# A Vibrational Signal Fault Diagnosis Rule Extraction Method Based on DST-ACI Discriminant Criterion

Authors: Yalun Zhang; Lin He; Guo Cheng
Journal: Shock and Vibration
 (2021)

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## Abstract

A fault diagnosis rule extraction method oriented to machine foot signal based on dynamic support threshold and association coefficient interestingness (DST-ACI) discriminant criterion is proposed in this paper. The new method includes three main innovations. First, the feature state coding method based onK-means clustering fully takes into account the imbalanced distribution of signal feature values due to the noise interference, and divide the signal feature values into several range intervals to generate the feature state code. Second, the frequent feature pattern mining method based on dynamic support threshold (DST) discriminant criterion can dynamically adjust support threshold according to the frequency of the feature states in each candidate pattern. Third, the fault diagnosis rule extraction method based on the association coefficient interestingness (ACI) discriminant criterion introduces a new metrics called ACI to evaluate the correlation between the pattern and the fault. Four types of fault simulation experiments were carried out, and the performance of the DST-ACI method was tested using the collected vibration signal. The results show that compared with the coding method based on equal-width discretization or equal-density discretization, the accuracy of the transactional dataset generated by the feature state coding method based on K-means clustering is higher. Compared with the frequent feature pattern mining method based on the constant support threshold criterion, the pattern mined by the DST-based criterion has generally higher support. Compared with the existing confidence-lift-based and confidence-improint-based fault diagnosis rule extraction frameworks, the positive correlation between the feature states and the fault type of the rules extracted based on the DST-ACI framework is generally stronger.


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## Body

## 1. Introduction

The health condition of the mechanical equipment can be evaluated by analyzing the vibration signals. Since the signal is a vector composed of a large number of vibration amplitudes, it is difficult to directly observe the hidden information in it. Therefore, researchers usually preprocess the signal first and extract some signal features that can be used for fault diagnosis. Then, based on these quantifiable features, the signal type and the equipment health will be discriminated further. These features come from researchers’ previous research experience, and can describe the volatility, periodicity, energy distribution, or waveform characteristics of the signal from different angles. After the feature extraction, the relationship between the signal feature value and the fault type can be used to construct a fault diagnosis rule of the form “when a signal feature value is within a certain range interval, it means that the equipment has a certain fault.”Association rule mining and frequent pattern extraction are the important technical methods to find the relationship between different events. The frequent pattern mining technique was first proposed by Agrawal et al. [1]. This technique first finds frequent itemset whose support is greater than the support threshold. Then, several inferences are generated according to the frequent itemset, and the inferences whose confidence is greater than the confidence threshold will be saved as association rules. Subsequently, Agrawal and Srikant, successively, proposed the famous apriori algorithm, apriori-Tid algorithm, and apriori-Hybird algorithm [2]. Since then, the researchers have developed a variety of improved Apriori-based algorithms. Park et al. proposed the DHP algorithm based on the Hash technique [3]. Savasere et al. proposed the partition algorithm to improve the Apriori algorithm based on the idea of partitioning [4, 5]. Bayardo and Agrawal introduced random sampling technology to the apriori algorithm [6]. By mining the frequent co-occurrence relationships between entities, researchers can conduct more in-depth research on the implicit connections between seemingly unrelated entities [7–9]. Then, the obtained knowledge can be put into industrial or commercial applications [10–12]. Therefore, the two technologies have been widely used in many fields such as disease diagnosis, traffic management, weather forecasting, software testing, and commercial marketing [13–19].With reference to the traditional association rule mining method based on the support-confidence-lift (SCL) discriminant framework, a fault diagnosis rule extraction method based on the same framework can be given.The first step is data preprocessing. Data preprocessing consists of feature extraction, feature state coding, and transactional dataset generation. Several nonrepeated feature state code can form afeature pattern, which represents a signal whose feature values all fall within the corresponding feature state coding range interval. Only when the support of a feature pattern is greater than the support threshold, this pattern can be judged as a frequent feature pattern. On this basis, only when the confidence and lift of the inference containing a frequent feature pattern are greater than the set threshold, the inference can be judged as a correct fault diagnosis rule. In the SCL-based framework, the support evaluates the adequacy of the sample that can support the conclusion, the confidence evaluates the reliability of the inference, and lift evaluates the correlation between the pattern and the fault.In order to collect the vibration signal of the equipment, the vibration acceleration sensor will be installed on the equipment. In general, the key locations close to the mechanical equipment core component or the vulnerable points that are prone to damage and aging will be selected as the sensor measurement points. However, in some specific environments, due to the design constraints of industrial land or equipment specifications, the space where the equipment is located is small, and the equipment distribution is dense. As a result, there is no space for arranging the signal transmission cable and sensor protective cover on the case or the position near the fault source. At this time, engineers can only install the sensors on the base or machine foot, so that the sensors do not take up space and is easy to load or unload.At present, in the research literature on fault diagnosis based on vibration signals, few scholars specifically study the installation location of the sensor. In the related literature, the sensor installation positions are all set at the key positions mentioned above [20–30]. For example, if engineers want to monitor bearings, sensors are often installed in the radial and vertical directions of the bearing seat. If engineers want to monitor the alignment of the shaft system, the sensor is often installed near the coupling. Almost no researchers set the measuring points on the machine feet specially. Furthermore, for many commonly used standard dataset in the field of fault diagnosis, the sensors used to collect the data are installed in the conventional measuring point positions, instead of the positions of the machine feet [31–36]. In addition, in some literature that considers severe background noise interference, only application scenarios with severe noise are introduced. However, the specific sensor placement location is not involved [37–43]. Therefore, there is almost no research on the fault diagnosis of the machine foot vibration signal.This special engineering application scenario brings the following three inconveniences.(1)
In different measurements, the value of the same feature calculated from the signal under the same working condition is not the same. Due to the noise interference, these values are distributed in a range interval, but the width of the interval is random. When the sensor is arranged at the machine foot, the measuring position is far away from the fault source. As a result, the noise interference is more serious. In the data preprocessing stage, it is necessary to map the continuous signal feature values to the unique code representing the different range intervals. If the range interval is simply divided according to the traditional method such as equal-width discretization or equal-density discretization, it will be possible to confuse the signal feature values of different fault types into one category and map them to the same feature state code. As a result, a large number of transactions in which the fault type does not match the feature state code will be generated, and the credibility of the extracted rules will be reduced.(2)
Due to the strong noise interference of the machine foot vibration signal, the noise feature and the fault signal feature that can reflect the fault characteristics are mixed with each other. Under these circumstances, it is very possible to extract the redundant pattern of a nonreference value. The redundant pattern refers to the pattern that contains a large number of noise feature states. The noise features are widely distributed in the features of all types of signals. Therefore, if a pattern contains a large number of noise feature states, the support of the pattern will be greater, and the traditional algorithm will misjudge the pattern with a higher occurrence frequency as the frequent feature pattern, but in fact the value of this pattern is not high.(3)
In the traditional frameworks, the lift is introduced to further evaluate the correlation between the feature pattern and the fault. Lift is a metrics to describe the promotion degree of a pattern to the occurrence probability of a fault. By introducing the lift, the so-called redundant rules (the contradictory “mutual exclusion rules”) can be prevented from being extracted. In order to further eliminate the influence of accidental factors, especially when the background noise interference of the machine foot measurement position is large, the user often hopes to extract the rules with the higher lift to further improve the correlation between the feature pattern and the fault. In this case, it is necessary to artificially set a lift threshold. However, the range of the traditional lift function has no upper bound. Especially in the case of a large number of transactions, the lift value of the rule is often very large. At this time, it is difficult for people to set an appropriate lift threshold for further rule screening.For the above reasons, this paper proposesa fault diagnosis rule extraction method based on the Dynamic Support Threshold and Association Coefficient Interestingness (DST-ACI) discriminant criterion for the machine foot measurement point signal. The main innovations of the proposed method are as follows:(1)
In order to divide the range of different features into different intervals and map densely clustered signal feature values that may represent the same fault to the same range interval as much as possible,a feature state coding method based on K-means clustering is proposed. This method uses the K-means clustering method to classify the densely clustered values of the same feature into one cluster, and each cluster represents a subrange of the feature value range. The feature values classified into the same cluster will be assigned a feature state code representing the cluster. The transactional dataset will be generated according to the conclusion of the feature state coding.This method takes into account the imbalance of signal feature value distribution. The relatively clustered feature values will map to the same range interval as much as possible, so that the converted feature state codes will be also as the same as possible. In this way, the matching accuracy between the feature state and the fault type in the generated transaction can be further improved, thereby improving the credibility of the extracted fault diagnosis rules.(2)
In order to reduce the number of the extracted redundant patterns,a frequent feature pattern mining method based on the dynamic threshold support (DST) discriminant criterion is proposed. In the process of frequent feature pattern mining, the DST of different candidate feature patterns can be dynamically adjusted based on the occurrence frequency of the feature state in the pattern.Since the noise features are widely distributed in all types of signals, the DST of the pattern that contains a large number of noise feature states will be large. Therefore, the redundant pattern that contains a large number of noise feature states will be filtered out to a certain extent. On the other side, if a pattern contains a large number of fault feature states, since the occurrence frequency of these feature states is relatively smaller, the DST will become smaller. By adopting this discriminant criterion, the feature pattern with relatively few occurrences but with strong cooccurrence of feature states can be extracted as much as possible.(3)
In order to meet the users’ needs to further improve the credibility of the extracted rules,a fault diagnosis rules extraction method based on the association coefficient interestingness (ACI) discriminant criterion is proposed. This method introduces a new metrics called the association coefficient interestingness, abbreviated as ACI. The user can intuitively distinguish the correlation between the feature pattern and fault through ACI. In addition, users can set the interestingness threshold between 0 and 1 to filter out the fault diagnosis rules with a higher lift, thereby further improving the credibility of the extracted rules.Furthermore, the concept ofminimum lift function is introduced to measure the reliability of different fault diagnosis rule discriminant criterions. It is mathematically proved that the minimum lift function of the rules extracted based on the ACI criterion are generally higher than the other two existing methods under the same threshold conditions. Moreover, the user can increase the minimum lift function value to any desired level by adjusting the association coefficient in ACI. Therefore, the ACI-based discriminant criterion can ensure that its minimum lift function is always larger than any other criterion by adjusting the association coefficient.


## 2. Fault Diagnosis Rules Extraction Method Based on DST-ACI Discriminant Criterion

This chapter includes four sections. The first three sections, respectively, introduce the three main innovations of the article, namely,Feature State Coding Method Based on K-means Clustering, Frequent Feature Pattern Mining Method Based on the Dynamic Support Threshold Discriminant Criterion, and Fault Diagnosis Rules Extraction Method Based on Association Coefficient Interestingness Discriminant Criterion. The fourth section mainly introduces the algorithm flow of extracting fault diagnosis rules. The logical and technical relationship between the three innovations is shown in Figure 1.Figure 1
The logical and technical relationship between the main innovations.


### 2.1. Feature State Coding Method Based onK-Means Clustering

This chapter introduces the whole process of using cluster-based feature state coding methods to complete data preprocessing in detail. The first part introduces the commonly used signal features and the calculation method. The second part introduces the feature state coding method based onK-means clustering. In this part, the K-means clustering algorithm will be used to divide the range of each feature into different intervals, and the feature values falling in the same range interval will be mapped to the feature state code representing the range interval. The third part introduces the process of generating total fault transactional datasets based on the clustering results. The three parts of this chapter and the logical relationship between them are shown in Figure 2.Figure 2
Process of using cluster-based feature state coding methods to complete data preprocessing.


#### 2.1.1. Signal Feature Extraction

The first step of feature state coding is to extract the feature parameter value of each sample signal. Signal features can reflect different characteristics of the signal [44–49]. The commonly used signal features are shown in Table 1.Table 1
Signal features.
Calculation formulaSignal featuresP1=maxxiMaximumP2=minxiMinimumP3=1/N∑i=1NxiMeanP4=P1−P2Peak to peakP5=1/N∑i=1NxiAbsolute averageNVarianceP7=P6Standard deviationP8=1/N−1P64∑i=1Nxi−x¯4KurtosisP9=1/N−1P63∑i=1Nxi−x¯3SkewnessP10=1/N∑i=1Nxi2Root mean squareP11=P10/P5Shape factorP12=P1/P10Crest factorP13=P1/P5Impulse factorP14=1/N∑i=1Nxi2Square root amplitudeP15=P1/P14Coefficient of variationP16=1/K∑k=1KykSpectral meanP17=1/K∑k=1Kyk−P162Spectral varianceP18=1/KP173∑k=1Kyk−P162Spectral skewnessP19=P18=1/KP172∑k=1Kyk−P164Spectral kurtosisP20=∑k=1Kfk⋅yk/∑k=1KykFrequency centerP21=∑k=1Kfk−P202⋅yk/∑k=1KykFrequency standard deviationP22=∑k=1Kfk2⋅yk/∑k=1KykFrequency RMSP23=∑k=1Kfk4⋅yk/∑k=1Kfk2⋅ykMain frequency band positionN is the number of sampling points, xi is the amplitude of the i-th sampling point, k is the number of frequency domain sequence points, fk is the k-th frequency value, and yk is the spectral value corresponding to fk.Suppose there areF different fault signal datasets, each dataset Df consists of Nf signal samples snf with fault type f, and each sample signal contains N sampling points, n=1,2,…,Nf, f=1,2,…,F. The values of the above-mentioned M signal feature can be extracted from each sample signal snf. The set of all the feature values of snf is defined as the signal feature value vector of the signal snf in the form of Pnf=Pnf1,Pnf2,…Pnfm,…,PnfM, m=1,2,…,M. Each type of fault dataset Df can generate Nf different signal feature value vectors.


#### 2.1.2. Feature State Coding Based onK-Means Clustering

Let the value of the featurePm of the signal snf be Pnfm. The feature value Pnfm in all the signal feature value vector should be coded into a single identifier Qnfm according to the range interval, which it belongs. The code Qnfm represents the range interval code in which the feature value Pnfm is located. In other words, feature state coding is to find a range interval division method to construct a mapping F⋅ from feature value Pnfm to feature the range interval code Qnfm, which is(1)F⋅:Pnfm⟶Qnfm.Suppose the range of a feature parameterPnfm is Ψm, and it needs to be divided into Lm range intervals Ψlm, then(2)Ψm=∪l=1LmΨlm.In traditional methods, equal-width discretization or equal-density discretization is often used to complete coding. The equal-width discretization will divide the feature value range into several equal-length intervals. The equal-density discretization maintain the number of the feature value in each interval the same. Then, the feature values falling in the same range interval will be uniformly mapped to the unique code representing the range interval.The above two coding methods are suitable for the situation where the data distribution is uniform and the data change is continuous or gentle. However, if the feature value distribution of the sample signal is imbalanced, the above coding methods are likely to classify feature values that do not belong to the same fault into the same range interval. In the industrial scenario studied in this paper, due to the influence of noise, the noise interference of the signal collected from the machine foot measuring point is more serious. Therefore, the distribution of the feature values will be more imbalanced. As a result, it is likely to classify the feature values that do not belong to the same fault into one category. The fault type and the feature state code will mismatch in many transactions, which seriously affects the credibility of the extracted fault diagnosis rules.For this reason, this paper uses theK-means clustering algorithm to find the optimal division of each feature range. The K-means clustering algorithm can adaptively classify the data according to the distribution characteristics of the data, and classify the densely distributed sample points into the same category. This method has been widely used in many fields such as medical diagnosis, traffic management, and image recognition [26–34].For them-th signal feature Pm, the feature state coding algorithm flowcharts based on K-means clustering are shown in Figure 3. In this paper, the number of clusters k is equal to the number of fault types that need to be classified, and the cluster center is equal to the average of the feature values of all samples.Figure 3
Algorithm flowcharts of feature state coding based onK-means clustering.It should be noted that all the feature coding method cannot be completely accurate, and the possibility of dividing feature values belong to different faults into the same range interval still exists. However, if an encoding method fully considers the distribution law of the feature values, the matching degree between the fault type and the feature state code in the same transaction will be higher, and the fault diagnosis rules extracted based on this coding method will generally be more accurate. In terms of the overall level, the extracted rule metrics will be better than the discretization methods that do not consider the feature value distribution. Therefore, the performance of different discretization methods can be compared based on the overall metrics level of the extracted rules.Finally, all the feature valuesPnfm of different sample signals will be divided into k parts. This paper takes the output as the final optimal range interval division, and the values of Pnfm that fall in different subsets Ψlm will be coded as corresponding interval numbers Qnfm. Qnfm is called the feature state code of feature value Pnfm.


