=Paper=
{{Paper
|id=Vol-3392/paper16
|storemode=property
|title=Analysis of Discrete Wavelet Spectra of Broand Signals
|pdfUrl=https://ceur-ws.org/Vol-3392/paper16.pdf
|volume=Vol-3392
|authors=Olha Oliinyk,Yuri Taranenko,Valerii Lopatin
|dblpUrl=https://dblp.org/rec/conf/cmis/OliinykTL23
}}
==Analysis of Discrete Wavelet Spectra of Broand Signals==
https://ceur-ws.org/Vol-3392/paper16.pdf
Analysis of Discrete Wavelet Spectra of Broadband Signals
Olha Oliinyka, Yuri Taranenkob and Valerii Lopatinc
a
College of Radio Electronics, st. Stepan Bandera, 18, Dnipro, 49000, Ukraine
b
Сompany «Likopak» st.Kachalova, 1, Dnipro, 49005, Ukraine
c
Poljakov Institute of Geotechnical Mechanics of the National Academy of Sciences of Ukraine
st. Simferopolska., 2a, Dnipro, 49005,Ukraine
Abstract
The article is devoted to the development of a mathematical model for autoregressive and
autocoherent analysis of discrete wavelet spectra with approbation on known experimental
data. The model is used to detect anomalies or emergency states of the object after wavelet
filtering of noise and zero crossing. The proposed method for processing noisy signals,
combining noise filtering and Mahalanobis proximity analysis, shows better machine
learning results than neural networks and other methods for detecting classification and
recognition anomalies. The wavelet spectra were processed using autoregressive and
autocoherent analyses. The possibility of using these functions as classification features for
identifying possible anomalous states of devices is shown. Before forming the wavelet
spectra, the procedure of direct with filtering and inverse transformation of the already
filtered signal is carried out. As a result, it seems possible to identify the characteristic
features of the signal - anomalies or emergency conditions. The main factor in the
classification of device status signals is the moment when changes in the signal spectrum
begin to develop, therefore, all analysis, both in the time and frequency domain, is tied either
to the number of samples by the time the signal was received, and in some cases even to the
extended date format - month number.
Keywords 1
Wavelet spectrum, autocoherence, autoregression, state anomaly, machine learning
1. Introduction
Computer simulation, processing, identification, and analysis of signals are among the most urgent
tasks facing intelligent systems today. The task of analyzing the state signals of complex objects
attracts the attention of specialists, since there is no single approach and method for identifying
signals in computerized control and monitoring systems. This problem has not been solved for
broadband signals, for which the frequency band used for signal transmission is much wider than the
minimum required for information transmission.
The most common approach to signal analysis is the search and comparison with a signal database
containing signal samples with which the obtained data are compared [1]. Comparison of signal
records with a reference base can be carried out in different ways: cross-correlation; distance
comparisons; cluster analysis; application of the likelihood ratio; hybrid expert methods [2]. These
methods are widely covered in the literature.
A well-known approach to solving this problem of analyzing and classifying signals is to find the
optimal feature space in which objects (signals) can most easily be separated using classical
classification algorithms [3, 4]. If the diagnostic analysis is carried out for a long period of operation
and is characterized by the appearance and development of increasing non-informative noise, the
known methods of signal analysis cannot be applied, since the comparison of signals will be incorrect
CMIS-2023 (The Sixth International Workshop on Computer Modeling and Intelligent Systems), 3 of May 2023, Zaporizhzhia, Ukraine
EMAIL: oleinik_o@ukr.net (O. Oliinyk); tatanen@ukr.net (Yu. Taranenko); vlop@ukr.net (V. Lopatin)
ORCID:0000-0003-2666-3825 (O. Oliinyk); ORCID: 0000-0003-2209-2244 (Yu. Taranenko);ORCID: 0000-0003-2448-0857 (V. Lopatin)
© 2023 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
[5]. The problem can be solved by analyzing the state signals of devices and identifying the
characteristic features of the signal that cause anomalous or emergency states of objects.
The implementation of this approach makes it possible to obtain correct data for the formation of a
comparative base of signals for known methods of signal identification, and to become an
independent analysis tool. Since in real conditions of monitoring and control it is not uncommon to
work with signals about which there are no a priori data [2], the use of classification of status signals
significantly expands the scope of diagnostic analysis of signals.
