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VIBRATION-BASED CONDITION MONITORING OF COMPRESSORS BY PRINCIPAL COMPONENT ANALYSIS AND DISCRIMINANT ANALYSIS
Primož Potočnik
University of Ljubljana, Faculty of Mechanical Engineering, 1000 Ljubljana, Slovenia
email: [email protected]
Edvard Govekar
University of Ljubljana, Faculty of Mechanical Engineering, 1000 Ljubljana, Slovenia
Vibration-based condition monitoring and fault detection approach for compressors built in re-
frigeration appliances is proposed. The method combines feature extraction and principal com-
ponent analysis (PCA), and compares unsupervised k-means clustering and discriminant analy-
sis (DA). The method is demonstrated on a case study based on a large dataset of 10.000 indus-
trially acquired vibration measurements of compressors during the production of refrigeration
appliances. The initial step of the proposed method is feature extraction, based on statistics, sta-
tistical moments, and spectral analysis. A selected single feature was applied for statistical de-
tection of initial compressor faults, based on which the initial compressor classes were defined
as ‘normal’, ‘noisy’, and ‘inactive’. In the next step, extracted features were transformed by
PCA and only the first two principal components, contributing over 90% of variability, were re-
tained for subsequent analysis. Three initial classes were applied to initialize DA. The results of
linear DA revealed many additional ‘noisy’ and ‘inactive’ samples that were not evident from a
single extracted feature. Furthermore, an additional cluster defining new class ‘unstable’ was
detected, indicating a new type of defect characterized by high vibration transients. Results of
DA reveal decision boundaries between all classes, and confirm the efficiency of the proposed
method. Finally, the results are compared also with an unsupervised k-means clustering which
shows that unsupervised clustering doesn’t provide appropriate decision boundaries. The pro-
posed DA-based approach detects compressors with defects and has the potential to detect novel
classes of unusual or faulty operation. The method can be effectively applied for industrial con-
dition monitoring of compressors.
1. Introduction
Condition monitoring (CM) of machines and products is an established and important part of
successful modern industrial production. In order to manufacture fault-less products, various non-
destructive condition monitoring approaches can be applied during the production process. Vibra-
tion signal analysis [1, 2] continues to be one of the most useful and popular CM methods, beside
other acoustic and acoustic emission based approaches [3]. Methods for the analysis of vibration
signals include statistical methods, wavelets [4], psychoacoustic approaches [5], neural and fuzzy
logic based methods [6], and many modern machine learning methods, such as support vector ma-
chines [7, 8].
In this paper, vibration-based condition monitoring and fault detection approach for compressors
built in refrigeration appliances is discussed. Whereas particular CM solutions for different produc-
tion stages of reciprocating compressors have already been proposed [9-14], in this paper solutions
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for final vibration-based inspection of compressors [15-17] are discussed. The existing approaches
provided in the literature are based on supervised learning, where appropriate sets of samples repre-
senting normal and faulty operation are available. In this paper, semi-supervised approach is con-
sidered, which is based only on a few samples of faulty compressors. The method is therefore high-
ly relevant for industrial application where expert-based sets of labelled sets are difficult and expen-
sive to obtain.
The proposed method is based on feature extraction, principal component analysis (PCA), and
discriminant analysis (DA). The method is demonstrated on a case study comprising 10,000 vibra-
tion measurements of compressors acquired during the industrial production of refrigeration appli-
ances. The method is initiated with feature extraction and then statistical detection of initial com-
pressor faults based on selected single feature. Initial results are translated into PCA space which
reveals clusters of compressors with different operating conditions. Finally, discriminant analysis is
applied to determine decision boundaries between various categories of compressors. Results are
compared also with an unsupervised k-means clustering which confirms superior performance of
the proposed DA-based approach.
2. Measurements
An industrial vibration-based condition monitoring system (Figure 1) was developed for compa-
ny Gorenje in order to provide measurements of compressors built into refrigeration appliances.
