizes the
basic principle.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}An example of two classes separated by SVM.}
\end{figure}


To describe the SVM algorithm, let's consider the set P that trains the
SVM classifier:
{\begin{equation}
\label{eq1}
P=(x_{i}, y_{i}),\quad x_{i}\in R^{m},\quad y_{i}\in {\{-1,1\}}_{i=1}^{n} \quad 
i=1,2,\ldots , n.
\end{equation}}\unskip

In the case of linearly separated data, where $x$ an input space
containing $m$ indicators of a $n$ training set samples, while $y_{i}$
is the desired output, and $\begin{cases} 
y_{i}=1\\
y_{j}=-1
\end{cases}$
the separating hyperplane $f(x)=0$. can be presented as:
{\begin{equation}
\label{eq2}
f(x)=w^{T}x+b=\sum _{i=1}^{n}w_{i}x_{i}+b=0
\end{equation}}\unskip
where $b$ is the bias scalar and $w$ is a weight vector. 

The hyperplane was defined as:
{\begin{equation}
\label{eq3}
y_{i}f(x_{i})=y_{i}(w^{T}x_{i}+b)-1\geq 1.
\end{equation}}\unskip
Soft margin are obtained by introducing a slack factor $\xi _{i}\geq 0$: 
{\begin{equation}
\label{eq4}
y_{i}(w^{T}x_{i}+b)\geq 1-\xi _{i}\quad (i=1,2,\ldots , n).
\end{equation}}\unskip

The following equations derived from a soft margin method may be
calculated to produce the hyperplane separating the two classes with
the maximum distance of ${2}/{\|w\|}$ is equivalent to minimizing
$\|w\|^{2}$ from the Euclidean ${1}/{\|w\|}$ further to hyperplane
$f(x)=0$.
{\begin{equation}
\label{eq5}
\text{minimise}
\left(\frac{1}{2}\|w\|^{2}+C\sum _{i=1}^{n}\xi
_{i}\right)
\end{equation}}\unskip
where $C$ is the regularization parameter that controls the trade-off
between the classification accuracy and the evaluation of the margin
maximization.

To overcome the constraints complexity induced by the slack variable,
the Lagrangian method (Equation~(\ref{eq5})) makes the problem easily solvable and can
be extendible to the kernel SVM
{\begin{equation}
\label{eq6}
\text{maximize} W(\alpha )={\sum }_{i=1}^{n} \alpha
_{i}-\frac{1}{2}{\sum }_{ij=1}^{n}\alpha _{i} \alpha
_{j}y_{i}y_{j}(x_{i}x_{j})
\end{equation}}\unskip
$\alpha $: Lagrange multiplier.

The kernel function $K$ is used in the case of nonlinear separability
to transform the input vectors in to a high dimensional space with
linear separability. Where, $(x_{i}, x_{j}) $ is replaced by
$K(x_{i}x_{j})$.
{\begin{equation}
\label{eq7}
\text{maximize} W(\alpha )=\sum _{i=1}^{n} \alpha
_{i}-\frac{1}{2}\sum _{ij=1}^{n}\alpha _{i} \alpha
_{j}y_{i}y_{j}K(x_{i}x_{j}).
\end{equation}}\unskip
The optimal classification function can be given:
{\begin{equation}
\label{eq8}
f(x)=\text{sign}\left(\sum _{ij=1}^{n}\alpha _{i}y_{i}K (x_{i}x_{j})+b\right)
\end{equation}}\unskip

Originally, SVMs were created for binary classes. But many real
problems have more than two classes. Multi-class SVMs is an extension
of the binary SVM classification issue as described by Wong 
et~al.~\cite{32}. Two multi-class SVMs methods, frequently employed are:
\begin{itemize}
\item One-Against-One method (OAO), involves binary SVM classifier
construction for all pairs of classes.
\item One-Against-All method (OAA), separates each class from all
others and constructs a combined classifier.
\end{itemize}
Hsu et~al.~\cite{33} compared the multi-class SVMs techniques in detail
on major experimental problems, noting that an OAO method is a
competitive and effective approach.

In bearing fault classification, several defects like inner race
defect, outer race defect, and ball defect occur during fault
diagnosis. However, in the case of simultaneous fault conditions, a
reliable diagnosis may be more difficult. There may be circumstances
when some defects remain stable while others get severe. A robust SVM
that can effectively distinguish between healthy and faulty conditions,
is thus required, also identifying the severity of damaged fault
conditions. In this paper, SVM method is applied to evaluate the impact
of angular measurements in an intelligent bearing condition monitoring
strategy and accurately and reliably give the presence of specific
defective bearing and can also distinguish different types of
defects~\cite{34}.