2. Analysis of the literature data and a formulation of the problem
The analysis shows that the most common types of signal errors are: stop of data transmission
from the measuring device, atypical data outliers, failures of measuring equipment, accumulation of
errors due to interference, data breaks, etc. [6]. In addition, the need for data processing in comparison
with the specified threshold signal values also precludes a qualitative analysis of information [6].
The use of filtering the data of the diagnostic analysis of the control object makes it possible to
eliminate noise, however, in the future, it is not possible to isolate from the data the signs of loss of
device operability. Numerous publications on this problem are devoted to the search for an algorithm
or a combination of data processing methods for selective filtering of information.
Under conditions of growth of non-informative noise, diagnostic signals of complex objects are
non-stationary complex signals, which are characterized by the presence of complex time
dependences of amplitude, frequency, and phase. Therefore, it seems logical to apply filtering using
the wavelet transform. In the general case, the wavelet transform is the decomposition of a signal into
a wavelet spectrum (signal decomposition) with subsequent processing (detailing) and reconstruction
[7-10]. The height of the signal peak and its location in the frequency spectrum are ideal
characteristics for classifiers (random forest, gradient boosting, logistic regression, etc.) used in
machine learning [11]. However, studies in [8-12] confirm that filtering using only the continuous
wavelet transform does not solve the problem of filtering high-frequency and low-frequency noise.
Signal classification using discrete wavelet transform (DWT) is as follows: DWT is used to
separate the signal into different frequency subbands [13]. If there are different frequency
characteristics in the signals, the characteristic features appear in one of the frequency subranges.
Thus, when generating features from each subrange and using a set of features as input data for a
classifier in machine learning (random forest, gradient boosting, logistic regression, etc.), the task of
identifying state signals for various types of signals is realized [14]. Therefore, classifier machine
learning using signal processing methods has wide application prospects.
The literature widely describes the application of various criteria that are used in the application of
classical classification algorithms. The work [15] considers the implementation of the block of
mathematical signal processing, which is implemented by deconstructing the signal into its frequency
subbands, from each subbands areas are generated that can be used as input data for the comparative
proximity of the series. Authors [16. 17] use the autocoherence function to determine the local in time
non-stationarity of a random process [18-20]. The above model of autocoherence is used both in the
spectral and in the correlation representation. This makes it possible to determine signal anomalies in
local frequency and time intervals.
Analysis of publications shows that a large amount of experimental data arrays makes it
impossible to use a single function for all arrays. For real-time data processing, other approaches
should be used, for example, the initial filtering of incoming data by filtering out obviously incorrect
measurements [6].Thus, research related to the development and improvement of signal analysis
methods for the subsequent application of machine learning methods is an urgent scientific task. The
purpose of this work is to develop a mathematical model for autoregressive and autocoherent analysis
of discrete wavelet spectra for the classification of noisy signals by machine learning.
3. Mathematical model of discrete wavelet spectra
This paper uses empirical data collected during experimental studies of reliability characteristics
using four bearings (B1-B4) on a loaded shaft (6000 pounds) rotating at a constant speed of 2000 rpm
[21]. The 2nd_test set containing one accelerometer was used for the analysis. The dataset is
formatted in separate files, each containing a 1-second snapshot of the vibration waveform recorded at
specific time intervals. Each file consists of 20480 points with a sampling frequency of 20 kHz [21].
The file name indicates the date of the data capture which occurred every 10 minutes. The known data
were taken in order to compare the obtained results on the processing of vibration monitoring data
performed by other authors [21].
3.1 Autocoherence for Stationarity Analysis of Discrete Spectra
If there is noise in the signal, then the series can become non-stationary and it is impossible to
extract the necessary information from it, which is usually contained in the low-frequency part of the
spectrum. Such a characteristic for the series of wavelet coefficients is autocoherence, which can be
determined from the well-known relation for the continuous wavelet spectrum:
1 +∞ 𝑡𝑡 − 𝑏𝑏 (1)
𝑊𝑊𝑋𝑋 = � 𝐼𝐼(𝑡𝑡)𝜑𝜑 � � 𝑑𝑑𝑑𝑑,
√𝑎𝑎 −∞ 𝑎𝑎
𝑡𝑡−𝑏𝑏
where 𝜑𝜑 � �– wavelet function; WX– the scale-time spectrum of the signal I(t); a – the scale factor,
𝑎𝑎
b – the signal shift along the time axis.