During the operation of an industrial production line, 10,000 vibration measurements of compres-
sors were acquired. Special custom-made flexible sensor head was developed for automated ad-
justment of accelerometer to various compressor types. Inside the sensor head, the accelerometer
PCB 352B was installed for high sensitivity vibration measurements. Each measurement was ac-
quired in the duration of 2 seconds with a sampling frequency 25.6 kHz. Figure 2 shows custom-
made flexible sensor head and its position on the compressor during vibration measurement. All
measurements were acquired without any labelled information about the quality of the measured
compressor. Therefore, this information has to be estimated based on the analysis of the measure-
ments.
Figure 1: Industrial data acquisition and condition monitoring system.
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Figure 2: Custom-made flexible sensor head and its position on the compressor during measurement.
3. Methods
The following methods were applied in this study: principal component analysis, discriminant
analysis, and k-means clustering.
3.1 Principal component analysis
The central idea of principal component analysis is to reduce the dimensionality of a data set
consisting of a large number of interrelated variables, while retaining as much as possible of the
variation present in the data set [18]. This is achieved by transforming initial variables to a new set
of variables, the principal components (PCs), which are uncorrelated, and which are ordered so that
the first few retain most of the variation in all of the original variables. In our case study, the set of
extracted features were first rescaled to zero mean and variance one, and then PCA analysis was
applied to extract principal components.
3.2 Discriminant analysis
The discriminant analysis (DA) is a classification method used to find discrimination boundaries
which separate (discriminate) two or more classes of objects [19, 20]. DA assumes that different
classes generate data based on different Gaussian distributions. In our research, linear discriminant
analysis (LDA) and quadratic discriminant analysis (QDA) [21] were applied. Both LDA and QDA
have been shown to rank among the top classifiers. The reason for such a good track record is most
probably the bias−variance trade-off where the data can only support simple decision boundaries
such as linear or quadratic, and the estimates provided via the Gaussian models are stable [22].
3.3 K-means clustering
The k-means algorithm is one of the most popular and commonly used clustering algorithms
employing a squared error criterion [23, 24]. The algorithm starts with a random initial partition and
keeps reassigning the patterns to k clusters based on the similarity between the pattern and the clus-
ter centres until a convergence criterion is met. The k-means algorithm is popular because it is easy
to implement. A major problem with this algorithm is that it is sensitive to the selection of the initial
partition and may converge to a local minimum of the criterion function value if the initial partition
is not properly chosen. The problem can be practically solved by several restarts of the algorithm
each time with new random seed and then taking the best solution as a final clustering result.
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4. Solution approach
The proposed solution approach is designed for industrial vibration-based condition monitoring
systems where no prior information regarding the quality of inspected products is available. There-
fore a supervised approach based on samples of normal and faulty products can not be applied, and
instead unsupervised or semi-supervised methods should be considered. Such situation is typical for
industrial condition monitoring where expert based acquisition of samples with normal and faulty
operation is difficult, expensive and time consuming. Consequently, we propose a semi-supervised
condition monitoring approach that consists of initial statistical isolation of representative faults,
followed by application of classification methods for more detailed recognition of faulty products.
The proposed solution approach, applied in our case study to the industrial condition monitoring of
compressors, consists of the following steps:
1. Feature extraction from vibration signals based on statistics, statistical moments, and spectral
analysis.
2. Application of a selected single feature z1 for statistical detection of initial compressor faults.
Based on statistical analysis and inspection of outliers, the initial compressor classes are defined
as ‘normal’, ‘noisy’, and ‘inactive’.
3. Transformation of extracted features by principal components analysis in order to reduce di-
mensionality. PCA results show that the first two principal components contribute over 90% of
variability and therefore only these two components are retained for subsequent analysis. The
visualization of measurements in 2-dimensional PCA space reveals clusters of compressors with
similar properties.
4. Application of discriminant analysis: initial statistically determined compressor faults are used
to initialize DA, and resulting decision boundaries determine regions of normal or faulty opera-
tion.
5. An additional cluster, defining a new class ‘unstable’ is visible in the projection in the PCA
space, indicating a new type of defect characterized by high vibration transients. DA is applied
with this new class included, and the final result of DA shows decision boundaries between all
four classes, namely ‘normal’, ‘noisy’, ‘inactive’, and ‘unstable’.