\section{Description of the dataset}\label{sec4}

In this study, we chose the bearing experiment data provided by the
University of Ottawa~\cite{35} and openly available in~\cite{36}, for
several reasons. First, the instrumentation of this database is
inexpensive. Second, the signals are sampled temporally with a very
high sampling rate of 200~kHz. Third, the diversity of physical
quantities measured (vibration, speed (encoder)). These offer the
possibility of applying several advanced signal processing methods and
consequently extracting more information on the health status of the
system regardless of the operating conditions (types of faults and/or
speed variation) and thus validating the effectiveness of the proposed
algorithm for non-stationary and general operating. 


The experimental setup shown in Figure~\ref{fig3}, is performed on a
SpectraQuest machinery fault simulator (MFS-PK5M), consisting of a
shaft supported on two ball bearings (ER16K) and controlled by an AC
drive. Table~\ref{tab1} summarizes the geometric parameters and the
Characteristic Defect Frequency (CDFs) of the tested bearings. The
healthy bearing is on the left, while the experimental bearing is on
the right with different health conditions, Healthy (H), faulty with an
Inner Race Defect (IRD), faulty with an Outer Race Defect (ORD), faulty
with Ball Defect (BD) and faulty with Combined Defects (CD). An
incremental encoder (EPC model 775) with 1024 cycles per revolution
measures the shaft rotational speed. In addition, an ICP accelerometer
(Model 623C01) was mounted on the housing bearing to obtain vibration
signals.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Experimental set-up~\cite{35}.}
\end{figure}

\begin{table}
\caption{\label{tab1}Structural parameters and CDFs of the tested
bearings (ER16K)}
\begin{tabular}{ccccccc}
\thead
Bearing type &amp; \parbox[t]{2.2cm}{\centering Pitch diameter (mm)} &amp;
\parbox[t]{2cm}{\centering Ball diameter (mm)} {\vspace*{2.5pt}}&amp;
\parbox[t]{1.3cm}{\centering Number of
balls} &amp; BPFI &amp; BPFO &amp; BSF \\
\endthead
ER16K &amp; 38.52 &amp; 7.94 &amp; 9 &amp; $5.43\times fr$ &amp; $3.57\times fr$ &amp; $2.32\times fr$ 
\botline
\end{tabular}
\tabnote{BPFI: Ball Pass Frequency Inner race,
BPFO: Ball Pass Frequency Outer race,
BSF: Ball (roller) Spin Frequency,
\textit{fr}: Rotating Frequency.}
\end{table}

The data are acquired by the NI data acquisition board (NI USB-6212
BNC). Each sampled dataset contains two channels, ``Channel\_1'' is the
vibration data and ``Channel\_2'' is the rotational speed data (TTL
signals). In all the experiments, vibration and speed signals are
sampled at 200 kHz for 10~s for each one. Signals are obtained in
four time-varying rotating speed conditions: increasing speed,
decreasing speed, increasing then decreasing speed, and decreasing then
increasing speed.

Table~\ref{tab2} shows how the dataset is numbered for the following
experimental conditions: health and varying speed conditions, for each
experimental setting, three trials are obtained, totaling 60 datasets,
for example, the dataset H-A-1 presents the vibration and TTL data
collected from a healthy bearing and an increased operating rotational
speed from 14.1~Hz to 29~Hz. The dataset O-D-3 presents the vibration
and TTL data collected from a faulty bearing with an outer race defect
and a decreasing operating rotational speed from 25.5~Hz to 15~Hz, then
increasing to 19.6~Hz. The description of the other datasets is
detailed in~\cite{35}.