To determine autocoherence, one should use the relation for wavelet coherence:
|𝑠𝑠(𝑎𝑎−1 𝑊𝑊𝑋𝑋𝑋𝑋 (𝑎𝑎, 𝑏𝑏))|2 (2)
𝑅𝑅2 (𝑎𝑎, 𝑏𝑏) = ,
𝑠𝑠(𝑎𝑎 −1 |𝑊𝑊𝑋𝑋 (𝑎𝑎, 2
𝑏𝑏)| )𝑠𝑠(𝑎𝑎 −1 |𝑊𝑊𝑌𝑌 (𝑎𝑎, 2
𝑏𝑏)| )
where s – smoothing operator [10]. Coherence is interpreted as the square of the correlation
coefficient, and its values range from 0 to 1.
Autocoherence is determined by a simple substitution in (2) of relations (1) and (3):
1 +∞ 𝑑𝑑𝑑𝑑(𝑡𝑡) 𝑡𝑡 − 𝑏𝑏 (3)
𝑊𝑊𝑌𝑌 = � 𝜑𝜑 � � 𝑑𝑑𝑑𝑑,
√𝑎𝑎 −∞ 𝑑𝑑𝑑𝑑 𝑎𝑎
It should be noted that the relation for autocoherence is used to estimate the stationarity of a
number of discrete wavelet coefficients.
3.2 Proximity of time series for the analysis of autoregression and
autocoherence
Let us designate the next two time series of the wavelet coefficients of the spectra as:
𝑁𝑁
𝑋𝑋𝑖𝑖 = �𝑎𝑎�𝑗𝑗,𝑘𝑘 �, �𝑑𝑑̃2,𝑘𝑘 �, … , �𝑑𝑑̃𝑗𝑗,𝑘𝑘 � … �𝑑𝑑� 𝚥𝚥,𝑘𝑘 �, 𝑘𝑘 = 1,2, … 𝑗𝑗 , 𝑗𝑗 = 1 … 𝐽𝐽,
2
𝑁𝑁 (4)
𝑌𝑌𝑖𝑖 = �𝑎𝑎�𝑗𝑗,𝑘𝑘 �, �𝑑𝑑̃2,𝑘𝑘 �, … , �𝑑𝑑̃𝑗𝑗,𝑘𝑘 � … �𝑑𝑑̃𝑗𝑗,𝑘𝑘 �, 𝑘𝑘 = 1,2, … 𝑗𝑗 , 𝑗𝑗 = 1 … 𝐽𝐽,
2
where Xi – series of wavelet coefficients at time ti; Yi – series of wavelet coefficients at time
ti+dt; dt – determined by the time for which a change in the spectrum will not lead to
equipment failure.
The correlation closeness of series (4) is determined from the relationship:
∑𝑁𝑁
𝑖𝑖=1 𝑋𝑋𝑖𝑖 𝑌𝑌𝑖𝑖 (5)
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 1 − .
𝑁𝑁 2 ∑ 𝑁𝑁
�∑𝑖𝑖=1 𝑋𝑋𝑖𝑖 𝑖𝑖=1 𝑌𝑌𝑖𝑖2
When the spectra series are close and completely identical proximity=0 when there is no
difference -proximity=1. On practice 0 ≤ proximity ≤ 1 .
3.3. Decomposition of the Time Series
The wavelet coefficients in the decomposition of the measuring signal f(t) are generally
determined from the following system of equations [22]:
(6)
aj,k = � f(t)φj,k (t)dt⎫
⎪
R
,
⎬
dj,k = � f(t)ψj,k (t)dt⎪
R ⎭
where R – the interval for determining the function f(t) of the signal; aj,k, dj,k – the coefficients of
approximation and detailing of the discrete wavelet decomposition of the signal, respectively; φj,k , ψj,k
– father and mother wavelets, respectively; j, k – he current level of the wavelet decomposition and
the ordinal number of the wavelet coefficient in the wavelet decomposition of the signal, respectively.
In calculations, instead of integrating (6), Mull's pyramidal algorithm is used to eliminate
methodological errors associated with quadrature formulas [23]. The approximating coefficients are
set at the conditionally zero level jo, j0,k, k=1,…N, where N=2m, m>1. Accuracy was assessed using
the predictive regression metrics method and an uncertainty matrix.