#### 2.1.3. Transactional Dataset Generation Based on the Clustering Result

After feature state coding, the signal feature value vectorPnf can be converted into the feature state code vector Qnf=Qnf1,Qnf2,…,Qnfm,…,QnfM. To generate the corresponding transaction Tnf, all the elements in Qnf need to be reordered and coded according to their occurrence frequency from small to large. Then, Qnfm can be updated with the its sort number Infm. Finally, the fault type code f of the signal snf will be added to the end of the updated feature state code vector. By this way, the transaction Tnf generated by the signal snf is obtained, denoted as Tnf=Inf1,Inf2,…,Infm,…,InfM,f. The set of all Tnf is called the transactional dataset of fault f, denoted as Tf. The set of all Tf is called the total fault transactional dataset T.


### 2.2. Frequent Feature Pattern Mining Method Based on the Dynamic Support Threshold Discriminant Criterion

After the total fault transactional dataset is generated, frequent feature patterns can be mined. This chapter is divided into two sections. The first section introduces the concept and calculation method of the dynamic support threshold. Then, the frequent feature pattern mining method based on the DST discriminant criterion and its advantages are analyzed. The second section mathematically proves that the DST criterion has downward closure property.


#### 2.2.1. Frequent Feature Pattern Mining Based on DST Discriminant Criterion

Suppose there is a candidate feature patternX=x1,x2,…,xk,…,xXf, k=1,2,…,Xf. Then, the dynamic support threshold (DST) θSX of pattern X can be calculated according to the following formula:(3)θsX=maxθmin,θavgX,where θmin is the minimum support threshold set by the user, and θavgX is the average threshold of pattern X. The calculation methods of the two thresholds are as follows:(4)θmin=α⋅countT,θavgX=β⋅avgX,where countT is the transaction number of total fault transactional dataset T, and avgX is the mean of the occurrence frequency of all elements in the pattern X. 0<α<1 is the support coefficient, and 0<β<1 is the average coefficient. Only when the support of pattern X meets the following conditions, X will be judged to be a frequent feature pattern,(5)supX=countX≥θsX,where countX is the occurrence frequency of the feature pattern X in the dataset T. It can be seen that the traditional constant support-based criterion emphasizes the “absolute occurrence frequency” of the feature pattern. However, the DST-based criterion emphasizes the “relative co-occurrence frequency” of the feature states in a pattern.The DST-based criterion has the following advantages:(1)
The support coefficientα measures the proportion of the transaction number that contains X to the transaction number of the total fault transactional dataset T. Therefore, the minimum support threshold θmin can ensure that the occurrence frequency of frequent feature patterns meets the bottom line of users’ psychological expectations.(2)
The average thresholdθavgX can be dynamically adjusted according to the occurrence frequency of each item in pattern X. If the occurrence frequency of the items in pattern X is generally large, then avgX and θavgX will increase. Conversely, avgX and θavgX will decrease. If most of the feature states in a candidate feature pattern comes from the noise signal, these feature states will frequently appear in the total fault transactional dataset. At this time, the DST corresponding to this kind of candidate feature pattern will increase, thereby filtering out more redundant patterns to a certain extent. Conversely, when most of the feature states in a candidate feature pattern come from fault signals, the occurrence frequency of these feature states in the total fault transactional dataset will be relatively less than that of the noise feature states. At this time, the DST corresponding to this pattern will be relatively low. In other words, the average threshold θavgX can ensure that each feature state in it have high co-occurrence. Each feature state in pattern X often occurs concentratedly, and rarely occurs separately when they do not occur at the same time.(3)
The discriminant criterion based on the DST has downward closure property, so the apriori-like algorithm can be used to mine the frequent feature pattern. The detailed proof will be given in Section2.2.2.


#### 2.2.2. Downward Closure Property Proof of DST-Based Discriminant Criterion

Theorem 1.
The DST-based discriminant criterion has downward closure property. When the patternY⊆X, if Y is not a frequent feature pattern, then X must not be a frequent feature pattern.Proof.
Suppose that the patternY⊆X, so xY|xY∈Y⊆xX|xX∈X and countY≥countX. Y is not a frequent feature pattern, so countY<θsY.
The feature pattern containingk feature states is called as feature k-pattern. Suppose that a frequent feature k-pattern Y=x1,x2,…xk…xY. The process of generating high-dimensional patterns from low-dimensional patterns is a process of constantly adding new items to the patterns. Therefore, in order to generate a high-dimensional superpattern X from the low-dimensional pattern Y, first add an item xY1−Y to the pattern Y to generate candidate feature (k + 1)-patterns Y1.
In the process of feature state coding, the feature states have been sorted and recoded according to their occurrence frequency. In addition, in the process of connecting two frequent featurek-patterns to generate a candidate feature (k + 1)-pattern, only the feature state with a larger code is allowed to become a suffix of the feature state with a smaller code. Therefore, the occurrence frequency countxk of any item xk in the frequent feature k-pattern Y must not be greater than the occurrence frequency countxY1−Y of the newly added item xY1−Y, which is(6)countxY1−Y≥countxk.
Therefore, the following inequality holds,(7)Y⋅countxY1−Y≥∑k=1Ycountxk,Y∑k=1Ycountxk+countxY1−Y≥Y+1∑k=1Ycountxk.
By shifting the term, the inequality becomes the following form:(8)∑k=1Ycountxk+countxkY+1≥∑k=1YcountxkY.
Therefore,(9)avgY1≥avgY.
Therefore, the average threshold of patternY and the average threshold of pattern Y1 meet the following inequality:(10)θavgY1≥θavgY.
Repeatedly adding a single item to patternY, then the higher-dimensional superpatterns Yrr=1,2,…,n will continuously generate until the feature pattern X is generated. Based on the above conclusion, the occurrence frequency of each newly added feature state code must meet the following recurrence inequality:(11)countxX−Yn≥countxYn−Yn−1≥⋯≥countxY1−Y≥countxk,where countxA−B represents the occurrence frequency of the new item added in pattern B to generate pattern A. Therefore, the following conclusions are established:(12)θavgX≥⋯≥θavgY2≥θavgY1≥θavgY.
Theθmin is the constant minimum support threshold artificially set. Therefore,(13)maxθmin,θavgY≤maxθmin,θavgX.
Finally, the following conclusions can be derived:(14)countX≤countY<θSY≤θsX.
In summary, the discriminant criterion based on the DST has downward closure property.


### 2.3. Fault Diagnosis Rule Extraction Method Based on Association Coefficient Interestingness Discriminant Criterion

After the frequent feature pattern mining is completed, fault diagnosis rules can be extracted based on the obtained results. This chapter includes three subsections. First, a function called the minimum lift function is introduced to evaluate the minimum credibility of the fault diagnosis rules extracted by different correlation discriminant criteria. Second, a new metrics called the association coefficient interestingness (hereinafter, referred to as ACI) for evaluating the correlation between patterns and fault is proposed. This metrics maps the range of lift to the interval (−1, 1), allowing users to set an interestingness threshold that meets their psychological expectations to further filter out more reliable fault diagnosis rules. Third, the following conclusions are proved mathematically: given the same threshold, the minimum lift function of the ACI-based discriminant criterion is greater than the two existing discriminant criteria, therefore, the extracted fault diagnosis rules are more credible.


#### 2.3.1. Minimum Lift Function

Suppose the correlation discriminant criterion between the patternX and fault f is interestX⟶f≥θI. Here, interestX⟶f is a metrics function that measures the correlation between the pattern X and the fault f, which can be called the interestingness function. 0<θI<1 is the interestingness threshold.When the interestingness of the ruleXL⟶fL is exactly equal to θI, the rule XL⟶fL has the lowest interestingness among all the rules that can be extracted by this interestingness discriminant criterion. Therefore, it can be considered that the credibility of XL⟶fL is the lowest among all the rules that can be extracted. The lift of rule XL⟶fL can be calculated as(15)liftXL⟶fL=PfL|XLPfL.It is easy to prove that when the user setsliftXL⟶fL as interestX⟶f, then liftXL⟶fL=θI. Bao and Zhang, respectively, proposed two different improved interestingness discriminant criteria in the literature [50, 51]. The liftXL⟶fL of these two criteria are the same, and both can be written as follows:(16)liftXL⟶fL=1+θr1−θr.TheliftXL⟶fL of the least credible rule XL⟶fL is a function of interestingness threshold θI. Therefore, the following definition is made: when the discriminant criterion interestX⟶f≥θI is used to determine the correlation between the pattern X and the fault f, the lift of the rule XL⟶fL with the smallest interestingness among all the rules that can be extracted is defined as the minimum lift function MLFθI of the criterion, which is denoted as(17)MLFθI=liftXL⟶fL.MLFθI is a function of θI. As explained above, the lift measures the promotion degree of the pattern X to the occurrence probability of the fault f. Obviously, under the same threshold condition, the greater the MLFθI of a discriminant criterion, the greater the lift of the least reliable fault diagnosis rule that can be extracted according to this criterion. Therefore, the lift of the extracted rules based on this criterion will be generally higher, and the overall credibility of the extracted rules will also be generally higher.The existing research shows that under the same threshold condition, the minimum lift function of the two discriminant criteria in literature [50, 51] is greater than that of the lift-based criterion.


#### 2.3.2. Fault Diagnosis Rule Extraction Based on ACI Discriminant Criterion

TheAssociation Coefficient Interestingness (ACI) proposed in this paper is as follows:(18)ACIX⟶f=liftX⟶f−1liftX⟶f+1p,where p>1 is the association coefficient and p≡1mod2. Only when the confidence and the ACI of the inference X⟶f meet the following two conditions, can it be determined that X⟶f is the fault diagnosis rule,(19)ACIX⟶f≥θI,confX⟶f≥θC,where θI is the minimum interestingness threshold and θC is the minimum confidence threshold. The advantages of this discriminant method are as follows:(1)
The monotonicity ofACIX⟶f is consistent with liftX⟶f. In addition, the correlation between the feature pattern X and the fault f can be directly judged according to the sign (plus or minus) of ACIX⟶f, the discriminant conclusion is compatible with liftX⟶f, and the discriminant method is intuitive and simple. When ACIX⟶f>0, liftX⟶f>1, the pattern X and the fault f are positively correlated. When ACIX⟶f=0, liftX⟶f=1, the pattern X and the fault f are independent of each other. When ACIX⟶f<0, liftX⟶f<1, the pattern X and the fault f are negatively correlated.(2)
ACI maps the lift to the range (−1, 1). Therefore, it is possible for the user to set an interestingness threshold in the range (−1, 1) to further filter the fault diagnosis rules with higher credibility, which solves the problem that it is difficult to set the proper threshold due to the unbounded lift.(3)
TheMLFθI of the ACI-based discriminant criterion is greater than the criteria proposed in the literature [50, 51] under the same threshold condition. The user can adjust the association coefficient in the ACI to adjust the change rate of ACI and make the MLFθI of the ACI-based discriminant criterion reach a sufficiently high level according to their actual needs. The next section will give the proof process of this conclusion.


#### 2.3.3. The Property of ACI and Its Proof

Conclusion 1.
Under the same threshold conditions,MLFθI of the ACI-based discriminant criterion is larger than those of the criteria proposed in the literature [50, 51].Proof.
When the patternX and the fault f are positively correlated, liftX⟶f>1. The following inequality holds:(20)0<liftX⟶f−1liftX⟶f+1=1−2liftX⟶f+1<1.
Suppose there is a fault diagnosis rule,XL⟶fL, whose ACIXL⟶fL is exactly equal to θI, that is,(21)ACIXL⟶fL=liftXL⟶fL−1liftXL⟶fL+1p=θI.
Then, the following conclusion holds:(22)liftXL⟶fL=PB|APB=1+θIp1−θIp.
TheMLFθI of the ACI-based criterion can be derived as follows:(23)MLFθI=1+θIp1−θIp.
If0<θI<1 and p≡1mod2, obviously, 0<θIp<1. Construct a function gx, the expression of gx is(24)gx=1+x1−x.
Whenx∈0,1, gx is a monotonically increasing function. Therefore, if 0<x1<x2<1, then gx1<gx2.
Construct a functionhx, the expression of hx is(25)hx=xp−x.
Whenp>1 and x∈0,1, hx>0. Therefore, xp>x. Thus, if 0<θI<1, then θIp>θI. Since gx is a monotonically increasing function, gθIp>gθI. Therefore, the following inequality can be obtained:(26)MLFθI=1+θIp1−θIp>1+θI1−θI.
It can be seen that the minimum lift functionMLFθI of ACI-based criterion proposed in this paper is larger than those of the criteria proposed in the literature [50, 51].
In addition, it can be found that the greater the association coefficientp, the greater the minimum lift function MLFθI. If p⟶+∞, then θrp⟶1, the following conclusion holds:(27)limp⟶∞fθr=1+θrp1−θrp=+∞.
Therefore, the change rate of ACI and the minimum lift functionMLFθI can be adjusted by changing the association coefficient p according to the users’ different psychological expectations, so that the credibility of the extracted rules can be increased to any expectation level. In other words, the discriminant criterion based on ACI can ensure that its minimum lift function is always larger than any other criterion by adjusting the association coefficient. This advantage guarantees the flexibility and versatility of the criteria based on ACI.
In addition, due top≡1mod2, for different association coefficients p, the monotonicity and the sign (plus or minus) of ACIX⟶f will not change.


### 2.4. Fault Diagnosis Rule Extraction Algorithm Based on DST-ACI Discriminant Criterion

Synthesizing the contents of the above three subsections, only when inferenceX⟶f meets the following three conditions can it be judged as the correct fault diagnosis rule:(28)supX=countX≥θsX=maxθmin,θavgX,confX⟶f=countXfcountX≥θC,ACIX⟶f=liftX⟶f−1liftX⟶f+1p≥θI.If the inferenceX⟶f is judged to be the correct fault diagnosis rule, it will be determined that the fault diagnosis rule “if the feature pattern X appears, the equipment fault type will be determined as f” is established.The algorithm used in this paper will be referred to as the DST-ACI-apriori algorithm. The detailed process of the algorithm is shown in Figure4.Figure 4
DST-ACI-apriori algorithm flowcharts.


## 2.1. Feature State Coding Method Based onK-Means Clustering

This chapter introduces the whole process of using cluster-based feature state coding methods to complete data preprocessing in detail. The first part introduces the commonly used signal features and the calculation method. The second part introduces the feature state coding method based onK-means clustering. In this part, the K-means clustering algorithm will be used to divide the range of each feature into different intervals, and the feature values falling in the same range interval will be mapped to the feature state code representing the range interval. The third part introduces the process of generating total fault transactional datasets based on the clustering results. The three parts of this chapter and the logical relationship between them are shown in Figure 2.Figure 2
Process of using cluster-based feature state coding methods to complete data preprocessing.