\begin{table}
\caption{\label{tab2}Dataset numbering}
\begin{tabular}{ccccc}
\thead
\morerows{1}{\raisebox{-14pt}{Bearing health conditions}}
&amp; \multicolumn{4}{c}{Speed varying conditions{\vspace*{2pt}}} \\\cline{2-5}
&amp; \parbox[t]{1.5cm}{\centering Increasing speed} &amp; 
\parbox[t]{1.5cm}{\centering Decreasing speed} {\vspace*{2pt}}&amp;
\parbox[t]{2.7cm}{\centering Increasing then  decreasing speed} &amp; 
\parbox[t]{2.7cm}{\centering Decreasing then  increasing speed} \\
\endthead
\morerows{2}{Healthy} &amp; H-A-1 &amp;  H-B-1 &amp; H-C-1 &amp;  H-D-1\\
&amp; H-A-2 &amp;  H-B-2 &amp; H-C-2 &amp;  H-D-2\\
&amp; H-A-3 &amp;  H-B-3 &amp; H-C-3 &amp;  H-D-3{\vspace*{5pt}}\\
\morerows{2}{Faulty (inner race defect)}&amp; I-A-1 &amp;  I-B-1 &amp; I-C-1 &amp;  I-D-1\\
&amp; I-A-2 &amp;  I-B-2 &amp; I-C-2 &amp;  I-D-2\\
&amp; I-A-3 &amp;  I-B-3 &amp; I-C-3 &amp;  I-D-3{\vspace*{5pt}}\\
\morerows{2}{Faulty (outer race defect)}&amp; O-A-1 &amp;  O-B-1 &amp; O-C-1 &amp;  O-D-1\\
&amp; O-A-2 &amp;  O-B-2 &amp; O-C-2 &amp;  O-D-2\\
&amp; O-A-3 &amp;  O-B-3 &amp; O-C-3 &amp;  O-D-3{\vspace*{5pt}}\\
\morerows{2}{Faulty (ball defect)}&amp; B-A-1 &amp;  B-B-1 &amp; B-C-1 &amp;  B-D-1\\
&amp; B-A-2 &amp;  B-B-2 &amp; B-C-2 &amp;  B-D-2\\
&amp; B-A-3 &amp;  B-B-3 &amp; B-C-3 &amp;  B-D-3 {\vspace*{5pt}} \\
\morerows{2}{Faulty (combined defect)}&amp; C-A-1 &amp;  C-B-1 &amp; C-C-1 &amp;  C-D-1\\
&amp; C-A-2 &amp;  C-B-2 &amp; C-C-2 &amp;  C-D-2\\
&amp; C-A-3 &amp;  C-B-3 &amp; C-C-3 &amp;  C-D-3
\botline
\end{tabular}
\end{table}


\section{Experimental part}\label{sec5}

Figure~\ref{fig4} shows the flowchart of the methodology employed in
this study. The encoder signals are used to resample vibration signals
in the angular domain. Then, different types of indicators are
extracted to construct three FVs. The first is composed from four
classic angular indicators. The second FV is composed from four order
spectrum indicators. The last is the combination of the previous FVs.
All FVs will then be used to train and test the SVM classifiers without
and with selection using the MRMR algorithm. The classification is
divided in two stages. First, the detection stage is carried out by
testing the Healthy (H) class with the Damaged class (D). Second, the
identification is performed to compare only all damaged classes (IRD,
ORD, BD, and CD).


\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Proposed method flowchart.}
\end{figure}


\subsection{Signal analysis}

To analyze bearing fault signals under non-stationary conditions,
angular resampling is necessary to eliminate speed variation influence.
We calculated the following indicators on the \mbox{angular} signals
($x_{i}$): RMS, kurtosis, Crest Factor (CF), Impulse Factor
(IF)~\cite{37,38}. The following equations present the mathematical
formulas of the obtained indicators respectively:
{\advance\jot by 4pt
\begin{eqnarray}\label{eq9}
&amp;\displaystyle x_{\mathrm{rms}}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}
(x_{i}-\overline{x})^{2}}\Seqnsplit
\label{eq10}
&amp;\displaystyle x_{\mathrm{kurt}}=\frac{\sum_{i=1}^{N}(x_{i}-
\overline{x})^{4}}{\left[\frac{1}{N}\sum _{i=1}^{N}(x_{i-}\overline{x})^{2}\right]^{2}}\Seqnsplit
\label{eq11}
&amp;\displaystyle \mathrm{CF}=\frac{\max |x_{i}-\overline{x}| }{x_{\mathrm{rms}}} \Seqnsplit
\label{eq12}
&amp;\displaystyle \mathrm{IF} =\frac{\max |x_{i}-\overline{x}|
}{\frac{1}{N}{\sum }_{i=1}^{N}|x_{i}| } .
\end{eqnarray}}\unskip

Figure~\ref{fig5} shows the acceleration signal of the CD, for increasing speed
then decreasing. The presence of defects causes a significant increase
in temporal signals energy and a presence of random shocks. From this
representation, we cannot determine the faults type of this bearing. On
the other hand, due to the variation of rotational speed, the periodic
pulses cannot be defined in time domain signals. Therefore, it is
impossible to judge the presence of a defect or identify its type.


\begin{figure}
\includegraphics{fig05}
{\vspace*{2pt}}
\caption{\label{fig5}(a) Faulty bearing temporal signal with a CD,
(Rotation speed increasing then decreasing), (b) The zoomed portion of
the signal indicated by a red box.}
{\vspace*{2pt}}
\end{figure}