Next, approximating aj0+1,k and detailing dj0+1,k coefficients are calculated, and from aj0+1,k
вычисляют aj0+1,2 k , dj0+1,2 k and so on. In the presence of noise with zero mathematical expectation
and standard deviation, the set of coefficients enclosed in arrays array ({}) for the maximum
decomposition level J will take the form of the formula:
N (7)
�a� j,k �, �d� 2,k �, … , �d� j,k � … �d� j,k �, k = 1,2, … j , j = 1 … J,
2
The set of nested arrays (7) is transformed into a 1D list using the special flatten function [24]:
N (8)
�a� j,k , d� 2,k , … d� j,k … d� j,k �, k = 1,2, … j , j = 1 … J,
2
3.4 Noise Reduction by Limiting Detail Factors
The filtered signal is determined from the relation [25]:
j (9)
̂f(x) = � a� j+j φj+j (t) + � ��F(λi )d� j+j ψj+j (t)� ,
,0,k 0,k 0,k 0,k
j=1 k
where F(λj) – threshold function from the list garotte, garrote, greater, hard, less, soft, for example,
hard(λj), λj – threshold at the j decomposition level; φj+j0,k (t), ψj+j0,k (t) – the “father” and “mother”
wavelets, respectively [25]. The universal threshold λj is determined from the relation [25]:
median��d1,k �� ⎫ (10)
σ= ,⎪
0,6742
⎬
λuniv
j = σ�2lnNj , ⎪
⎭
where λjuniv – universal threshold; Nj – he number of detail coefficients d1,k at the j decomposition
level; median(│d1,k│)– median from the array of detail coefficients.
A common threshold can be used as the average of λjuniv or both series (4):
1
N (11)
λX = λY = � λi,
n
i=1
The universal threshold λuniv j = σ�2lnNj of limiting the wavelet coefficients of the spectrum
depends on the dispersion and the number of wavelet coefficients of detail, therefore, when analyzing
discrete spectra, it is an essential characteristic of high-frequency noise.
Depending on the nature of the signal, additional mathematical processing can be applied to
discrete wavelet coefficients [25].
4. Regression Analysis of Real Data Based
Applying the proposed model of the method of regression analysis of discrete spectra, it is possible
to clearly identify the date 2004-02-19 06:12:39 of a possible defect for the B3 device (Figure 1).
Identification occurs both by the number of accumulated maxima - 10 exceeding the threshold of 0.5
correlation proximity, and by the maximum value.
Figure 1: Autoregression of discrete wavelet spectra obtained from relations (4), (5) with filtering by
a common threshold (11)
5. Coherent analysis of real data based on the developed mathematical
model
The analysis shows that the proximity of the autocoherence of the spectra is the highest - unity for
the B3 device and a minimum of autocoherence. Thus, the loss of stationarity is achieved on 2004-02-
19 06:12:39 (Figure 2). Not despite the fact that the number of peaks -91 is less than that of the device
B2.
Figure 2: Autocoherence of discrete wavelet spectra obtained by relations (1), (2), (3), (4), (5) with
common threshold filtering (11)
6. Using the Mahalanobis algorithm to analyze discrete wavelet spectra
To detect signal anomalies, we use zero-crossing analysis, which has practically not been used in
relation to discrete wavelet spectra. This approach is designed to detect chirp and non-linear chirp
crossover signals. These include the average value of the derivative, the zero crossing frequency, and
the average transition rate of a given signal amplitude [25]. Combined with noise filtering and
Mahalanobis proximity analysis, zero-crossing analysis shows better machine learning results than
neural networks and other classification and recognition anomaly detection methods.
Features of the Mahalanobis algorithm consist in the use of a covariance matrix according to the
relation [26]:
(12)
DM (x�) = �(x� − μ�x )T S −1 �y� − μ�y �,
where x�, y� – multidimensional series of wavelet coefficients; �μ��𝑋𝑋�, ���
μ𝑌𝑌 – mathematical expectations; S–
he covariance matrix.
The covariance matrix S and its inverse matrix are symmetric and positive definite. Therefore, the
transformation of the spectral series of wavelet coefficients from multidimensional to one-
dimensional is not required. At the same time, wavelet decomposition into levels during filtering and
zero-crossing dramatically increase the number of identification features. The increase in the number
of features only due to zero-crossing and filtering can be seen by comparing the graphs of the training
sets (Figure 3).