### 2.1.1. Signal Feature Extraction

The first step of feature state coding is to extract the feature parameter value of each sample signal. Signal features can reflect different characteristics of the signal [44–49]. The commonly used signal features are shown in Table 1.Table 1
Signal features.
Calculation formulaSignal featuresP1=maxxiMaximumP2=minxiMinimumP3=1/N∑i=1NxiMeanP4=P1−P2Peak to peakP5=1/N∑i=1NxiAbsolute averageNVarianceP7=P6Standard deviationP8=1/N−1P64∑i=1Nxi−x¯4KurtosisP9=1/N−1P63∑i=1Nxi−x¯3SkewnessP10=1/N∑i=1Nxi2Root mean squareP11=P10/P5Shape factorP12=P1/P10Crest factorP13=P1/P5Impulse factorP14=1/N∑i=1Nxi2Square root amplitudeP15=P1/P14Coefficient of variationP16=1/K∑k=1KykSpectral meanP17=1/K∑k=1Kyk−P162Spectral varianceP18=1/KP173∑k=1Kyk−P162Spectral skewnessP19=P18=1/KP172∑k=1Kyk−P164Spectral kurtosisP20=∑k=1Kfk⋅yk/∑k=1KykFrequency centerP21=∑k=1Kfk−P202⋅yk/∑k=1KykFrequency standard deviationP22=∑k=1Kfk2⋅yk/∑k=1KykFrequency RMSP23=∑k=1Kfk4⋅yk/∑k=1Kfk2⋅ykMain frequency band positionN is the number of sampling points, xi is the amplitude of the i-th sampling point, k is the number of frequency domain sequence points, fk is the k-th frequency value, and yk is the spectral value corresponding to fk.Suppose there areF different fault signal datasets, each dataset Df consists of Nf signal samples snf with fault type f, and each sample signal contains N sampling points, n=1,2,…,Nf, f=1,2,…,F. The values of the above-mentioned M signal feature can be extracted from each sample signal snf. The set of all the feature values of snf is defined as the signal feature value vector of the signal snf in the form of Pnf=Pnf1,Pnf2,…Pnfm,…,PnfM, m=1,2,…,M. Each type of fault dataset Df can generate Nf different signal feature value vectors.


### 2.1.2. Feature State Coding Based onK-Means Clustering

Let the value of the featurePm of the signal snf be Pnfm. The feature value Pnfm in all the signal feature value vector should be coded into a single identifier Qnfm according to the range interval, which it belongs. The code Qnfm represents the range interval code in which the feature value Pnfm is located. In other words, feature state coding is to find a range interval division method to construct a mapping F⋅ from feature value Pnfm to feature the range interval code Qnfm, which is(1)F⋅:Pnfm⟶Qnfm.Suppose the range of a feature parameterPnfm is Ψm, and it needs to be divided into Lm range intervals Ψlm, then(2)Ψm=∪l=1LmΨlm.In traditional methods, equal-width discretization or equal-density discretization is often used to complete coding. The equal-width discretization will divide the feature value range into several equal-length intervals. The equal-density discretization maintain the number of the feature value in each interval the same. Then, the feature values falling in the same range interval will be uniformly mapped to the unique code representing the range interval.The above two coding methods are suitable for the situation where the data distribution is uniform and the data change is continuous or gentle. However, if the feature value distribution of the sample signal is imbalanced, the above coding methods are likely to classify feature values that do not belong to the same fault into the same range interval. In the industrial scenario studied in this paper, due to the influence of noise, the noise interference of the signal collected from the machine foot measuring point is more serious. Therefore, the distribution of the feature values will be more imbalanced. As a result, it is likely to classify the feature values that do not belong to the same fault into one category. The fault type and the feature state code will mismatch in many transactions, which seriously affects the credibility of the extracted fault diagnosis rules.For this reason, this paper uses theK-means clustering algorithm to find the optimal division of each feature range. The K-means clustering algorithm can adaptively classify the data according to the distribution characteristics of the data, and classify the densely distributed sample points into the same category. This method has been widely used in many fields such as medical diagnosis, traffic management, and image recognition [26–34].For them-th signal feature Pm, the feature state coding algorithm flowcharts based on K-means clustering are shown in Figure 3. In this paper, the number of clusters k is equal to the number of fault types that need to be classified, and the cluster center is equal to the average of the feature values of all samples.Figure 3
Algorithm flowcharts of feature state coding based onK-means clustering.It should be noted that all the feature coding method cannot be completely accurate, and the possibility of dividing feature values belong to different faults into the same range interval still exists. However, if an encoding method fully considers the distribution law of the feature values, the matching degree between the fault type and the feature state code in the same transaction will be higher, and the fault diagnosis rules extracted based on this coding method will generally be more accurate. In terms of the overall level, the extracted rule metrics will be better than the discretization methods that do not consider the feature value distribution. Therefore, the performance of different discretization methods can be compared based on the overall metrics level of the extracted rules.Finally, all the feature valuesPnfm of different sample signals will be divided into k parts. This paper takes the output as the final optimal range interval division, and the values of Pnfm that fall in different subsets Ψlm will be coded as corresponding interval numbers Qnfm. Qnfm is called the feature state code of feature value Pnfm.


### 2.1.3. Transactional Dataset Generation Based on the Clustering Result

After feature state coding, the signal feature value vectorPnf can be converted into the feature state code vector Qnf=Qnf1,Qnf2,…,Qnfm,…,QnfM. To generate the corresponding transaction Tnf, all the elements in Qnf need to be reordered and coded according to their occurrence frequency from small to large. Then, Qnfm can be updated with the its sort number Infm. Finally, the fault type code f of the signal snf will be added to the end of the updated feature state code vector. By this way, the transaction Tnf generated by the signal snf is obtained, denoted as Tnf=Inf1,Inf2,…,Infm,…,InfM,f. The set of all Tnf is called the transactional dataset of fault f, denoted as Tf. The set of all Tf is called the total fault transactional dataset T.


## 2.1.1. Signal Feature Extraction

The first step of feature state coding is to extract the feature parameter value of each sample signal. Signal features can reflect different characteristics of the signal [44–49]. The commonly used signal features are shown in Table 1.Table 1
Signal features.
Calculation formulaSignal featuresP1=maxxiMaximumP2=minxiMinimumP3=1/N∑i=1NxiMeanP4=P1−P2Peak to peakP5=1/N∑i=1NxiAbsolute averageNVarianceP7=P6Standard deviationP8=1/N−1P64∑i=1Nxi−x¯4KurtosisP9=1/N−1P63∑i=1Nxi−x¯3SkewnessP10=1/N∑i=1Nxi2Root mean squareP11=P10/P5Shape factorP12=P1/P10Crest factorP13=P1/P5Impulse factorP14=1/N∑i=1Nxi2Square root amplitudeP15=P1/P14Coefficient of variationP16=1/K∑k=1KykSpectral meanP17=1/K∑k=1Kyk−P162Spectral varianceP18=1/KP173∑k=1Kyk−P162Spectral skewnessP19=P18=1/KP172∑k=1Kyk−P164Spectral kurtosisP20=∑k=1Kfk⋅yk/∑k=1KykFrequency centerP21=∑k=1Kfk−P202⋅yk/∑k=1KykFrequency standard deviationP22=∑k=1Kfk2⋅yk/∑k=1KykFrequency RMSP23=∑k=1Kfk4⋅yk/∑k=1Kfk2⋅ykMain frequency band positionN is the number of sampling points, xi is the amplitude of the i-th sampling point, k is the number of frequency domain sequence points, fk is the k-th frequency value, and yk is the spectral value corresponding to fk.Suppose there areF different fault signal datasets, each dataset Df consists of Nf signal samples snf with fault type f, and each sample signal contains N sampling points, n=1,2,…,Nf, f=1,2,…,F. The values of the above-mentioned M signal feature can be extracted from each sample signal snf. The set of all the feature values of snf is defined as the signal feature value vector of the signal snf in the form of Pnf=Pnf1,Pnf2,…Pnfm,…,PnfM, m=1,2,…,M. Each type of fault dataset Df can generate Nf different signal feature value vectors.


## 2.1.2. Feature State Coding Based onK-Means Clustering

Let the value of the featurePm of the signal snf be Pnfm. The feature value Pnfm in all the signal feature value vector should be coded into a single identifier Qnfm according to the range interval, which it belongs. The code Qnfm represents the range interval code in which the feature value Pnfm is located. In other words, feature state coding is to find a range interval division method to construct a mapping F⋅ from feature value Pnfm to feature the range interval code Qnfm, which is(1)F⋅:Pnfm⟶Qnfm.Suppose the range of a feature parameterPnfm is Ψm, and it needs to be divided into Lm range intervals Ψlm, then(2)Ψm=∪l=1LmΨlm.In traditional methods, equal-width discretization or equal-density discretization is often used to complete coding. The equal-width discretization will divide the feature value range into several equal-length intervals. The equal-density discretization maintain the number of the feature value in each interval the same. Then, the feature values falling in the same range interval will be uniformly mapped to the unique code representing the range interval.The above two coding methods are suitable for the situation where the data distribution is uniform and the data change is continuous or gentle. However, if the feature value distribution of the sample signal is imbalanced, the above coding methods are likely to classify feature values that do not belong to the same fault into the same range interval. In the industrial scenario studied in this paper, due to the influence of noise, the noise interference of the signal collected from the machine foot measuring point is more serious. Therefore, the distribution of the feature values will be more imbalanced. As a result, it is likely to classify the feature values that do not belong to the same fault into one category. The fault type and the feature state code will mismatch in many transactions, which seriously affects the credibility of the extracted fault diagnosis rules.For this reason, this paper uses theK-means clustering algorithm to find the optimal division of each feature range. The K-means clustering algorithm can adaptively classify the data according to the distribution characteristics of the data, and classify the densely distributed sample points into the same category. This method has been widely used in many fields such as medical diagnosis, traffic management, and image recognition [26–34].For them-th signal feature Pm, the feature state coding algorithm flowcharts based on K-means clustering are shown in Figure 3. In this paper, the number of clusters k is equal to the number of fault types that need to be classified, and the cluster center is equal to the average of the feature values of all samples.Figure 3
Algorithm flowcharts of feature state coding based onK-means clustering.It should be noted that all the feature coding method cannot be completely accurate, and the possibility of dividing feature values belong to different faults into the same range interval still exists. However, if an encoding method fully considers the distribution law of the feature values, the matching degree between the fault type and the feature state code in the same transaction will be higher, and the fault diagnosis rules extracted based on this coding method will generally be more accurate. In terms of the overall level, the extracted rule metrics will be better than the discretization methods that do not consider the feature value distribution. Therefore, the performance of different discretization methods can be compared based on the overall metrics level of the extracted rules.Finally, all the feature valuesPnfm of different sample signals will be divided into k parts. This paper takes the output as the final optimal range interval division, and the values of Pnfm that fall in different subsets Ψlm will be coded as corresponding interval numbers Qnfm. Qnfm is called the feature state code of feature value Pnfm.


## 2.1.3. Transactional Dataset Generation Based on the Clustering Result

After feature state coding, the signal feature value vectorPnf can be converted into the feature state code vector Qnf=Qnf1,Qnf2,…,Qnfm,…,QnfM. To generate the corresponding transaction Tnf, all the elements in Qnf need to be reordered and coded according to their occurrence frequency from small to large. Then, Qnfm can be updated with the its sort number Infm. Finally, the fault type code f of the signal snf will be added to the end of the updated feature state code vector. By this way, the transaction Tnf generated by the signal snf is obtained, denoted as Tnf=Inf1,Inf2,…,Infm,…,InfM,f. The set of all Tnf is called the transactional dataset of fault f, denoted as Tf. The set of all Tf is called the total fault transactional dataset T.


## 2.2. Frequent Feature Pattern Mining Method Based on the Dynamic Support Threshold Discriminant Criterion

After the total fault transactional dataset is generated, frequent feature patterns can be mined. This chapter is divided into two sections. The first section introduces the concept and calculation method of the dynamic support threshold. Then, the frequent feature pattern mining method based on the DST discriminant criterion and its advantages are analyzed. The second section mathematically proves that the DST criterion has downward closure property.


### 2.2.1. Frequent Feature Pattern Mining Based on DST Discriminant Criterion

Suppose there is a candidate feature patternX=x1,x2,…,xk,…,xXf, k=1,2,…,Xf. Then, the dynamic support threshold (DST) θSX of pattern X can be calculated according to the following formula:(3)θsX=maxθmin,θavgX,where θmin is the minimum support threshold set by the user, and θavgX is the average threshold of pattern X. The calculation methods of the two thresholds are as follows:(4)θmin=α⋅countT,θavgX=β⋅avgX,where countT is the transaction number of total fault transactional dataset T, and avgX is the mean of the occurrence frequency of all elements in the pattern X. 0<α<1 is the support coefficient, and 0<β<1 is the average coefficient. Only when the support of pattern X meets the following conditions, X will be judged to be a frequent feature pattern,(5)supX=countX≥θsX,where countX is the occurrence frequency of the feature pattern X in the dataset T. It can be seen that the traditional constant support-based criterion emphasizes the “absolute occurrence frequency” of the feature pattern. However, the DST-based criterion emphasizes the “relative co-occurrence frequency” of the feature states in a pattern.The DST-based criterion has the following advantages:(1)
The support coefficientα measures the proportion of the transaction number that contains X to the transaction number of the total fault transactional dataset T. Therefore, the minimum support threshold θmin can ensure that the occurrence frequency of frequent feature patterns meets the bottom line of users’ psychological expectations.(2)
The average thresholdθavgX can be dynamically adjusted according to the occurrence frequency of each item in pattern X. If the occurrence frequency of the items in pattern X is generally large, then avgX and θavgX will increase. Conversely, avgX and θavgX will decrease. If most of the feature states in a candidate feature pattern comes from the noise signal, these feature states will frequently appear in the total fault transactional dataset. At this time, the DST corresponding to this kind of candidate feature pattern will increase, thereby filtering out more redundant patterns to a certain extent. Conversely, when most of the feature states in a candidate feature pattern come from fault signals, the occurrence frequency of these feature states in the total fault transactional dataset will be relatively less than that of the noise feature states. At this time, the DST corresponding to this pattern will be relatively low. In other words, the average threshold θavgX can ensure that each feature state in it have high co-occurrence. Each feature state in pattern X often occurs concentratedly, and rarely occurs separately when they do not occur at the same time.(3)
The discriminant criterion based on the DST has downward closure property, so the apriori-like algorithm can be used to mine the frequent feature pattern. The detailed proof will be given in Section2.2.2.