The angular resampling of the vibration signal in an integer number of
samples per revolution allows it possible to solve the problems of
periodicity and to overcome blurring effect induced by the speed
variation. Figure~\ref{fig6} presents the angular signal of the CD, with
increasing then decreasing speed conditions. One can easily notice the
presence of periodic shocks as well as a significant increase in
amplitude compared to the temporal signal (Figure~\ref{fig5}).



\begin{figure}
\includegraphics{fig06}
{\vspace*{1pt}}
\caption{\label{fig6}(a) Faulty bearing angular signal with a CD,
(Rotation speed increasing then decreasing), (b) The zoomed portion of
the signal indicated by a red box.}
{\vspace*{1pt}}
\end{figure}


\subsection{Order spectrum analysis}\label{sec5.2}

Order spectrum analysis has proven to be one of the most useful
techniques in CM for all rotating machines. Additionally, the Power
Spectral Density (PSD) is a highly helpful technique used to detect
faults in rotating machines~\cite{39}. Many time windows have been
developed to minimize the spectral leakage problem induced by PSD. The
Hanning window is the most used in spectral analyzes~\cite{40}. This
window should be adopted as a norm because it provides a good balance
between amplitude accuracy and frequency resolution, which minimizes
transient errors in periodic repetitions of the time signal~\cite{41},
and help to minimize the edge effect by reducing the side lobes in the
frequency domain.

In this study, we have applied the PSD on an integer number of
revolutions, to extract features from the spectrum domain, using the
Hanning window, whose length is equal to the length of the signal.

Figure~\ref{fig7} presents the angular signal PSD in the case of IRD, with
rotation speed increasing from 12.5 Hz to 27.8 Hz. The BPFI is directly
observed in the 5.43 order and its harmonics, which causes a periodic
phenomenon and a significant increase in the fault peak amplitude. We
note that this fault is the most dominant on the signal in the
different fault conditions.


\begin{figure}
\includegraphics{fig07}
{\vspace*{1pt}}
\caption{\label{fig7}(a) Angular signal PSD presentation with Inner
Race Defect (Rotation speed increasing from 12.5 Hz to 27.8 Hz), (b)
The zoomed portion of the signal indicated by a red box.}
{\vspace*{1pt}}
\end{figure}


Figure~\ref{fig8} shows the order spectrum of the signal with an ORD.
The peak corresponding to this defect is located at order 3.57, which
is the fundamental characteristic frequency (BPFO). The energy of the
first harmonic in the healthy case (H) is greater than this case (ORD)
for all signals, although we took the same number of points for all
database signals before the construction of the order spectrum
indicators. Many researches discussed this phenomenon, it can be
explained by the theory of the ``healing phenomenon'' when the defect is
well advanced~\cite{42,43,44,45}.


\begin{figure}
\includegraphics{fig08}
\caption{\label{fig8}Angular signal PSD presentation with ORD (Rotation
speed increasing from 14.8~Hz to 27.1 Hz).}
\end{figure}


Figure~\ref{fig9} shows the angular signal PSD for CD. We distinguish the
presence of orders 3.57 and 5.43 corresponding to the outer race (BPFO)
and the inner race defects (BPFI) respectively. It can be seen that the
inner race fault (BPFI), presents a significant increase in the main
order levels and that some of its harmonics are modulated by the shaft
rotation. This defect can be considered among the most intensive
defect. However, the order spectrum focused on the rotational frequency
of the ball at order 2.32 does not clearly show the presence of the
defect, we notice a modest increase in amplitude compared to the
healthy state (Figure~\ref{fig10}).


\begin{figure}
\includegraphics{fig09}
\caption{\label{fig9}(a) Angular signal PSD presentation with CD
(Rotation speed increasing from 13~Hz to 27.9~Hz). (b) The zoomed portion
of the signal indicated by a red box.}
\end{figure}


\begin{figure}
{\vspace*{2pt}}
\includegraphics{fig10}
\caption{\label{fig10}(a) Angular signal PSD presentation with BD
(Rotation speed increasing from 13~Hz to 27.9 Hz). (b) The zoomed portion
of the signal indicated by a green box.}
{\vspace*{2pt}}
\end{figure}

The precedent figures allow a possibility to keep track with precision
of the order components of the powers associated with the various
conditions, whether a healthy state, inner race defect, outer race
defect, a ball defect, or combined defects.