Figure 3: Graph of the training set signals in the trouble-free period of operation without the use of
wavelet noise filtering and zero-crossing
From the graph (Figure 4) it follows that the method of regression analysis of discrete spectra
clearly identifies the date 2004-02-19 06:12:39 of a possible defect for device B3 both by the number
of accumulated maxima– 10 exceeding the threshold of 0.5 correlation closeness, and by the
maximum value.
Figure 4: Graph of the training set signals in the trouble-free period of operation using wavelet
noise filtering and zero-crossing
The graphs of test set signals also have a significant difference (Figure 5, Figure 6).
Figure 5: Graph of test set signals during detection of anomalies or emergency conditions without
wavelet noise filtering and zero crossing
Figure 6: Graph of test set signals in the period of detection of anomalies or emergency conditions
after wavelet filtering of noise and zero crossing
When determining anomalies using the described approach, a clear identification of signs of
anomalies and emergency conditions of the object is observed, as evidenced by a strong oscillation of
the Mahalanobis distance in the threshold zone (Figure 7, Figure 8).
Figure 7: Changing the Mahalanobis distance in the threshold zone without signal processing
Figure 8: Changing the Mahalanobis distance in the threshold zone using Filtering Wavelet and Zero
Crossing
7. The Results of the Classifier Using the Machine Learning Method
To test the proposed method for classifying state signals using the obtained features, the classifiers
described by the following algorithms were studied: gradient method, linear (SVM), logistic
regression, nearest neighbors, decision tree (Table 1) on the resulting computer model.
The previously obtained wavelet data analysis model was used for machine learning with the
gradient method signal classifier (as the most commonly used). Time series of wavelet coefficients
db38 of cosine proximity (autocoherent proximity) were taken as features. The results of machine
learning of the model confirm the effectiveness of the proposed algorithm (Table 1).
To confirm the effectiveness of the developed method for classifying noisy signals using the
machine learning method, we will process the same signal without using the established signs of state
classification (Figure 9, a) and using the developed classifier (Figure 9, b).
Table 1
Efficiency of application of algorithms for machine learning of classifiers
№ Algorithm Accuracy on the training Accuracy on the test Studying time
name sample at the level of sample at the level of with varying degrees of
detail detail detail
0 0,3 0,4 0 0,3 0,4 0 0,3 0,4
1 logistic
regression 0,949 0,934 0,916 0,912 0,907 0,920 0,148 0,016 0,023
2 linear
(SVM) 1,000 1,000 0,996 0,910 0,904 0,926 0,083 0,057 0,041
3 nearest
neighbors 0,931 0,938 0,916 0,899 0,900 0,916 0,045 0,003 0,006
4 gradient
method 1.000 1,000 1,000 0,891 0,959 0,905 1,907 1,091 1,964
5 decision
tree 1,000 1,000 1,000 0,838 0,950 0,855 0,039 0,024 0,019
a b
Figure 9: Comparative analysis of the results of signal processing without (a) and with (b) the use of
the developed method for classifying state signals
The use of noise filtering with its own time-dependent universal threshold makes it possible to
increase the accuracy of predicting the occurrence of object defects.
8. Conclusions
1. A new method for classifying noisy signals using machine learning has been developed.
Proposed method for processing noisy signals that combines noise filtering and Mahalanobis
proximity analysis. The essence of the method lies in the following: the data obtained after
mathematical processing for all frequency subranges of the signal and a given moment of time
constitute time series, filtering each series from noise with its own, time-dependent, universal
threshold. The time series filtered in this way are used as input signals to the machine learning
classifier and characterize the change in the state of the control object over time. The use of this
classifier implements pre-processing of the signal in order to eliminate noise and allows you to
highlight the signs of inoperable states of devices in dynamics.
2. The list of used functions for processing wavelet spectra has been extended with two more ones:
by cosine proximity of the autocoherence spectra of discrete wavelet spectra and by the cosine
proximity function of wavelet spectra of signals following one after another in time. The possibility of
using these functions as classification features for identifying possible anomalous states of devices
has been confirmed.
3. A new criterion for classifying the loss of uptime is based on the presence of local minima
depending on the sum of squares of the wavelet detailing coefficients at the level of the wavelet
decomposition.
4. The developed classification method was tested using a computational experiment on a known
data set obtained using an accelerometer, all stages of the classification of an object's inoperable state
signal are shown. The obtained results of machine learning using the signal classifier gradient method
confirm the effectiveness of the application of new signs of signal classification in wavelet analysis
(the accuracy on the test set was 0.96).
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