### 2.2.2. Downward Closure Property Proof of DST-Based Discriminant Criterion

Theorem 1.
The DST-based discriminant criterion has downward closure property. When the patternY⊆X, if Y is not a frequent feature pattern, then X must not be a frequent feature pattern.Proof.
Suppose that the patternY⊆X, so xY|xY∈Y⊆xX|xX∈X and countY≥countX. Y is not a frequent feature pattern, so countY<θsY.
The feature pattern containingk feature states is called as feature k-pattern. Suppose that a frequent feature k-pattern Y=x1,x2,…xk…xY. The process of generating high-dimensional patterns from low-dimensional patterns is a process of constantly adding new items to the patterns. Therefore, in order to generate a high-dimensional superpattern X from the low-dimensional pattern Y, first add an item xY1−Y to the pattern Y to generate candidate feature (k + 1)-patterns Y1.
In the process of feature state coding, the feature states have been sorted and recoded according to their occurrence frequency. In addition, in the process of connecting two frequent featurek-patterns to generate a candidate feature (k + 1)-pattern, only the feature state with a larger code is allowed to become a suffix of the feature state with a smaller code. Therefore, the occurrence frequency countxk of any item xk in the frequent feature k-pattern Y must not be greater than the occurrence frequency countxY1−Y of the newly added item xY1−Y, which is(6)countxY1−Y≥countxk.
Therefore, the following inequality holds,(7)Y⋅countxY1−Y≥∑k=1Ycountxk,Y∑k=1Ycountxk+countxY1−Y≥Y+1∑k=1Ycountxk.
By shifting the term, the inequality becomes the following form:(8)∑k=1Ycountxk+countxkY+1≥∑k=1YcountxkY.
Therefore,(9)avgY1≥avgY.
Therefore, the average threshold of patternY and the average threshold of pattern Y1 meet the following inequality:(10)θavgY1≥θavgY.
Repeatedly adding a single item to patternY, then the higher-dimensional superpatterns Yrr=1,2,…,n will continuously generate until the feature pattern X is generated. Based on the above conclusion, the occurrence frequency of each newly added feature state code must meet the following recurrence inequality:(11)countxX−Yn≥countxYn−Yn−1≥⋯≥countxY1−Y≥countxk,where countxA−B represents the occurrence frequency of the new item added in pattern B to generate pattern A. Therefore, the following conclusions are established:(12)θavgX≥⋯≥θavgY2≥θavgY1≥θavgY.
Theθmin is the constant minimum support threshold artificially set. Therefore,(13)maxθmin,θavgY≤maxθmin,θavgX.
Finally, the following conclusions can be derived:(14)countX≤countY<θSY≤θsX.
In summary, the discriminant criterion based on the DST has downward closure property.


## 2.2.1. Frequent Feature Pattern Mining Based on DST Discriminant Criterion

Suppose there is a candidate feature patternX=x1,x2,…,xk,…,xXf, k=1,2,…,Xf. Then, the dynamic support threshold (DST) θSX of pattern X can be calculated according to the following formula:(3)θsX=maxθmin,θavgX,where θmin is the minimum support threshold set by the user, and θavgX is the average threshold of pattern X. The calculation methods of the two thresholds are as follows:(4)θmin=α⋅countT,θavgX=β⋅avgX,where countT is the transaction number of total fault transactional dataset T, and avgX is the mean of the occurrence frequency of all elements in the pattern X. 0<α<1 is the support coefficient, and 0<β<1 is the average coefficient. Only when the support of pattern X meets the following conditions, X will be judged to be a frequent feature pattern,(5)supX=countX≥θsX,where countX is the occurrence frequency of the feature pattern X in the dataset T. It can be seen that the traditional constant support-based criterion emphasizes the “absolute occurrence frequency” of the feature pattern. However, the DST-based criterion emphasizes the “relative co-occurrence frequency” of the feature states in a pattern.The DST-based criterion has the following advantages:(1)
The support coefficientα measures the proportion of the transaction number that contains X to the transaction number of the total fault transactional dataset T. Therefore, the minimum support threshold θmin can ensure that the occurrence frequency of frequent feature patterns meets the bottom line of users’ psychological expectations.(2)
The average thresholdθavgX can be dynamically adjusted according to the occurrence frequency of each item in pattern X. If the occurrence frequency of the items in pattern X is generally large, then avgX and θavgX will increase. Conversely, avgX and θavgX will decrease. If most of the feature states in a candidate feature pattern comes from the noise signal, these feature states will frequently appear in the total fault transactional dataset. At this time, the DST corresponding to this kind of candidate feature pattern will increase, thereby filtering out more redundant patterns to a certain extent. Conversely, when most of the feature states in a candidate feature pattern come from fault signals, the occurrence frequency of these feature states in the total fault transactional dataset will be relatively less than that of the noise feature states. At this time, the DST corresponding to this pattern will be relatively low. In other words, the average threshold θavgX can ensure that each feature state in it have high co-occurrence. Each feature state in pattern X often occurs concentratedly, and rarely occurs separately when they do not occur at the same time.(3)
The discriminant criterion based on the DST has downward closure property, so the apriori-like algorithm can be used to mine the frequent feature pattern. The detailed proof will be given in Section2.2.2.


## 2.2.2. Downward Closure Property Proof of DST-Based Discriminant Criterion

Theorem 1.
The DST-based discriminant criterion has downward closure property. When the patternY⊆X, if Y is not a frequent feature pattern, then X must not be a frequent feature pattern.Proof.
Suppose that the patternY⊆X, so xY|xY∈Y⊆xX|xX∈X and countY≥countX. Y is not a frequent feature pattern, so countY<θsY.
The feature pattern containingk feature states is called as feature k-pattern. Suppose that a frequent feature k-pattern Y=x1,x2,…xk…xY. The process of generating high-dimensional patterns from low-dimensional patterns is a process of constantly adding new items to the patterns. Therefore, in order to generate a high-dimensional superpattern X from the low-dimensional pattern Y, first add an item xY1−Y to the pattern Y to generate candidate feature (k + 1)-patterns Y1.
In the process of feature state coding, the feature states have been sorted and recoded according to their occurrence frequency. In addition, in the process of connecting two frequent featurek-patterns to generate a candidate feature (k + 1)-pattern, only the feature state with a larger code is allowed to become a suffix of the feature state with a smaller code. Therefore, the occurrence frequency countxk of any item xk in the frequent feature k-pattern Y must not be greater than the occurrence frequency countxY1−Y of the newly added item xY1−Y, which is(6)countxY1−Y≥countxk.
Therefore, the following inequality holds,(7)Y⋅countxY1−Y≥∑k=1Ycountxk,Y∑k=1Ycountxk+countxY1−Y≥Y+1∑k=1Ycountxk.
By shifting the term, the inequality becomes the following form:(8)∑k=1Ycountxk+countxkY+1≥∑k=1YcountxkY.
Therefore,(9)avgY1≥avgY.
Therefore, the average threshold of patternY and the average threshold of pattern Y1 meet the following inequality:(10)θavgY1≥θavgY.
Repeatedly adding a single item to patternY, then the higher-dimensional superpatterns Yrr=1,2,…,n will continuously generate until the feature pattern X is generated. Based on the above conclusion, the occurrence frequency of each newly added feature state code must meet the following recurrence inequality:(11)countxX−Yn≥countxYn−Yn−1≥⋯≥countxY1−Y≥countxk,where countxA−B represents the occurrence frequency of the new item added in pattern B to generate pattern A. Therefore, the following conclusions are established:(12)θavgX≥⋯≥θavgY2≥θavgY1≥θavgY.
Theθmin is the constant minimum support threshold artificially set. Therefore,(13)maxθmin,θavgY≤maxθmin,θavgX.
Finally, the following conclusions can be derived:(14)countX≤countY<θSY≤θsX.
In summary, the discriminant criterion based on the DST has downward closure property.


## 2.3. Fault Diagnosis Rule Extraction Method Based on Association Coefficient Interestingness Discriminant Criterion

After the frequent feature pattern mining is completed, fault diagnosis rules can be extracted based on the obtained results. This chapter includes three subsections. First, a function called the minimum lift function is introduced to evaluate the minimum credibility of the fault diagnosis rules extracted by different correlation discriminant criteria. Second, a new metrics called the association coefficient interestingness (hereinafter, referred to as ACI) for evaluating the correlation between patterns and fault is proposed. This metrics maps the range of lift to the interval (−1, 1), allowing users to set an interestingness threshold that meets their psychological expectations to further filter out more reliable fault diagnosis rules. Third, the following conclusions are proved mathematically: given the same threshold, the minimum lift function of the ACI-based discriminant criterion is greater than the two existing discriminant criteria, therefore, the extracted fault diagnosis rules are more credible.


### 2.3.1. Minimum Lift Function

Suppose the correlation discriminant criterion between the patternX and fault f is interestX⟶f≥θI. Here, interestX⟶f is a metrics function that measures the correlation between the pattern X and the fault f, which can be called the interestingness function. 0<θI<1 is the interestingness threshold.When the interestingness of the ruleXL⟶fL is exactly equal to θI, the rule XL⟶fL has the lowest interestingness among all the rules that can be extracted by this interestingness discriminant criterion. Therefore, it can be considered that the credibility of XL⟶fL is the lowest among all the rules that can be extracted. The lift of rule XL⟶fL can be calculated as(15)liftXL⟶fL=PfL|XLPfL.It is easy to prove that when the user setsliftXL⟶fL as interestX⟶f, then liftXL⟶fL=θI. Bao and Zhang, respectively, proposed two different improved interestingness discriminant criteria in the literature [50, 51]. The liftXL⟶fL of these two criteria are the same, and both can be written as follows:(16)liftXL⟶fL=1+θr1−θr.TheliftXL⟶fL of the least credible rule XL⟶fL is a function of interestingness threshold θI. Therefore, the following definition is made: when the discriminant criterion interestX⟶f≥θI is used to determine the correlation between the pattern X and the fault f, the lift of the rule XL⟶fL with the smallest interestingness among all the rules that can be extracted is defined as the minimum lift function MLFθI of the criterion, which is denoted as(17)MLFθI=liftXL⟶fL.MLFθI is a function of θI. As explained above, the lift measures the promotion degree of the pattern X to the occurrence probability of the fault f. Obviously, under the same threshold condition, the greater the MLFθI of a discriminant criterion, the greater the lift of the least reliable fault diagnosis rule that can be extracted according to this criterion. Therefore, the lift of the extracted rules based on this criterion will be generally higher, and the overall credibility of the extracted rules will also be generally higher.The existing research shows that under the same threshold condition, the minimum lift function of the two discriminant criteria in literature [50, 51] is greater than that of the lift-based criterion.


### 2.3.2. Fault Diagnosis Rule Extraction Based on ACI Discriminant Criterion

TheAssociation Coefficient Interestingness (ACI) proposed in this paper is as follows:(18)ACIX⟶f=liftX⟶f−1liftX⟶f+1p,where p>1 is the association coefficient and p≡1mod2. Only when the confidence and the ACI of the inference X⟶f meet the following two conditions, can it be determined that X⟶f is the fault diagnosis rule,(19)ACIX⟶f≥θI,confX⟶f≥θC,where θI is the minimum interestingness threshold and θC is the minimum confidence threshold. The advantages of this discriminant method are as follows:(1)
The monotonicity ofACIX⟶f is consistent with liftX⟶f. In addition, the correlation between the feature pattern X and the fault f can be directly judged according to the sign (plus or minus) of ACIX⟶f, the discriminant conclusion is compatible with liftX⟶f, and the discriminant method is intuitive and simple. When ACIX⟶f>0, liftX⟶f>1, the pattern X and the fault f are positively correlated. When ACIX⟶f=0, liftX⟶f=1, the pattern X and the fault f are independent of each other. When ACIX⟶f<0, liftX⟶f<1, the pattern X and the fault f are negatively correlated.(2)
ACI maps the lift to the range (−1, 1). Therefore, it is possible for the user to set an interestingness threshold in the range (−1, 1) to further filter the fault diagnosis rules with higher credibility, which solves the problem that it is difficult to set the proper threshold due to the unbounded lift.(3)
TheMLFθI of the ACI-based discriminant criterion is greater than the criteria proposed in the literature [50, 51] under the same threshold condition. The user can adjust the association coefficient in the ACI to adjust the change rate of ACI and make the MLFθI of the ACI-based discriminant criterion reach a sufficiently high level according to their actual needs. The next section will give the proof process of this conclusion.


### 2.3.3. The Property of ACI and Its Proof

Conclusion 1.
Under the same threshold conditions,MLFθI of the ACI-based discriminant criterion is larger than those of the criteria proposed in the literature [50, 51].Proof.
When the patternX and the fault f are positively correlated, liftX⟶f>1. The following inequality holds:(20)0<liftX⟶f−1liftX⟶f+1=1−2liftX⟶f+1<1.
Suppose there is a fault diagnosis rule,XL⟶fL, whose ACIXL⟶fL is exactly equal to θI, that is,(21)ACIXL⟶fL=liftXL⟶fL−1liftXL⟶fL+1p=θI.
Then, the following conclusion holds:(22)liftXL⟶fL=PB|APB=1+θIp1−θIp.
TheMLFθI of the ACI-based criterion can be derived as follows:(23)MLFθI=1+θIp1−θIp.
If0<θI<1 and p≡1mod2, obviously, 0<θIp<1. Construct a function gx, the expression of gx is(24)gx=1+x1−x.
Whenx∈0,1, gx is a monotonically increasing function. Therefore, if 0<x1<x2<1, then gx1<gx2.
Construct a functionhx, the expression of hx is(25)hx=xp−x.
Whenp>1 and x∈0,1, hx>0. Therefore, xp>x. Thus, if 0<θI<1, then θIp>θI. Since gx is a monotonically increasing function, gθIp>gθI. Therefore, the following inequality can be obtained:(26)MLFθI=1+θIp1−θIp>1+θI1−θI.
It can be seen that the minimum lift functionMLFθI of ACI-based criterion proposed in this paper is larger than those of the criteria proposed in the literature [50, 51].
In addition, it can be found that the greater the association coefficientp, the greater the minimum lift function MLFθI. If p⟶+∞, then θrp⟶1, the following conclusion holds:(27)limp⟶∞fθr=1+θrp1−θrp=+∞.
Therefore, the change rate of ACI and the minimum lift functionMLFθI can be adjusted by changing the association coefficient p according to the users’ different psychological expectations, so that the credibility of the extracted rules can be increased to any expectation level. In other words, the discriminant criterion based on ACI can ensure that its minimum lift function is always larger than any other criterion by adjusting the association coefficient. This advantage guarantees the flexibility and versatility of the criteria based on ACI.
In addition, due top≡1mod2, for different association coefficients p, the monotonicity and the sign (plus or minus) of ACIX⟶f will not change.


## 2.3.1. Minimum Lift Function

Suppose the correlation discriminant criterion between the patternX and fault f is interestX⟶f≥θI. Here, interestX⟶f is a metrics function that measures the correlation between the pattern X and the fault f, which can be called the interestingness function. 0<θI<1 is the interestingness threshold.When the interestingness of the ruleXL⟶fL is exactly equal to θI, the rule XL⟶fL has the lowest interestingness among all the rules that can be extracted by this interestingness discriminant criterion. Therefore, it can be considered that the credibility of XL⟶fL is the lowest among all the rules that can be extracted. The lift of rule XL⟶fL can be calculated as(15)liftXL⟶fL=PfL|XLPfL.It is easy to prove that when the user setsliftXL⟶fL as interestX⟶f, then liftXL⟶fL=θI. Bao and Zhang, respectively, proposed two different improved interestingness discriminant criteria in the literature [50, 51]. The liftXL⟶fL of these two criteria are the same, and both can be written as follows:(16)liftXL⟶fL=1+θr1−θr.TheliftXL⟶fL of the least credible rule XL⟶fL is a function of interestingness threshold θI. Therefore, the following definition is made: when the discriminant criterion interestX⟶f≥θI is used to determine the correlation between the pattern X and the fault f, the lift of the rule XL⟶fL with the smallest interestingness among all the rules that can be extracted is defined as the minimum lift function MLFθI of the criterion, which is denoted as(17)MLFθI=liftXL⟶fL.MLFθI is a function of θI. As explained above, the lift measures the promotion degree of the pattern X to the occurrence probability of the fault f. Obviously, under the same threshold condition, the greater the MLFθI of a discriminant criterion, the greater the lift of the least reliable fault diagnosis rule that can be extracted according to this criterion. Therefore, the lift of the extracted rules based on this criterion will be generally higher, and the overall credibility of the extracted rules will also be generally higher.The existing research shows that under the same threshold condition, the minimum lift function of the two discriminant criteria in literature [50, 51] is greater than that of the lift-based criterion.