Several approaches have been developed for automating the detection of
bearings defects, for example, Shiroishi et~al.~\cite{46}, proposed
three frequency domain features: Spectrum peak Ratio Outer (SPRO),
Spectrum Peak Ratio Inner (SPRI), and Spectrum Peak Ratio Rolling
element (SPRR) to identifying bearing defects. These indicators proved
their effectiveness only in stationary conditions. Farhat 
et~al.~\cite{47}, adapted the precedent features for the angular field
using the envelope order analysis technique to eliminate the speed
variation. The updated features: SPROa, SPRIa, and SPRRa are obtained
by dividing the sum of the defect peaks (fundamental and harmonics) by
the sum of all peaks in the order spectrum. This method takes a long
computational cost to extract the features. However, the first harmonic
of a defect in the order spectrum is sufficient to detect the defect in
the bearing. 

In this work we propose to use only the level of the first harmonic for
each frequency defect, the Level of Inner Race Frequency Defect
(LIRFD), the Level of Outer Race Frequency Defect (LORFD), and the
Level of Ball Frequency Defect (LBFD), we also take the Sum of the
levels of the 40th first harmonics of Orders (SO) as an indicator,
because the majority of the bearing faults generating a modulation
effect~\cite{12}. Renaudin et~al.~\cite{48} found that the positions of
these peaks will remain fixed despite variations in speed, whereas the
amplitudes vary differently from one frequency channel to another. In
addition, this indicator does not require a long computation time. We
employed a computer equipped with an Intel(R) Core (TM) i7-10750H CPU,
running on Windows 11 Pro (version 24H2), with 16.0 GB of RAM to
construct the Feature Vectors (FVs). To evaluate the computational
efficiency of our methodology, we measured the time required for both
angular resampling and extracting all indicators. For a single signal,
the entire process was completed in less than 0.2 s. When
processing the entire database, the total computational time was under
18~s. These results highlight the efficiency and speed of the
proposed method, confirming its suitability for real-time applications
and high-dimensional time series data analysis. Furthermore, the
results demonstrate that the proposed method outperforms techniques
found in the literature source for bearing defect diagnosis under
various conditions, such as those based on time-frequency features
extracted through Wavelet Packet Transform (WPT) and dimension
reduction using Multi-Weight Singular Value Decomposition (MWSVD),
where the reduced dimensionality features are used in Support Vector
Machines (SVM) for bearing fault diagnosis~\cite{49}. This performance
advantage is attributed to the method's ability to efficiently handle
complex datasets while maintaining computational{\unskip\break} tractability.

\subsection{Feature vectors (FVs) construction}

One of the most vital steps in the pattern recognition process is
feature vector construction. The objective of this step is twofold:
firstly, the reduction of the dimensionality of the data provided to
the classifier while keeping the maximum information relating to the
different classes of functioning. Secondly, converting vibration
signals into monitoring indicators which enables a more efficient use
of classification~\cite{50}.

From the angular signals cited in the previous section, we calculated 8
indicators which are summarized in Table~\ref{tab3}. In total, 3 FVs were
constructed from different domains in order to determine the one that
provides the best classification performance in the bearing fault
detection and identification stage. The three FVs used are:
\begin{itemize}
\item FV composed from the angular domain indicators
\item FV composed from the order spectrum domain indicators
\item FV composed from a mixture of angular and order spectrum domain
indicators.
\end{itemize}


\begin{table}
\caption{\label{tab3}Features description}
\begin{tabular}{ccccccc}
\thead
Indicators &amp; Domain &amp; H &amp; IRD &amp; ORD &amp; BD &amp; CD \\ 
\endthead
1 &amp; \morerows{3}{Angular}  &amp; \multicolumn{5}{l}{RMS} \\
2 &amp;  &amp; \multicolumn{5}{l}{Kurtosis} \\
3 &amp;  &amp; \multicolumn{5}{l}{Crest Factor (CF)} \\
4 &amp;  &amp; \multicolumn{5}{l}{Impulse Factor (IF)} {\vspace*{5pt}}\\ 
5 &amp; \morerows{3}{Spectrum}  &amp; \multicolumn{5}{l}{LIRFD} \\
6 &amp;  &amp; \multicolumn{5}{l}{LORFD} \\
7 &amp;  &amp; \multicolumn{5}{l}{LBFD} \\
8 &amp;  &amp; \multicolumn{5}{l}{SO}  
\botline
\end{tabular}
\end{table}


To keep the same order of magnitude and the same physical units, all
FVs are normalized with center 0 and standard deviation 1 using the
standard MATLAB function ``normalize'' specified with a ``z-score''
option.