6. Finally, the results of discriminant analysis are compared also with an unsupervised k-means
clustering which shows that unsupervised clustering doesn’t provide appropriate decision
boundaries.
5. Results
The results of the proposed solution approach to the industrial vibration-based condition moni-
toring of compressors are presented in this section.
5.1 Feature extraction
Feature extraction step provides initial data compression from time domain vibration signals into
the set of features characterizing each measurement. In our study the following features were ex-
tracted from vibration signals:
a) 10 statistical features:
log M1 (logarithmic mean absolute value of the vibration signal)
log M2, log M4, log M6 (logarithmic moments),
kurtosis,
run var M2 (running variance of M2, calculated across 0.1 second segments),
perc 75, perc 90, perc 95, perc 98 (percentiles of absolute value of the signal).
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b) 17 features based on spectral analysis (components of power spectrum):
P1 (0 Hz), P2 (800 Hz), P3 (1600 Hz), …, P17 (12800 Hz).
The set of 27 extracted features (z1… z27) for each compressor provided initial information for
subsequent analysis.
5.2 Statistical detection of initial faults
Figure 3 shows selected single feature z1 (log M1) for n = 10,000 compressors with marked ini-
tial outliers representing ‘noisy’ and ‘inactive’ faults, and a set of median samples representing
‘normal’ compressors. These samples represent initial classes of compressors of different qualities.
Figure 3: Selected single feature z1 with marked initial compressor classes.
5.3 PCA transform
Extracted features were compressed into the principal components, and the first two components
contribute over 90% of variability, therefore only these two components were used for subsequent
analysis. Figure 4 shows measurements in resulting 2-dimensional PCA space which reveals clus-
ters of compressors with similar properties.
5.4 Application of discriminant analysis
Based on initially defined compressor classes, LDA is applied to determine decision boundaries
between ‘normal’ class and each faulty class. Figure 5 presents initial samples, defining classes
‘normal’, ‘noisy’, and ‘inactive’, and resulting LDA boundaries that reveal many additional ‘noisy’
and ‘inactive’ samples that are not evident from a single extracted feature.
Figure 4: Samples in the first two PCA axes with
marked initial compressor classes.
Figure 5: Linear discriminant analysis with class
borders for three classes of compressors.
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5.5 Detecting additional ‘unstable’ class
Figure 6 presents initial sample of a new class ‘unstable’, and the resulting decision boundaries
defined by LDA. This final result represents decision strategy capable of classifying the quality of a
compressor according to one of the four classes: ‘normal’, ‘noisy’, ‘inactive’, and ‘unstable’. It is
also possible to apply QDA to isolate just two classes of compressors, namely ‘normal’ and ‘faulty’
that includes all irregular compressor operations, as presented in Figure 7.
Figure 6: Linear discriminant analysis with class
borders for four classes of compressors (inclu-
ding the additional ‘unstable’ class).
Figure 7: Quadratic discriminant analysis with class
border between ‘normal’ and ‘faulty’ compressors.
5.6 Comparison of results with k-means clustering
For comparison with DA, k-means clustering results are shown in Figure 8. Results represent un-
supervised approach which does not provide appropriate decision boundaries, as demonstrated by
semi-supervised approach based on DA described above.
Figure 8: Classification results obtained by k-means clustering.
6. Conclusions
The paper presents a semi-supervised approach to industrial condition monitoring based on anal-
ysis of vibration signals. The proposed method is appropriate for industrial situations where very
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limited or no prior information about product classes is available, and therefore statistical methods
have to be applied first in order to acquire initial faulty conditions. Based on initial defined classes
of normal and abnormal operation, the methods proposed in this paper can be applied to determine
condition of products by defining decision boundaries using principal component analysis and dis-
criminant analysis. The method has been succesfully applied to an industrial case study, namely the
condition monitoring of compressors built in refrigeration appliances. The method detects compres-
sors with defects and has the potential to detect novel classes of unusual or faulty operation.
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