## 2.3.2. Fault Diagnosis Rule Extraction Based on ACI Discriminant Criterion

TheAssociation Coefficient Interestingness (ACI) proposed in this paper is as follows:(18)ACIX⟶f=liftX⟶f−1liftX⟶f+1p,where p>1 is the association coefficient and p≡1mod2. Only when the confidence and the ACI of the inference X⟶f meet the following two conditions, can it be determined that X⟶f is the fault diagnosis rule,(19)ACIX⟶f≥θI,confX⟶f≥θC,where θI is the minimum interestingness threshold and θC is the minimum confidence threshold. The advantages of this discriminant method are as follows:(1)
The monotonicity ofACIX⟶f is consistent with liftX⟶f. In addition, the correlation between the feature pattern X and the fault f can be directly judged according to the sign (plus or minus) of ACIX⟶f, the discriminant conclusion is compatible with liftX⟶f, and the discriminant method is intuitive and simple. When ACIX⟶f>0, liftX⟶f>1, the pattern X and the fault f are positively correlated. When ACIX⟶f=0, liftX⟶f=1, the pattern X and the fault f are independent of each other. When ACIX⟶f<0, liftX⟶f<1, the pattern X and the fault f are negatively correlated.(2)
ACI maps the lift to the range (−1, 1). Therefore, it is possible for the user to set an interestingness threshold in the range (−1, 1) to further filter the fault diagnosis rules with higher credibility, which solves the problem that it is difficult to set the proper threshold due to the unbounded lift.(3)
TheMLFθI of the ACI-based discriminant criterion is greater than the criteria proposed in the literature [50, 51] under the same threshold condition. The user can adjust the association coefficient in the ACI to adjust the change rate of ACI and make the MLFθI of the ACI-based discriminant criterion reach a sufficiently high level according to their actual needs. The next section will give the proof process of this conclusion.


## 2.3.3. The Property of ACI and Its Proof

Conclusion 1.
Under the same threshold conditions,MLFθI of the ACI-based discriminant criterion is larger than those of the criteria proposed in the literature [50, 51].Proof.
When the patternX and the fault f are positively correlated, liftX⟶f>1. The following inequality holds:(20)0<liftX⟶f−1liftX⟶f+1=1−2liftX⟶f+1<1.
Suppose there is a fault diagnosis rule,XL⟶fL, whose ACIXL⟶fL is exactly equal to θI, that is,(21)ACIXL⟶fL=liftXL⟶fL−1liftXL⟶fL+1p=θI.
Then, the following conclusion holds:(22)liftXL⟶fL=PB|APB=1+θIp1−θIp.
TheMLFθI of the ACI-based criterion can be derived as follows:(23)MLFθI=1+θIp1−θIp.
If0<θI<1 and p≡1mod2, obviously, 0<θIp<1. Construct a function gx, the expression of gx is(24)gx=1+x1−x.
Whenx∈0,1, gx is a monotonically increasing function. Therefore, if 0<x1<x2<1, then gx1<gx2.
Construct a functionhx, the expression of hx is(25)hx=xp−x.
Whenp>1 and x∈0,1, hx>0. Therefore, xp>x. Thus, if 0<θI<1, then θIp>θI. Since gx is a monotonically increasing function, gθIp>gθI. Therefore, the following inequality can be obtained:(26)MLFθI=1+θIp1−θIp>1+θI1−θI.
It can be seen that the minimum lift functionMLFθI of ACI-based criterion proposed in this paper is larger than those of the criteria proposed in the literature [50, 51].
In addition, it can be found that the greater the association coefficientp, the greater the minimum lift function MLFθI. If p⟶+∞, then θrp⟶1, the following conclusion holds:(27)limp⟶∞fθr=1+θrp1−θrp=+∞.
Therefore, the change rate of ACI and the minimum lift functionMLFθI can be adjusted by changing the association coefficient p according to the users’ different psychological expectations, so that the credibility of the extracted rules can be increased to any expectation level. In other words, the discriminant criterion based on ACI can ensure that its minimum lift function is always larger than any other criterion by adjusting the association coefficient. This advantage guarantees the flexibility and versatility of the criteria based on ACI.
In addition, due top≡1mod2, for different association coefficients p, the monotonicity and the sign (plus or minus) of ACIX⟶f will not change.


## 2.4. Fault Diagnosis Rule Extraction Algorithm Based on DST-ACI Discriminant Criterion

Synthesizing the contents of the above three subsections, only when inferenceX⟶f meets the following three conditions can it be judged as the correct fault diagnosis rule:(28)supX=countX≥θsX=maxθmin,θavgX,confX⟶f=countXfcountX≥θC,ACIX⟶f=liftX⟶f−1liftX⟶f+1p≥θI.If the inferenceX⟶f is judged to be the correct fault diagnosis rule, it will be determined that the fault diagnosis rule “if the feature pattern X appears, the equipment fault type will be determined as f” is established.The algorithm used in this paper will be referred to as the DST-ACI-apriori algorithm. The detailed process of the algorithm is shown in Figure4.Figure 4
DST-ACI-apriori algorithm flowcharts.


## 3. Experiments

This chapter includes four sections. The first section introduces the fault simulation experiment equipment and the parts used to simulate the fault. The second section introduces the data format of the collected vibration signal and the parameter values used in the algorithm. The third section shows the experimental conclusions in detail. In the fourth section, using the same experimental data as the input, the performance between the DST-ACI method and other existing fault diagnosis rule extraction methods is compared.


### 3.1. Fault Simulation Experiment Equipment

The fault simulation experiment platform used for fault simulation experiments is shown in Figure5.Figure 5
Fault simulation experiment platform.The experiment platform is composed of a drive motor, a conveyor belt, a rotating shaft, a shafting aluminum plate, a bearing seat, a movable base, and other parts. The three magnetic vibration acceleration sensors are used to collect the signals. For the acceleration sensor, the sensitivity is 100 mv/g, the frequency response range is 0.1 Hz∼10 kHz, and the measuring range is −10 g∼+10 g.The data acquisition equipment used in the experiment is shown in Figure6. The CPU is 4-core 1.91 GHz, the Internet access is 1 gigabit ethernet access, and the RAM is 4 GB.Figure 6
Data acquisition equipment.For the acceleration monitoring module of the data acquisition equipment, the input channel is 4-channel ICP sensor input, the precision is 24 bit, and the sampling rate is optional from 2.56 kHz to 256 kHz.The computer used to save the data is shown in Figure7.Figure 7
The computer to save the data.In the experiment, four types of typical faults, namely, rotor imbalance fault, shafting misalignment fault, bearing inner ring fault, and bearing outer ring fault will be simulated by different fault simulation methods.


#### 3.1.1. Imbalance Fault Simulation Experiment

A threaded hole is set on the shafting aluminum plate of the fault simulation experiment platform. The imbalanced counterweight bolt will be screwed into the threaded hole on the aluminum plate to simulate the rotor imbalance fault. The imbalanced counterweight bolt and its installation method are shown in Figure8. The sensor installation position is shown in Figure 9.Figure 8
Imbalanced counterweight bolt.Figure 9
Sensor installation position for the imbalance fault simulation experiment.The No.1 and No.2 measuring points are the positions where the sensor is installed under the ideal measurement conditions. The two measuring points are close to the shafting aluminum plate and the rotating shaft, and the signals collected at the two positions are clearer. No.3 measuring point is located on the base of the experiment platform and has no direct contact with the shafting aluminum plate or the rotating shaft. It can be used to simulate the data acquisition conditions when the sensor is placed on the machine foot. Therefore, the data used in this paper comes from the sensor installed at No.3 measuring point.In the following, for each failure simulation test, the two ideal sensor installation positions will be recorded as No.1 and No.2 measuring points, and the sensor to collect the experiment data will be recorded as No.3. The location and number of the measuring point will be shown in the corresponding figure.


#### 3.1.2. Misalignment Fault Simulation Experiment

After the shafting is aligned, the front mandrel screw of the experiment platform remains stationary, and the rear mandrel screw is screwed in to one side, pushing a corner of the movable base to slightly deviate, causing a shafting misalignment fault. The mandrel screw is as shown in Figure10. The sensor installation position is shown in Figure 11.Figure 10
Mandrel screws to move the base.Figure 11
Sensor installation position for misalignment fault simulation experiment.


#### 3.1.3. Bearing Inner Ring Fault Simulation Experiment

A normal bearing is installed in the bearing seat at the end of the experiment platform, and it can be replaced with a inner ring fault bearing. The contact surface between the inner ring and the ball of the fault bearing is machined with a wear groove along the axial direction to simulate the damage of the inner ring. The faulty inner ring bearing used in the experiment is shown in Figure12. The sensor installation position is shown in Figure 13.Figure 12
Inner ring fault bearing used in the experiment.Figure 13
Sensor installation position for bearing inner ring fault simulation experiment.For the inner ring fault bearing, the inner diameter of the inner ring is 2.62 cm, the outer diameter of the inner ring is 3.11 cm, the ball diameter is 0.50 cm, the inner diameter of the outer ring is 4.31 cm, and the outer diameter of the outer ring is 5.20 cm. The wear groove machined on the inner ring has a width of 2 mm and a depth of 1 mm.


#### 3.1.4. Bearing Outer Ring Fault Simulation Experiment

The normal bearing in the bearing seat at the end of the experiment platform can also be replaced with a outer ring fault bearing. The contact surface between the outer ring and the ball of the fault bearing is machined with wear grooves along the axial direction to simulate the damage of the outer ring. The faulty outer ring bearing used in the experiment is shown in Figure14.Figure 14
Outer ring fault bearing used in the experiment.The parameters of the inner ring and outer ring of the outer ring fault bearing are the same as those of the inner ring fault bearing. The location of the wear groove is on the contact surface between the outer ring and the ball. The sensor installation position of the outer ring fault bearing simulation experiment is the same as the inner ring fault bearing simulation experiment.


### 3.2. Experiment Parameters

In each fault simulation experiment, the data sampling frequency is 25600 Hz, 320, 000 acceleration amplitude data points are collected, and the unit is m2/s. Since four types of faults are simulated, the total sample signal dataset includes 1, 280, 000 acceleration amplitude data points. The sampling points in each type of fault simulation experiment are divided into a group of 2000 points, and the entire collected signal sequence can be divided into 160 groups of signal subsequences. The 23 signal features in Table 1 will be extracted from the collected vibration signals. Therefore, each type of the fault simulation experiment will generate 160 transactions. Each transaction contains 23 feature state codes and 1 fault type code.The corresponding parameter settings involved in the algorithm are as follows: association coefficientp=3, minimum support threshold θmin=100, minimum confidence threshold θC=0.7, minimum interestingness threshold θI=0.2, and average coefficient β=0.6.


### 3.3. Experimental Results

Input the data collected by the above-mentioned fault simulation experiment into the DST-ACI-apriori algorithm, and then the fault diagnosis rules that meet the DST-ACI criterion can be extracted.


#### 3.3.1. Results of Frequent Feature Pattern Mining

A total of 461 frequent feature patterns are extracted. The various metric values are shown in Table2.Table 2
Results of frequent feature pattern mining.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 179401118201.7848k = 274267106162.7432k = 3120263104149.4083k = 4111259104138.2613k = 557232104127.6316k = 616222104119.0625k = 74119104108.2500The frequent featurek-pattern contains k feature state codes. The support boxplot of different frequent feature k-patterns is shown in Figure 15.Figure 15
Support boxplot.In the above figure, thex-axis is the k value of the frequent feature k-pattern, and the y-axis is the support. According to the above table and boxplot, the following conclusions can be obtained:(1)
Whenk = 1, there are frequent feature patterns with support close to 400. When k > 1, the support of all the frequent feature patterns are smaller than 300. When k = 1, there is only one feature state code in the pattern, so the support of frequent feature pattern is generally greater than that of the pattern with a larger k. Therefore, it is easy to have the special cases with very large support among them.After checking the result, it is found that only one pattern has a support greater than 400, and the support of this pattern is 401. In addition, there is another pattern whose support of is very close to 400, and the support of this another pattern is 392. However, with the exception of the two frequent feature 1-patterns mentioned above, the support of all the other patterns is smaller than 300.Obviously, the feature states in the two patterns frequently appear in at least three types of fault transactional dataset. The two feature states are widely distributed in multiple types of fault transactional datasets. Therefore, they can be considered as the noise feature states with a low reference value. After further data inspection, it is found that in the frequent feature 2-patterns and the frequent feature patterns with higher dimensions, the above two feature states no longer appear. It can be seen that the frequent feature pattern mining method based on the DST criterion can indeed effectively filter out the noise feature state.(2)
Whenk = 1, the dispersion degree of the support of the frequent feature 1-pattern is very large. The possible reason is that when discriminating the frequent feature 1-patterns, only the support of the pattern is considered, and there is no other restrictions on the discrimination of the frequent feature patterns. In this case, the pattern will be judged as a frequent feature 1-pattern as long as its support exceeds the support threshold. Therefore, when k = 1, there will be various types of frequent feature 1-patterns, and the support dispersion degree of these patterns is difficult to limit.(3)
Ask continues to increase, the support dispersion degree of the frequent feature patterns gradually decreases. On the one hand, as the value of k become larger, the number of frequent feature patterns mined also decrease, and the support difference of patterns with the same dimension will also become smaller. This leads to a more concentrated support distribution of the frequent feature patterns. On the other hand, when k keeps increasing, DST will not only check the “absolute occurrence frequency” of the feature states, but also further check the “relative co-occurrence frequency” between each feature state in the same pattern. Therefore, in the high-dimensional frequent feature patterns, there will be no more significant “unsocial” noise feature state patterns like the mining frequent feature 1-patterns.