\subsection{Feature selection by MRMR algorithm}

Feature selection has been widely studied in areas such as machine
learning, pattern recognition, data mining, and signal processing.
Selected features are used to reduce the data dimensionality by
choosing an optimal subset of features that increases classification
accuracy. Regarding class/label classification, the challenge is to
find the feature subset with the lowest cardinality that maintains the
information found in the whole feature set $X=\{X^{(1)},
X^{(2)},\ldots , X^{(d)}\}$ with class set $C=\{c_{1},
c_{2},\ldots ,c_{m}\}${.} The two most common feature selection
techniques are filters and wrappers~\cite{51}. Filters approaches are
the most used because they quantify intrinsic feature values such as
feature variance, feature relevance and also take reduced computation
time compared to wrapper methods. Consequently, we propose to apply a
filter type selection method called MRMR algorithm which means Minimum
Redundancy Maximum Relevance Algorithm. This method uses the mutual
information of predictor variables and response variables 
(Equation~(\ref{eq13})) to
identify an optimal collection of mutually and maximally different
features in order to maximize the relevance of a set of features with
respect to the response variable~\cite{52,53}.

The mutual information \textit{I} of the discrete random variables
\textit{X} and \textit{Z} is defined as:
{\begin{equation}
\label{eq13}
I(X,Z)=\sum
_{i,j}P(X=x_{i},Z=z_{j})\text{log}\frac{P(X=x_{i},Z=z_{j})}
{P(X=x_{i}),P(Z=z_{j})}
\end{equation}}\unskip
where $I=0$ if $X$ and $Z$ are independent, and $I$ equal $X$ entropy
if $X$ and $Z$ are both random variables.

The MRMR is used to identify an optimal set $S$ of features that
maximizes the relevance $V_{S}$ of $S$ for a response variable y, and
reduces the redundancy $W_{S}$ of $S$.

Using $I,$ $V_{S}$ and $W_{S}$ are expressed in the following
Equations~(\ref{eq14})  and (\ref{eq15}): 
{\begin{eqnarray}\label{eq14}
&amp;\displaystyle V_{S}=\frac{1}{|S|}\sum _{x\in S}I(x,y) \Seqnsplit
\label{eq15}
&amp;\displaystyle W_{S}=\frac{1}{|S|^{2}}\sum _{x, z\in S}I(x, z) 
\end{eqnarray}}\unskip
$| S| $ presents the number of features in $S$.

The MRMR algorithm uses the Mutual Information Quotient {(MIQ)} value
Equation (\ref{eq16}) to rank features via the forward addition scheme
which requires $O(|\Omega|{\cdot}|S|)$ calculations. Where
{${\Omega}$} is the full feature set and $S$ is the optimal set.
{\begin{equation}
\label{eq16}
\mathrm{MIQ}_{x}=\sfrac{V_{x}}{W_{x}}
\end{equation}}\unskip
$V_{x}$ and $W_{x}$ are the relevance and redundancy of a feature:
{\begin{eqnarray}
\label{eq17}
&amp; \displaystyle V_{x}=I(x,y) \Seqnsplit
\label{eq18}
&amp;\displaystyle W_{x}=\frac{1}{| S| }\sum _{z\in S}I(x,z) .
\end{eqnarray}}\unskip

The MRMR algorithms rank features as follows:
\begin{enumerate}[(1)]
\item~Select the feature with $\max _{x\in \Omega } V_{x}$ and insert
the selected indicator into an empty set $S.$\label{mark-1.}

\item Find the features in the complement of $S$, $S^{c}.$ That have
nonzero relevance and zero redundancy. If $S^{c}$ lacks a feature with
nonzero relevance and zero redundancy, go to step 4. Otherwise, choose
the feature that is most relevant $\max _{x\in S^{c},W_{x}=0} V_{x}$.

\item~Repeat Step 2 until the redundancy for all features in $S^{c}$ is
not zero.

\item Select the feature in $S^{c}$ with the largest {MIQ}
value, nonzero relevance, and nonzero redundancy, and add it to the set
$S$. 
{\begin{equation} \label{eq19} 
\max _{x\in S^{c}} \mathrm{MIQ}_{x}=\max
_{x\in S^{c}} \frac{I(x,y)}{\frac{1}{| S| }\sum
_{z\in S}I(x,z)} 
\end{equation}}\unskip
\item Repeat Step 4 until the significance for all features in $S^{c}$
is zero.
\item In a random pattern, add the features without relevance to $S$.
\end{enumerate}
The MRMR selection algorithm is applied to select the best indicators
from the combined FV to verify that the combination of classic angular
and order spectrum indicators improves the performance of the
classification. Table~\ref{tab4} presents the ranked indicators for the
detection and identification stages. In the detection stage, it can be
observed that the spectral indicators LIRFD and LORFD are the most
relevant indicators, while the crest factor and the LBFD are ranked
third and fourth indicator respectively. The other indicators are
ranked in the last positions. In the identification stage, the five
relevant indicators are respectively the RMS, SO, LORFD, kurtosis, and
LIRFD. The rest of indicators listed in order of relevance are: IF,
LBFD and CF. From these results, one can logically suppose that the
combination of classic angular and order spectrum indicators gives the
best classification performance because the classic indicators allow to
follow the general health state of the machine which allows to easily
detect the presence of an anomaly, while the spectral indicators offer
the identification of the fault type and origin, thus we confirm the
result gives by the Mohanty reference in~\cite{38}.