#### 3.3.2. Result of Fault Diagnosis Rule Extraction

A total of 175 fault diagnosis rules are extracted. The number, maximum confidence, minimum confidence, average confidence, maximum lift, minimum lift, average lift, maximum ACI, minimum ACI, and average ACI of the extracted fault diagnosis rules are shown in Table3.Table 3
Result of fault diagnosis rule extraction.
Feature number in the ruleNumber of the rulesConfidence maximum/minimum/averageLift maximum/minimum/averageACI maximum/minimum/averagek = 1221/0.9557/0.986584/3.8228/3.94630.216/0.20051/0.21133k = 2251/0.98052/0.990424/3.9221/3.96170.216/0.20923/0.21267k = 3421/0.98052/0.991154/3.9221/3.96460.216/0.20923/0.21293k = 4471/0.98052/0.990844/3.9221/3.96330.216/0.20923/0.21282k = 5281/0.98077/0.988544/3.9231/3.95420.216/0.20932/0.21202k = 681/0.98077/0.985764/3.9231/3.9430.216/0.20932/0.21105k = 730.98113/0.98077/0.980893.9245/3.9231/3.92360.20945/0.20932/0.20936According to the above table, it can be seen that the confidence of the fault diagnosis rules extracted based on the DST-ACI discriminant criteria are all higher than 0.98, up to 1. Furthermore, the lift and ACI of the extracted rules are also maintained at a high numerical level. The confidence is essentially accuracy. This result shows that the fault diagnosis rules extracted by the DST-ACI discriminant criterion have a high credibility, the mined frequent feature patterns have a strong positive correlation with the faults, and the reliability and accuracy of the conclusions are high.The metrics of a diagnosis rule with the highest confidence are displayed in Table4.Table 4
Fault diagnosis rules with the highest confidence.
Feature numberFault diagnosis rulesConfidenceLiftACIk = 162⟶ 20000140.216k = 257 66⟶ 10000140.216k = 349 53 57⟶ 10000140.216k = 445 49 53 57⟶ 10000140.216k = 517 29 45 49 53⟶ 10000140.216k = 629 45 49 53 57 66⟶ 10000140.216k = 74 19 23 27 39 63 67⟶ 300000.981133.92450.20945In the above table, the number before the arrow is the feature state code, and the number greater than 10000 after the arrow is the fault type code. Among them, 10000 represents the shafting misalignment fault, 20000 represents the rotor imbalance fault, 30000 represents the bearing inner ring fault, and 40000 represents the bearing outer ring fault. In fact, after data analysis, among all the extracted rules, the number of fault diagnosis rules related to the shafting misalignment fault is the largest, followed by the bearing inner ring fault and the rotor imbalance fault, and the bearing outer ring fault is the least.The confidence boxplot, lift boxplot, and ACI boxplot of different fault diagnosis rules withk feature is shown in Figures 16–18.Figure 16
Confidence boxplot.Figure 17
Lift boxplot.Figure 18
ACI boxplot.In the above figures, thex-axis represents the k value of the frequent feature k-pattern, and the y-axis represents confidence, lift, and ACI, respectively. According to the above figures, the following conclusions can be drawn:(1)
The data distribution characteristics of the three metrics of confidence, lift, and ACI are similar. According to the calculation formula, there is a coefficient between the confidence and lift. This coefficient is only related to the transaction number of the fault transactional dataset. When the fault transactional dataset is generated, it can be considered that confidence multiplied by this coefficient is equal to lift. Therefore, there is a linear relationship between the lift and the confidence. Therefore, the data distribution characteristics of confidence and lift are similar. In addition, ACI is a function with lift as a variable, so the data distribution characteristics of ACI and lift are also similar.(2)
Whenk = 1, the dispersion degree of confidence, lift, and ACI of the extracted rule is the largest. When k = 7, the dispersion degree of confidence, lift, and ACI of the extracted rule is the smallest. When k = 1, there are more frequent feature 1-patterns. Therefore, the number of fault diagnosis rules that can be extracted is more, and the metrics difference between different rules is also large. When k reaches a certain level, the number of frequent feature k-patterns will be significantly reduced. In addition, the larger the k is, the fewer fault diagnosis rules can be extracted, and the metrics difference between different rules will be gradually reduced.


### 3.4. Performance Comparison Experiment

In this section, using the same experimental data as the input, the performance between the DST-ACI method and other existing fault diagnosis rule extraction methods will be compared. This section mainly compares the performance differences with different feature state coding methods, different frequent feature pattern discriminant criteria, and different fault diagnosis rule discriminant criteria.


#### 3.4.1. Performance Comparison of Different Feature State Coding Methods

This paper proposes a feature state coding method based onK-means clustering. The other two commonly used coding methods are equal-width discretization coding and equal-density discretization coding. In order to test the performance differences, using the same collected signal data, three different transactional datasets are generated according to the above three methods. Then, three transactional datasets will be used to mine the frequent feature pattern and extract fault diagnosis rules, respectively. The experiment uses the SCL framework, and the threshold is the same as those in Section 3.2. The frequent feature pattern mining results are shown in Tables 5–7.Table 5
Frequent feature pattern mining results based on the equal-density discretization coding method.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 173160160160k = 219160108140.9474k = 311160108137.2727k = 45154127133.6k = 51127127127Table 6
Frequent feature pattern mining results based on the equal-width discretization coding method.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 178532118293.0256k = 215449106199.1333k = 36116106112.1667k = 41108108108Table 7
Frequent feature pattern mining results based on theK-means clustering coding method.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 179401118201.7848k = 276267102161.1711k = 3150263102141.54k = 4165259102129.1152k = 5105232102118.981k = 640222102111.225k = 79119104106k = 81104104104To show the performance difference of different methods more clearly, the key data in the table are shown in Figure19.Figure 19
Average support of different feature state coding methods.According to the above tables and figures, the following conclusions can be drawn:(1)
Based on the equal-density discretization coding method, a total of 109 frequent feature patterns are mined, andk is at most 5. Based on the equal-width discretization coding method, a total of 100 frequent feature patterns are mined, and k is at most 4. Based on the K-means clustering coding method, a total of 625 frequent feature patterns are mined, and k is at most 8. It can be seen that the coding method based on K-means clustering can mine more frequent feature patterns that meet the requirements under the same constraint conditions.(2)
Whenk > 5, the other two discretization coding methods can no longer extract frequent feature patterns, while the K-means clustering method can still mine frequent feature patterns. In addition, the maximum support, minimum support, and average support of the mined frequent feature patterns are maintained at a relatively high numerical level.(3)
Overall, the number of frequent feature patterns mined based onK-means clustering coding method is about 6 times those of the other two coding methods. However, most of the metrics values are maintained at a similar level with the other two discretization coding methods and even have obvious advantages. It can be seen that the frequent feature patterns mined by the coding method based on K-means clustering ensure the accuracy and reliability of the conclusions under the premise of a significant increase in the results’ number. This phenomenon also proves that the transaction generated by this method is more accurate than the other two methods, and the coding method based on K-means clustering is more in line with the matching relationship between the feature value and the fault type.The fault diagnosis rule extraction results are shown in Tables8–10.Table 8
Fault diagnosis rules extraction results based on the equal-density discretization coding method.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1661/0.725/0.868374/2.9/3.4735k = 2191/0.7375/0.851894/2.95/3.4076k = 3110.98148/0.7375/0.785053.9259/2.95/3.1402k = 450.7874/0.75188/0.776063.1496/3.0075/3.1043k = 510.7874/0.7874/0.78743.1496/3.1496/3.1496Table 9
Fault diagnosis rule extraction results based on the equal-width discretization coding method.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1141/0.86286/0.960554/3.4514/3.8422k = 281/0.98148/0.997694/3.9259/3.9907k = 361/1/14/4/4k = 411/1/14/4/4Table 10
Fault diagnosis rule extraction results based on theK-means clustering coding method.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1471/0.73196/0.909554/2.9278/3.6382k = 2491/0.84746/0.931914/3.3898/3.7276k = 31121/0.84746/0.932454/3.3898/3.7298k = 41371/0.84746/0.935284/3.3898/3.7411k = 5961/0.84874/0.942254/3.395/3.769k = 6391/0.85124/0.954614/3.405/3.8185k = 791/0.86555/0.970144/3.4622/3.8806k = 810.98077/0.98077/0.980773.9231/3.9231/3.9231In order to show the performance difference of different methods more clearly, the key data in the table are shown in Figures20 and 21.Figure 20
Average confidence of different feature state coding methods.Figure 21
Average lift of different feature state coding methods.According to the above tables and figures, the following conclusions can be drawn:(1)
Based on the equal-density discretization coding method, a total of 102 rules are extracted, andk is at most 5. Based on the equal-width discretization coding method, a total of 29 rules are extracted, and k is at most 4. Based on the K-means clustering coding method, a total of 490 rules are extracted, and k is at most 8. The K-means clustering coding method can extract the largest number of fault diagnosis rules, and the number of feature states contained in the rules is also the largest. This result shows that the K-means clustering coding method can extract more available fault diagnosis rules while ensuring that the extracted rules meet the threshold constraints.(2)
The maximum confidence and the maximum lift of the fault diagnosis rules extracted by theK-means clustering coding method are not lower than the results of the equal-density discretization method. The minimum confidence, the average confidence, the minimum lift, and the average lift are all significantly greater than the results of the equal-density-based coding method. It can be seen that the transactional datasets generated by the K-means clustering coding method are more accurate than the transactional datasets generated by the coding method based on equal-density discretization, so more reliable and credible fault diagnosis rules can be extracted.(3)
Whenk > 5, the other two discretization coding methods can no longer extract the fault diagnosis rules, and the K-means clustering coding method can still extract the fault diagnosis rules. In addition, when k < 8, the maximum confidence of the extracted fault diagnosis rule can reach 1, and the maximum lift can reach 4. When k = 8, the confidence of the extracted fault diagnosis rule can reach 0.98077, and the lift can reach 3.9231. It can be seen that the transaction generated based on the K-means clustering coding method can not only ensure that the number of the extracted fault diagnosis rules is larger, but also that the mined rules have the higher metrics level. It proves that the K-means clustering coding method has better usability and reliability.(4)
In general, the number of fault diagnosis rules extracted by theK-means clustering coding method is roughly equivalent to 5 times the number of rules extracted based on the equal-density discretization coding method, and 17 times the number of rules extracted based on the equal-width discretization coding method. However, most of the metrics values are maintained at a similar level with the other two discretization coding methods and even have obvious advantages. It can be seen that the fault diagnosis rules extracted by the K-means clustering coding method ensure the accuracy and reliability of the conclusions under the premise of a significant increase in the results’ number. This result also proves that the transactions generated by this coding method are more accurate than the other two methods, and the K-means clustering coding method is more in line with the matching relationship between the feature value and the fault type.


#### 3.4.2. Performance Comparison of Different Frequent Feature Pattern Discriminant Criterion

In order to test the performance difference of different frequent feature pattern mining methods, the methods based on constant support threshold is used to mine the frequent feature patterns by the same collected signal dataset. The minimum support threshold is the same as in Section 7.2. The test results are shown in Tables11 and 12.Table 11
Frequent feature pattern mining results based on the constant support threshold discriminant criterion.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 179401118201.7848k = 276267102161.1711k = 3150263102141.54k = 4165259102129.1152k = 5105232102118.981k = 640222102111.225k = 79119104106k = 81104104104Table 12
Frequent feature pattern mining results based on the DST discriminant criterion.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 179401118201.7848k = 274267106162.7432k = 3120263104149.4083k = 4111259104138.2613k = 557232104127.6316k = 616222104119.0625k = 74119104108.2500To show the performance difference of different methods more clearly, the key data in the table are shown in Figure22.Figure 22
Average support of different frequent feature pattern discriminant criterion.According to the above tables and figure, the following conclusions can be drawn:(1)
Based on the constant support threshold discriminant criterion, a total of 625 frequent feature patterns are mined, andk is at most 8. Based on the DST criterion, a total of 461 frequent feature patterns are mined, and k is at most 7.(2)
In general, under the same support threshold condition, the number of frequent feature patterns mined by the DST-based discriminant criterion is roughly equivalent to 2/3 of the number of frequent feature patterns mined based on the constant support threshold discriminant criterion. However, the minimum support and average support of the frequent feature patterns mined by the DST-based discriminant criterion are both greater than those of the constant support threshold discriminant criterion. The maximum support of frequent feature patterns mined by the method based on the DST criterion is the same as the maximum support of frequent feature patterns mined based on the constant support threshold criterion.It can be seen that, on the basis of testing the support of the pattern, the DST-based discriminant criterion further tests the “relatively frequent co-occurrence” between each feature states in the pattern. Therefore, the patterns with stronger co-occurrence of the feature state and higher support can be extracted.


#### 3.4.3. Performance Comparison of Different Fault Diagnosis Rule Discriminant Criterion

In order to test the performance of different fault diagnosis rule extraction frameworks, using the same collected signal dataset, the three frameworks are used to extract fault diagnosis rules, respectively.The methods used for comparison are confidence-lift-based method and confidence-improint-based method [50]. The threshold is the same as those in Section 3.2. The results are shown in Tables 13–15.Table 13
Fault diagnosis rule extraction results based on the confidence-lift framework.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1471/0.73196/0.909554/2.9278/3.6382k = 2471/0.84746/0.929844/3.3898/3.7194k = 3821/0.84746/0.927024/3.3898/3.7081k = 4831/0.84746/0.934374/3.3898/3.7375k = 5481/0.85124/0.936274/3.405/3.7451k = 6151/0.85124/0.928934/3.405/3.7157k = 740.98113/0.86555/0.952053.9245/3.4622/3.8082Table 14
Fault diagnosis rule extraction results based on the confidence-improint framework.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1471/0.73196/0.909554/2.9278/3.6382k = 2471/0.84746/0.929844/3.3898/3.7194k = 3821/0.84746/0.927024/3.3898/3.7081k = 4831/0.84746/0.934374/3.3898/3.7375k = 5481/0.85124/0.936274/3.405/3.7451k = 6151/0.85124/0.928934/3.405/3.7157k = 740.98113/0.86555/0.952053.9245/3.4622/3.8082Table 15
Fault diagnosis rules extraction results based on the confidence-ACI framework.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1221/0.9557/0.986584/3.8228/3.9463k = 2251/0.98052/0.990424/3.9221/3.9617k = 3421/0.98052/0.991154/3.9221/3.9646k = 4471/0.98052/0.990844/3.9221/3.9633k = 5281/0.98077/0.988544/3.9231/3.9542k = 681/0.98077/0.985764/3.9231/3.943k = 730.98113/0.98077/0.980893.9245/3.9231/3.9236In order to show the performance difference of different methods more clearly, the key data in the table are shown in Figures23 and 24.Figure 23
Average confidence of different fault diagnosis rule discriminant criterion.Figure 24
Average lift of different fault diagnosis rule discriminant criterion.According to the above tables and figures, the following conclusions can be drawn:(1)
The confidence-lift framework extracts a total of 326 fault diagnosis rules, andk is at most 7. The confidence-improint framework extracts a total of 326 fault diagnosis rules, and k is at most 7. The confidence-ACI framework extracts a total of 175 frequent feature patterns, and k is at most 7.(2)
The rules extracted based on the confidence-lift framework are exactly the same as those extracted based on the confidence-improint framework. It means that the improint-based discriminant criterion does not actually play any role. When the number of transactions in the transactional datasets is large, the lift of the inference will generally be very large. If a small value is set as the interestingness threshold in the confidence-improint framework, the improint of almost all the inferences will be greater than the threshold. Under this condition, the improint-based criterion degenerates to the lift-based criterion and loses its meaning.(3)
Regardless of the value ofk, the maximum confidence and maximum lift of the fault diagnosis rules extracted based on the confidence-ACI framework are not less than those of the other two frameworks. Moreover, the minimum confidence, the average confidence, the minimum lift, and the average lift of the fault diagnosis rules extracted based on the confidence-ACI framework are all greater than those of the other two frameworks. This result proves that the positive correlation between the feature states and the fault type of the fault diagnosis rules extracted based on the confidence-ACI framework is generally stronger.


## 3.1. Fault Simulation Experiment Equipment

The fault simulation experiment platform used for fault simulation experiments is shown in Figure5.Figure 5
Fault simulation experiment platform.The experiment platform is composed of a drive motor, a conveyor belt, a rotating shaft, a shafting aluminum plate, a bearing seat, a movable base, and other parts. The three magnetic vibration acceleration sensors are used to collect the signals. For the acceleration sensor, the sensitivity is 100 mv/g, the frequency response range is 0.1 Hz∼10 kHz, and the measuring range is −10 g∼+10 g.The data acquisition equipment used in the experiment is shown in Figure6. The CPU is 4-core 1.91 GHz, the Internet access is 1 gigabit ethernet access, and the RAM is 4 GB.Figure 6
Data acquisition equipment.For the acceleration monitoring module of the data acquisition equipment, the input channel is 4-channel ICP sensor input, the precision is 24 bit, and the sampling rate is optional from 2.56 kHz to 256 kHz.The computer used to save the data is shown in Figure7.Figure 7
The computer to save the data.In the experiment, four types of typical faults, namely, rotor imbalance fault, shafting misalignment fault, bearing inner ring fault, and bearing outer ring fault will be simulated by different fault simulation methods.


### 3.1.1. Imbalance Fault Simulation Experiment

A threaded hole is set on the shafting aluminum plate of the fault simulation experiment platform. The imbalanced counterweight bolt will be screwed into the threaded hole on the aluminum plate to simulate the rotor imbalance fault. The imbalanced counterweight bolt and its installation method are shown in Figure8. The sensor installation position is shown in Figure 9.Figure 8
Imbalanced counterweight bolt.Figure 9
Sensor installation position for the imbalance fault simulation experiment.The No.1 and No.2 measuring points are the positions where the sensor is installed under the ideal measurement conditions. The two measuring points are close to the shafting aluminum plate and the rotating shaft, and the signals collected at the two positions are clearer. No.3 measuring point is located on the base of the experiment platform and has no direct contact with the shafting aluminum plate or the rotating shaft. It can be used to simulate the data acquisition conditions when the sensor is placed on the machine foot. Therefore, the data used in this paper comes from the sensor installed at No.3 measuring point.In the following, for each failure simulation test, the two ideal sensor installation positions will be recorded as No.1 and No.2 measuring points, and the sensor to collect the experiment data will be recorded as No.3. The location and number of the measuring point will be shown in the corresponding figure.


### 3.1.2. Misalignment Fault Simulation Experiment

After the shafting is aligned, the front mandrel screw of the experiment platform remains stationary, and the rear mandrel screw is screwed in to one side, pushing a corner of the movable base to slightly deviate, causing a shafting misalignment fault. The mandrel screw is as shown in Figure10. The sensor installation position is shown in Figure 11.Figure 10
Mandrel screws to move the base.Figure 11
Sensor installation position for misalignment fault simulation experiment.


### 3.1.3. Bearing Inner Ring Fault Simulation Experiment

A normal bearing is installed in the bearing seat at the end of the experiment platform, and it can be replaced with a inner ring fault bearing. The contact surface between the inner ring and the ball of the fault bearing is machined with a wear groove along the axial direction to simulate the damage of the inner ring. The faulty inner ring bearing used in the experiment is shown in Figure12. The sensor installation position is shown in Figure 13.Figure 12
Inner ring fault bearing used in the experiment.Figure 13
Sensor installation position for bearing inner ring fault simulation experiment.For the inner ring fault bearing, the inner diameter of the inner ring is 2.62 cm, the outer diameter of the inner ring is 3.11 cm, the ball diameter is 0.50 cm, the inner diameter of the outer ring is 4.31 cm, and the outer diameter of the outer ring is 5.20 cm. The wear groove machined on the inner ring has a width of 2 mm and a depth of 1 mm.


### 3.1.4. Bearing Outer Ring Fault Simulation Experiment

The normal bearing in the bearing seat at the end of the experiment platform can also be replaced with a outer ring fault bearing. The contact surface between the outer ring and the ball of the fault bearing is machined with wear grooves along the axial direction to simulate the damage of the outer ring. The faulty outer ring bearing used in the experiment is shown in Figure14.Figure 14
Outer ring fault bearing used in the experiment.The parameters of the inner ring and outer ring of the outer ring fault bearing are the same as those of the inner ring fault bearing. The location of the wear groove is on the contact surface between the outer ring and the ball. The sensor installation position of the outer ring fault bearing simulation experiment is the same as the inner ring fault bearing simulation experiment.


## 3.1.1. Imbalance Fault Simulation Experiment

A threaded hole is set on the shafting aluminum plate of the fault simulation experiment platform. The imbalanced counterweight bolt will be screwed into the threaded hole on the aluminum plate to simulate the rotor imbalance fault. The imbalanced counterweight bolt and its installation method are shown in Figure8. The sensor installation position is shown in Figure 9.Figure 8
Imbalanced counterweight bolt.Figure 9
Sensor installation position for the imbalance fault simulation experiment.The No.1 and No.2 measuring points are the positions where the sensor is installed under the ideal measurement conditions. The two measuring points are close to the shafting aluminum plate and the rotating shaft, and the signals collected at the two positions are clearer. No.3 measuring point is located on the base of the experiment platform and has no direct contact with the shafting aluminum plate or the rotating shaft. It can be used to simulate the data acquisition conditions when the sensor is placed on the machine foot. Therefore, the data used in this paper comes from the sensor installed at No.3 measuring point.In the following, for each failure simulation test, the two ideal sensor installation positions will be recorded as No.1 and No.2 measuring points, and the sensor to collect the experiment data will be recorded as No.3. The location and number of the measuring point will be shown in the corresponding figure.


## 3.1.2. Misalignment Fault Simulation Experiment

After the shafting is aligned, the front mandrel screw of the experiment platform remains stationary, and the rear mandrel screw is screwed in to one side, pushing a corner of the movable base to slightly deviate, causing a shafting misalignment fault. The mandrel screw is as shown in Figure10. The sensor installation position is shown in Figure 11.Figure 10
Mandrel screws to move the base.Figure 11
Sensor installation position for misalignment fault simulation experiment.


## 3.1.3. Bearing Inner Ring Fault Simulation Experiment

A normal bearing is installed in the bearing seat at the end of the experiment platform, and it can be replaced with a inner ring fault bearing. The contact surface between the inner ring and the ball of the fault bearing is machined with a wear groove along the axial direction to simulate the damage of the inner ring. The faulty inner ring bearing used in the experiment is shown in Figure12. The sensor installation position is shown in Figure 13.Figure 12
Inner ring fault bearing used in the experiment.Figure 13
Sensor installation position for bearing inner ring fault simulation experiment.For the inner ring fault bearing, the inner diameter of the inner ring is 2.62 cm, the outer diameter of the inner ring is 3.11 cm, the ball diameter is 0.50 cm, the inner diameter of the outer ring is 4.31 cm, and the outer diameter of the outer ring is 5.20 cm. The wear groove machined on the inner ring has a width of 2 mm and a depth of 1 mm.


## 3.1.4. Bearing Outer Ring Fault Simulation Experiment

The normal bearing in the bearing seat at the end of the experiment platform can also be replaced with a outer ring fault bearing. The contact surface between the outer ring and the ball of the fault bearing is machined with wear grooves along the axial direction to simulate the damage of the outer ring. The faulty outer ring bearing used in the experiment is shown in Figure14.Figure 14
Outer ring fault bearing used in the experiment.The parameters of the inner ring and outer ring of the outer ring fault bearing are the same as those of the inner ring fault bearing. The location of the wear groove is on the contact surface between the outer ring and the ball. The sensor installation position of the outer ring fault bearing simulation experiment is the same as the inner ring fault bearing simulation experiment.


## 3.2. Experiment Parameters

In each fault simulation experiment, the data sampling frequency is 25600 Hz, 320, 000 acceleration amplitude data points are collected, and the unit is m2/s. Since four types of faults are simulated, the total sample signal dataset includes 1, 280, 000 acceleration amplitude data points. The sampling points in each type of fault simulation experiment are divided into a group of 2000 points, and the entire collected signal sequence can be divided into 160 groups of signal subsequences. The 23 signal features in Table 1 will be extracted from the collected vibration signals. Therefore, each type of the fault simulation experiment will generate 160 transactions. Each transaction contains 23 feature state codes and 1 fault type code.The corresponding parameter settings involved in the algorithm are as follows: association coefficientp=3, minimum support threshold θmin=100, minimum confidence threshold θC=0.7, minimum interestingness threshold θI=0.2, and average coefficient β=0.6.


## 3.3. Experimental Results

Input the data collected by the above-mentioned fault simulation experiment into the DST-ACI-apriori algorithm, and then the fault diagnosis rules that meet the DST-ACI criterion can be extracted.


### 3.3.1. Results of Frequent Feature Pattern Mining

A total of 461 frequent feature patterns are extracted. The various metric values are shown in Table2.Table 2
Results of frequent feature pattern mining.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 179401118201.7848k = 274267106162.7432k = 3120263104149.4083k = 4111259104138.2613k = 557232104127.6316k = 616222104119.0625k = 74119104108.2500The frequent featurek-pattern contains k feature state codes. The support boxplot of different frequent feature k-patterns is shown in Figure 15.Figure 15
Support boxplot.In the above figure, thex-axis is the k value of the frequent feature k-pattern, and the y-axis is the support. According to the above table and boxplot, the following conclusions can be obtained:(1)
Whenk = 1, there are frequent feature patterns with support close to 400. When k > 1, the support of all the frequent feature patterns are smaller than 300. When k = 1, there is only one feature state code in the pattern, so the support of frequent feature pattern is generally greater than that of the pattern with a larger k. Therefore, it is easy to have the special cases with very large support among them.After checking the result, it is found that only one pattern has a support greater than 400, and the support of this pattern is 401. In addition, there is another pattern whose support of is very close to 400, and the support of this another pattern is 392. However, with the exception of the two frequent feature 1-patterns mentioned above, the support of all the other patterns is smaller than 300.Obviously, the feature states in the two patterns frequently appear in at least three types of fault transactional dataset. The two feature states are widely distributed in multiple types of fault transactional datasets. Therefore, they can be considered as the noise feature states with a low reference value. After further data inspection, it is found that in the frequent feature 2-patterns and the frequent feature patterns with higher dimensions, the above two feature states no longer appear. It can be seen that the frequent feature pattern mining method based on the DST criterion can indeed effectively filter out the noise feature state.(2)
Whenk = 1, the dispersion degree of the support of the frequent feature 1-pattern is very large. The possible reason is that when discriminating the frequent feature 1-patterns, only the support of the pattern is considered, and there is no other restrictions on the discrimination of the frequent feature patterns. In this case, the pattern will be judged as a frequent feature 1-pattern as long as its support exceeds the support threshold. Therefore, when k = 1, there will be various types of frequent feature 1-patterns, and the support dispersion degree of these patterns is difficult to limit.(3)
Ask continues to increase, the support dispersion degree of the frequent feature patterns gradually decreases. On the one hand, as the value of k become larger, the number of frequent feature patterns mined also decrease, and the support difference of patterns with the same dimension will also become smaller. This leads to a more concentrated support distribution of the frequent feature patterns. On the other hand, when k keeps increasing, DST will not only check the “absolute occurrence frequency” of the feature states, but also further check the “relative co-occurrence frequency” between each feature state in the same pattern. Therefore, in the high-dimensional frequent feature patterns, there will be no more significant “unsocial” noise feature state patterns like the mining frequent feature 1-patterns.


### 3.3.2. Result of Fault Diagnosis Rule Extraction

A total of 175 fault diagnosis rules are extracted. The number, maximum confidence, minimum confidence, average confidence, maximum lift, minimum lift, average lift, maximum ACI, minimum ACI, and average ACI of the extracted fault diagnosis rules are shown in Table3.Table 3
Result of fault diagnosis rule extraction.
Feature number in the ruleNumber of the rulesConfidence maximum/minimum/averageLift maximum/minimum/averageACI maximum/minimum/averagek = 1221/0.9557/0.986584/3.8228/3.94630.216/0.20051/0.21133k = 2251/0.98052/0.990424/3.9221/3.96170.216/0.20923/0.21267k = 3421/0.98052/0.991154/3.9221/3.96460.216/0.20923/0.21293k = 4471/0.98052/0.990844/3.9221/3.96330.216/0.20923/0.21282k = 5281/0.98077/0.988544/3.9231/3.95420.216/0.20932/0.21202k = 681/0.98077/0.985764/3.9231/3.9430.216/0.20932/0.21105k = 730.98113/0.98077/0.980893.9245/3.9231/3.92360.20945/0.20932/0.20936According to the above table, it can be seen that the confidence of the fault diagnosis rules extracted based on the DST-ACI discriminant criteria are all higher than 0.98, up to 1. Furthermore, the lift and ACI of the extracted rules are also maintained at a high numerical level. The confidence is essentially accuracy. This result shows that the fault diagnosis rules extracted by the DST-ACI discriminant criterion have a high credibility, the mined frequent feature patterns have a strong positive correlation with the faults, and the reliability and accuracy of the conclusions are high.The metrics of a diagnosis rule with the highest confidence are displayed in Table4.Table 4
Fault diagnosis rules with the highest confidence.
Feature numberFault diagnosis rulesConfidenceLiftACIk = 162⟶ 20000140.216k = 257 66⟶ 10000140.216k = 349 53 57⟶ 10000140.216k = 445 49 53 57⟶ 10000140.216k = 517 29 45 49 53⟶ 10000140.216k = 629 45 49 53 57 66⟶ 10000140.216k = 74 19 23 27 39 63 67⟶ 300000.981133.92450.20945In the above table, the number before the arrow is the feature state code, and the number greater than 10000 after the arrow is the fault type code. Among them, 10000 represents the shafting misalignment fault, 20000 represents the rotor imbalance fault, 30000 represents the bearing inner ring fault, and 40000 represents the bearing outer ring fault. In fact, after data analysis, among all the extracted rules, the number of fault diagnosis rules related to the shafting misalignment fault is the largest, followed by the bearing inner ring fault and the rotor imbalance fault, and the bearing outer ring fault is the least.The confidence boxplot, lift boxplot, and ACI boxplot of different fault diagnosis rules withk feature is shown in Figures 16–18.Figure 16
Confidence boxplot.Figure 17
Lift boxplot.Figure 18
ACI boxplot.In the above figures, thex-axis represents the k value of the frequent feature k-pattern, and the y-axis represents confidence, lift, and ACI, respectively. According to the above figures, the following conclusions can be drawn:(1)
The data distribution characteristics of the three metrics of confidence, lift, and ACI are similar. According to the calculation formula, there is a coefficient between the confidence and lift. This coefficient is only related to the transaction number of the fault transactional dataset. When the fault transactional dataset is generated, it can be considered that confidence multiplied by this coefficient is equal to lift. Therefore, there is a linear relationship between the lift and the confidence. Therefore, the data distribution characteristics of confidence and lift are similar. In addition, ACI is a function with lift as a variable, so the data distribution characteristics of ACI and lift are also similar.(2)
Whenk = 1, the dispersion degree of confidence, lift, and ACI of the extracted rule is the largest. When k = 7, the dispersion degree of confidence, lift, and ACI of the extracted rule is the smallest. When k = 1, there are more frequent feature 1-patterns. Therefore, the number of fault diagnosis rules that can be extracted is more, and the metrics difference between different rules is also large. When k reaches a certain level, the number of frequent feature k-patterns will be significantly reduced. In addition, the larger the k is, the fewer fault diagnosis rules can be extracted, and the metrics difference between different rules will be gradually reduced.