\begin{table}
\caption{\label{tab4}Ranked features}
\begin{tabular}{ccc}
\thead
Indicator rank &amp; Detection &amp; Identification \\ 
\endthead
1 &amp; LIRFD &amp; RMS \\ 
2 &amp; LORFD &amp; SO \\ 
3 &amp; Crest Factor (CF) &amp; LORFD \\ 
4 &amp; LBFD &amp; Kurtosis \\ 
5 &amp; Kurtosis &amp; LIRFD \\ 
6 &amp; Impulse Factor (IF) &amp; Impulse Factor (IF) \\ 
7 &amp; RMS &amp; LBFD \\ 
8 &amp; SO &amp; Crest Factor (CF) 
\botline
\end{tabular}
\end{table}

\subsection{Results and discussions}

We have different types of faults, so it is essential not only to
detect but also to identify these faults. For this, a cubic SVM
classifier is used for each of the two diagnostic stages. The detection
stage, where the data set is split into two classes, the healthy and
damaged conditions. The classification in this stage is performed using
a 5-fold Cross-Validation (CV). The identification stage in which the
data set consists only of faulty operating cases (4 classes). Here, we
applied the One-Against-One approach for multi-class classification
using a 4-fold CV. In total, ${N(N-1)/2}$ classifiers are
built and each one trains the data of two classes (\textit{N} is the
number of classes). The Cross-Validation Accuracy (CVA) is obtained by
averaging the $k$ individual{\unskip\break} accuracies:
{\begin{equation}
\label{eq20}
\mathrm{CVA}=\sfrac{1}{k}{\sum }_{j=1}^{k}A_{j}
\end{equation}}\unskip
where \textit{k} (4 in our case) is the number of folds used, 
and $A_j$ is the accuracy measure of each fold, $j=1,\ldots, k$.


Table~\ref{tab5} illustrates the classification performance for the
three FVs: angular, spectral, and combined FV (angular and spectral).
It can be noted that the best classification performance is obtained
using the combined FV with a success percentage of 98.3\% for the
detection stage and 97.9\% for the identification stage. While the
performances obtained with the other two FVs are lower. For the angular
FV, the performances reach 90\% for the detection stage and 93.8\% for
the identification stage. For the spectral FV, the classification
performance is equal to 93.3\% for the detection stage and 89.6\% for
the identification stage.


\begin{table}                                           
\caption{\label{tab5}SVM classification performances without selection}
\begin{tabular}{lcc}
\thead
Domain &amp; &amp; \\ 
\endthead
\morerows{3}{Angular}  &amp; \multicolumn{2}{l}{FV (04 indicators)} \\
 &amp;  &amp; Success (\%) \\
 &amp; Detection  &amp; 90 \\
 &amp; Identification  &amp; 93.8 {\vspace*{5pt}}\\ 
\morerows{2}{Order spectrum}  &amp; \multicolumn{2}{l}{FV (04 indicators)} \\
 &amp; Detection  &amp; 93.3 \\
 &amp; Identification  &amp; 89.6 {\vspace*{5pt}}\\ 
\morerows{2}{Angular \&amp; order spectrum}  &amp; \multicolumn{2}{l}{FV (08 indicators)} \\
 &amp; Detection  &amp; \textbf{98.3} \\
 &amp; Identification  &amp; \textbf{97.9} 
 \botline
\end{tabular}
\end{table}



The confusion matrices of the detection and identification stages using
the combined FV are displayed in Figures~\ref{fig11}a,b
respectively. It can be seen that in both stages only one sample was
misclassified. It is a sample of healthy class which is classified in
faulty class for the detection stage and a sample of ball defect class
which is classified with outer race fault class for the identification
stage.


\begin{figure}
\includegraphics{fig11}
{\vspace*{2pt}}
\caption{\label{fig11}Confusion matrices of combined FV.}
{\vspace*{2pt}}
\end{figure}



Table~\ref{tab6} indicates the SVM classifier performances after feature
selection of the combined FV using the MRMR algorithm. The
classification rate achieves a 100\% for the detection and
identification stages using only 7 and 4 indicators respectively.



\begin{table}
\caption{\label{tab6}SVM classification performances with MRMR selection algorithm}
\begin{tabular}{cccc}
\thead
Domain &amp; &amp;&amp; \\ 
\endthead
\morerows{3}{Angular \&amp; spectral}  &amp; \morerows{1}{Detection}  &amp; Success (\%) &amp; \textbf{100} \\
 &amp;  &amp; Number of selected indicators &amp; 7 {\vspace*{5pt}}\\
 &amp; \morerows{1}{Identification}  &amp; Success (\%) &amp; \textbf{100} \\
 &amp;  &amp; Number of selected indicators &amp; 4 
 \botline
\end{tabular}
{\vspace*{4pt}}
\end{table}



Figures~\ref{fig12}a,b show the confusion matrices of the
detection and identification stages using the combined FVs after
feature selection. It can be seen that all the samples are well
classified which gives perfect performance for detecting and
identifying the bearing faults.



\begin{figure}
\includegraphics{fig12}
\caption{\label{fig12}Confusion matrices of combined FV after features
selection.}
\end{figure}

The results of the previous section demonstrate that the proposed
algorithm can accurately distinguish between different bearing fault
conditions especially in non-stationary{\unskip\break} conditions.

In the preprocessing step, the angular resampling of the vibration
signals shows several advantages. Firstly, it eliminates the shift of
the spectrum induced by the speed variation. Moreover, it delivers the
same number of points per revolution contrary to time signals in
non-stationary conditions, which keeps the angular acceleration
homogeneity. The angular resampling is also effective in minimizing
noise in the vibration signals. Thus, it will improve the relevance of
monitoring{\unskip\break} indicators.

In the classification step, the proposed spectral indicators (LIRFD,
LORFD, LBFD, SO) show their effectiveness compared to other features
used in the literature as said by Farhat et~al., in the
referenceh~\cite{47}. The performance obtained is 93.3\% for the
detection step and 89.6\% for the identification step. Moreover, the
classic angular indicators used (RMS, kurtosis, IF, CF) have almost the
same classification results as those achieved with the spectral
indicators.

The combination of angular and spectral features in one FV increases
the performance of the classification for the two stages of detection
and identification with 98.3\% and 97.9\% of success respectively. This
combination of indicators exploits the information complementarity
because the statistical features are effective for monitoring the
machine's health state and spectral domain features are effective for
fault location. The MRMR feature selection algorithm allows to have a
perfect classification performance with 100\% of success and also
minimizes the size of FV which reduces significantly computational
costs.

\section{Conclusion}\label{sec6}

In this research, we propose a new intelligent bearing fault diagnosis
method in non-stationary conditions based on angular resampling of
acceleration signals and SVM. 

In the signal preprocessing step, the acceleration signals are
angularly resampled using the speed signals delivered by an incremental
encoder. The instants corresponding to each angular position are
determined by cubic interpolation. In the FV construction step, we used
four classic angular indicators (RMS, Kurtosis, IF, CF) and we proposed
four indicators calculated from the order spectrum (LIRFD, LORFD, LBFD
and SO). In the classification step, the SVM is trained to classify the
defect categories for the detection and identification stages using
three different FVs. The first is calculated from the angular domain,
the second is extracted from the order spectrum domain, to take
advantage of the information complementarity we have combined the two
FVs in one FV.

The results indicate that the angular resampling leads to more relevant
indicators and combining these features increases classification
performance to reach approximately 98\%. On the other hand, the MRMR
selection algorithm allows the reduction of the FV size which has a
double advantage. Firstly, the classification performances will achieve
100\% accuracy for the detection and identification stages. Secondly,
the computation time of the classification algorithm will be reduced.
The proposed method proves its efficiency for bearing fault diagnosis
in variable speed conditions. Moreover, this approach will be tested on
more complex systems operating under non-stationary conditions, such as
wind turbine gearboxes and also this method could be used to build an
online diagnostic system.

\section*{Data availability statement}

Data that support the findings of this study are available in a public,
open access repository: Mendeley Data, V2, 
doi:\href{https://doi.org/10.17632/v43hmbwxpm.2}{10.17632/v43hmbwxpm.2}.
Huang, Huan; Baddour, Natalie (2019), ``Bearing Vibration Data under
Time-varying Rotational Speed Conditions''.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that  could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Acknowledgments}

This work was achieved at the laboratories LMPA (IOMP, S\'{e}tif -1-
University, Algeria) and LASPI (Lyon University, France). The authors
would like to thank the Algerian Ministry of Higher Education and
Scientific Research for their financial and technical support in the
framework of the program PNE 2019--2020. A thought and a great
recognition for Dr.\ Hafida Mahgoun who contributed extremely and
rigorously at the beginning of the project and who unfortunately
deceased before the completion of this work.

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