## 3.3.1. Results of Frequent Feature Pattern Mining

A total of 461 frequent feature patterns are extracted. The various metric values are shown in Table2.Table 2
Results of frequent feature pattern mining.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 179401118201.7848k = 274267106162.7432k = 3120263104149.4083k = 4111259104138.2613k = 557232104127.6316k = 616222104119.0625k = 74119104108.2500The frequent featurek-pattern contains k feature state codes. The support boxplot of different frequent feature k-patterns is shown in Figure 15.Figure 15
Support boxplot.In the above figure, thex-axis is the k value of the frequent feature k-pattern, and the y-axis is the support. According to the above table and boxplot, the following conclusions can be obtained:(1)
Whenk = 1, there are frequent feature patterns with support close to 400. When k > 1, the support of all the frequent feature patterns are smaller than 300. When k = 1, there is only one feature state code in the pattern, so the support of frequent feature pattern is generally greater than that of the pattern with a larger k. Therefore, it is easy to have the special cases with very large support among them.After checking the result, it is found that only one pattern has a support greater than 400, and the support of this pattern is 401. In addition, there is another pattern whose support of is very close to 400, and the support of this another pattern is 392. However, with the exception of the two frequent feature 1-patterns mentioned above, the support of all the other patterns is smaller than 300.Obviously, the feature states in the two patterns frequently appear in at least three types of fault transactional dataset. The two feature states are widely distributed in multiple types of fault transactional datasets. Therefore, they can be considered as the noise feature states with a low reference value. After further data inspection, it is found that in the frequent feature 2-patterns and the frequent feature patterns with higher dimensions, the above two feature states no longer appear. It can be seen that the frequent feature pattern mining method based on the DST criterion can indeed effectively filter out the noise feature state.(2)
Whenk = 1, the dispersion degree of the support of the frequent feature 1-pattern is very large. The possible reason is that when discriminating the frequent feature 1-patterns, only the support of the pattern is considered, and there is no other restrictions on the discrimination of the frequent feature patterns. In this case, the pattern will be judged as a frequent feature 1-pattern as long as its support exceeds the support threshold. Therefore, when k = 1, there will be various types of frequent feature 1-patterns, and the support dispersion degree of these patterns is difficult to limit.(3)
Ask continues to increase, the support dispersion degree of the frequent feature patterns gradually decreases. On the one hand, as the value of k become larger, the number of frequent feature patterns mined also decrease, and the support difference of patterns with the same dimension will also become smaller. This leads to a more concentrated support distribution of the frequent feature patterns. On the other hand, when k keeps increasing, DST will not only check the “absolute occurrence frequency” of the feature states, but also further check the “relative co-occurrence frequency” between each feature state in the same pattern. Therefore, in the high-dimensional frequent feature patterns, there will be no more significant “unsocial” noise feature state patterns like the mining frequent feature 1-patterns.


## 3.3.2. Result of Fault Diagnosis Rule Extraction

A total of 175 fault diagnosis rules are extracted. The number, maximum confidence, minimum confidence, average confidence, maximum lift, minimum lift, average lift, maximum ACI, minimum ACI, and average ACI of the extracted fault diagnosis rules are shown in Table3.Table 3
Result of fault diagnosis rule extraction.
Feature number in the ruleNumber of the rulesConfidence maximum/minimum/averageLift maximum/minimum/averageACI maximum/minimum/averagek = 1221/0.9557/0.986584/3.8228/3.94630.216/0.20051/0.21133k = 2251/0.98052/0.990424/3.9221/3.96170.216/0.20923/0.21267k = 3421/0.98052/0.991154/3.9221/3.96460.216/0.20923/0.21293k = 4471/0.98052/0.990844/3.9221/3.96330.216/0.20923/0.21282k = 5281/0.98077/0.988544/3.9231/3.95420.216/0.20932/0.21202k = 681/0.98077/0.985764/3.9231/3.9430.216/0.20932/0.21105k = 730.98113/0.98077/0.980893.9245/3.9231/3.92360.20945/0.20932/0.20936According to the above table, it can be seen that the confidence of the fault diagnosis rules extracted based on the DST-ACI discriminant criteria are all higher than 0.98, up to 1. Furthermore, the lift and ACI of the extracted rules are also maintained at a high numerical level. The confidence is essentially accuracy. This result shows that the fault diagnosis rules extracted by the DST-ACI discriminant criterion have a high credibility, the mined frequent feature patterns have a strong positive correlation with the faults, and the reliability and accuracy of the conclusions are high.The metrics of a diagnosis rule with the highest confidence are displayed in Table4.Table 4
Fault diagnosis rules with the highest confidence.
Feature numberFault diagnosis rulesConfidenceLiftACIk = 162⟶ 20000140.216k = 257 66⟶ 10000140.216k = 349 53 57⟶ 10000140.216k = 445 49 53 57⟶ 10000140.216k = 517 29 45 49 53⟶ 10000140.216k = 629 45 49 53 57 66⟶ 10000140.216k = 74 19 23 27 39 63 67⟶ 300000.981133.92450.20945In the above table, the number before the arrow is the feature state code, and the number greater than 10000 after the arrow is the fault type code. Among them, 10000 represents the shafting misalignment fault, 20000 represents the rotor imbalance fault, 30000 represents the bearing inner ring fault, and 40000 represents the bearing outer ring fault. In fact, after data analysis, among all the extracted rules, the number of fault diagnosis rules related to the shafting misalignment fault is the largest, followed by the bearing inner ring fault and the rotor imbalance fault, and the bearing outer ring fault is the least.The confidence boxplot, lift boxplot, and ACI boxplot of different fault diagnosis rules withk feature is shown in Figures 16–18.Figure 16
Confidence boxplot.Figure 17
Lift boxplot.Figure 18
ACI boxplot.In the above figures, thex-axis represents the k value of the frequent feature k-pattern, and the y-axis represents confidence, lift, and ACI, respectively. According to the above figures, the following conclusions can be drawn:(1)
The data distribution characteristics of the three metrics of confidence, lift, and ACI are similar. According to the calculation formula, there is a coefficient between the confidence and lift. This coefficient is only related to the transaction number of the fault transactional dataset. When the fault transactional dataset is generated, it can be considered that confidence multiplied by this coefficient is equal to lift. Therefore, there is a linear relationship between the lift and the confidence. Therefore, the data distribution characteristics of confidence and lift are similar. In addition, ACI is a function with lift as a variable, so the data distribution characteristics of ACI and lift are also similar.(2)
Whenk = 1, the dispersion degree of confidence, lift, and ACI of the extracted rule is the largest. When k = 7, the dispersion degree of confidence, lift, and ACI of the extracted rule is the smallest. When k = 1, there are more frequent feature 1-patterns. Therefore, the number of fault diagnosis rules that can be extracted is more, and the metrics difference between different rules is also large. When k reaches a certain level, the number of frequent feature k-patterns will be significantly reduced. In addition, the larger the k is, the fewer fault diagnosis rules can be extracted, and the metrics difference between different rules will be gradually reduced.


## 3.4. Performance Comparison Experiment

In this section, using the same experimental data as the input, the performance between the DST-ACI method and other existing fault diagnosis rule extraction methods will be compared. This section mainly compares the performance differences with different feature state coding methods, different frequent feature pattern discriminant criteria, and different fault diagnosis rule discriminant criteria.


### 3.4.1. Performance Comparison of Different Feature State Coding Methods

This paper proposes a feature state coding method based onK-means clustering. The other two commonly used coding methods are equal-width discretization coding and equal-density discretization coding. In order to test the performance differences, using the same collected signal data, three different transactional datasets are generated according to the above three methods. Then, three transactional datasets will be used to mine the frequent feature pattern and extract fault diagnosis rules, respectively. The experiment uses the SCL framework, and the threshold is the same as those in Section 3.2. The frequent feature pattern mining results are shown in Tables 5–7.Table 5
Frequent feature pattern mining results based on the equal-density discretization coding method.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 173160160160k = 219160108140.9474k = 311160108137.2727k = 45154127133.6k = 51127127127Table 6
Frequent feature pattern mining results based on the equal-width discretization coding method.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 178532118293.0256k = 215449106199.1333k = 36116106112.1667k = 41108108108Table 7
Frequent feature pattern mining results based on theK-means clustering coding method.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 179401118201.7848k = 276267102161.1711k = 3150263102141.54k = 4165259102129.1152k = 5105232102118.981k = 640222102111.225k = 79119104106k = 81104104104To show the performance difference of different methods more clearly, the key data in the table are shown in Figure19.Figure 19
Average support of different feature state coding methods.According to the above tables and figures, the following conclusions can be drawn:(1)
Based on the equal-density discretization coding method, a total of 109 frequent feature patterns are mined, andk is at most 5. Based on the equal-width discretization coding method, a total of 100 frequent feature patterns are mined, and k is at most 4. Based on the K-means clustering coding method, a total of 625 frequent feature patterns are mined, and k is at most 8. It can be seen that the coding method based on K-means clustering can mine more frequent feature patterns that meet the requirements under the same constraint conditions.(2)
Whenk > 5, the other two discretization coding methods can no longer extract frequent feature patterns, while the K-means clustering method can still mine frequent feature patterns. In addition, the maximum support, minimum support, and average support of the mined frequent feature patterns are maintained at a relatively high numerical level.(3)
Overall, the number of frequent feature patterns mined based onK-means clustering coding method is about 6 times those of the other two coding methods. However, most of the metrics values are maintained at a similar level with the other two discretization coding methods and even have obvious advantages. It can be seen that the frequent feature patterns mined by the coding method based on K-means clustering ensure the accuracy and reliability of the conclusions under the premise of a significant increase in the results’ number. This phenomenon also proves that the transaction generated by this method is more accurate than the other two methods, and the coding method based on K-means clustering is more in line with the matching relationship between the feature value and the fault type.The fault diagnosis rule extraction results are shown in Tables8–10.Table 8
Fault diagnosis rules extraction results based on the equal-density discretization coding method.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1661/0.725/0.868374/2.9/3.4735k = 2191/0.7375/0.851894/2.95/3.4076k = 3110.98148/0.7375/0.785053.9259/2.95/3.1402k = 450.7874/0.75188/0.776063.1496/3.0075/3.1043k = 510.7874/0.7874/0.78743.1496/3.1496/3.1496Table 9
Fault diagnosis rule extraction results based on the equal-width discretization coding method.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1141/0.86286/0.960554/3.4514/3.8422k = 281/0.98148/0.997694/3.9259/3.9907k = 361/1/14/4/4k = 411/1/14/4/4Table 10
Fault diagnosis rule extraction results based on theK-means clustering coding method.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1471/0.73196/0.909554/2.9278/3.6382k = 2491/0.84746/0.931914/3.3898/3.7276k = 31121/0.84746/0.932454/3.3898/3.7298k = 41371/0.84746/0.935284/3.3898/3.7411k = 5961/0.84874/0.942254/3.395/3.769k = 6391/0.85124/0.954614/3.405/3.8185k = 791/0.86555/0.970144/3.4622/3.8806k = 810.98077/0.98077/0.980773.9231/3.9231/3.9231In order to show the performance difference of different methods more clearly, the key data in the table are shown in Figures20 and 21.Figure 20
Average confidence of different feature state coding methods.Figure 21
Average lift of different feature state coding methods.According to the above tables and figures, the following conclusions can be drawn:(1)
Based on the equal-density discretization coding method, a total of 102 rules are extracted, andk is at most 5. Based on the equal-width discretization coding method, a total of 29 rules are extracted, and k is at most 4. Based on the K-means clustering coding method, a total of 490 rules are extracted, and k is at most 8. The K-means clustering coding method can extract the largest number of fault diagnosis rules, and the number of feature states contained in the rules is also the largest. This result shows that the K-means clustering coding method can extract more available fault diagnosis rules while ensuring that the extracted rules meet the threshold constraints.(2)
The maximum confidence and the maximum lift of the fault diagnosis rules extracted by theK-means clustering coding method are not lower than the results of the equal-density discretization method. The minimum confidence, the average confidence, the minimum lift, and the average lift are all significantly greater than the results of the equal-density-based coding method. It can be seen that the transactional datasets generated by the K-means clustering coding method are more accurate than the transactional datasets generated by the coding method based on equal-density discretization, so more reliable and credible fault diagnosis rules can be extracted.(3)
Whenk > 5, the other two discretization coding methods can no longer extract the fault diagnosis rules, and the K-means clustering coding method can still extract the fault diagnosis rules. In addition, when k < 8, the maximum confidence of the extracted fault diagnosis rule can reach 1, and the maximum lift can reach 4. When k = 8, the confidence of the extracted fault diagnosis rule can reach 0.98077, and the lift can reach 3.9231. It can be seen that the transaction generated based on the K-means clustering coding method can not only ensure that the number of the extracted fault diagnosis rules is larger, but also that the mined rules have the higher metrics level. It proves that the K-means clustering coding method has better usability and reliability.(4)
In general, the number of fault diagnosis rules extracted by theK-means clustering coding method is roughly equivalent to 5 times the number of rules extracted based on the equal-density discretization coding method, and 17 times the number of rules extracted based on the equal-width discretization coding method. However, most of the metrics values are maintained at a similar level with the other two discretization coding methods and even have obvious advantages. It can be seen that the fault diagnosis rules extracted by the K-means clustering coding method ensure the accuracy and reliability of the conclusions under the premise of a significant increase in the results’ number. This result also proves that the transactions generated by this coding method are more accurate than the other two methods, and the K-means clustering coding method is more in line with the matching relationship between the feature value and the fault type.


### 3.4.2. Performance Comparison of Different Frequent Feature Pattern Discriminant Criterion

In order to test the performance difference of different frequent feature pattern mining methods, the methods based on constant support threshold is used to mine the frequent feature patterns by the same collected signal dataset. The minimum support threshold is the same as in Section 7.2. The test results are shown in Tables11 and 12.Table 11
Frequent feature pattern mining results based on the constant support threshold discriminant criterion.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 179401118201.7848k = 276267102161.1711k = 3150263102141.54k = 4165259102129.1152k = 5105232102118.981k = 640222102111.225k = 79119104106k = 81104104104Table 12
Frequent feature pattern mining results based on the DST discriminant criterion.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 179401118201.7848k = 274267106162.7432k = 3120263104149.4083k = 4111259104138.2613k = 557232104127.6316k = 616222104119.0625k = 74119104108.2500To show the performance difference of different methods more clearly, the key data in the table are shown in Figure22.Figure 22
Average support of different frequent feature pattern discriminant criterion.According to the above tables and figure, the following conclusions can be drawn:(1)
Based on the constant support threshold discriminant criterion, a total of 625 frequent feature patterns are mined, andk is at most 8. Based on the DST criterion, a total of 461 frequent feature patterns are mined, and k is at most 7.(2)
In general, under the same support threshold condition, the number of frequent feature patterns mined by the DST-based discriminant criterion is roughly equivalent to 2/3 of the number of frequent feature patterns mined based on the constant support threshold discriminant criterion. However, the minimum support and average support of the frequent feature patterns mined by the DST-based discriminant criterion are both greater than those of the constant support threshold discriminant criterion. The maximum support of frequent feature patterns mined by the method based on the DST criterion is the same as the maximum support of frequent feature patterns mined based on the constant support threshold criterion.It can be seen that, on the basis of testing the support of the pattern, the DST-based discriminant criterion further tests the “relatively frequent co-occurrence” between each feature states in the pattern. Therefore, the patterns with stronger co-occurrence of the feature state and higher support can be extracted.


### 3.4.3. Performance Comparison of Different Fault Diagnosis Rule Discriminant Criterion

In order to test the performance of different fault diagnosis rule extraction frameworks, using the same collected signal dataset, the three frameworks are used to extract fault diagnosis rules, respectively.The methods used for comparison are confidence-lift-based method and confidence-improint-based method [50]. The threshold is the same as those in Section 3.2. The results are shown in Tables 13–15.Table 13
Fault diagnosis rule extraction results based on the confidence-lift framework.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1471/0.73196/0.909554/2.9278/3.6382k = 2471/0.84746/0.929844/3.3898/3.7194k = 3821/0.84746/0.927024/3.3898/3.7081k = 4831/0.84746/0.934374/3.3898/3.7375k = 5481/0.85124/0.936274/3.405/3.7451k = 6151/0.85124/0.928934/3.405/3.7157k = 740.98113/0.86555/0.952053.9245/3.4622/3.8082Table 14
Fault diagnosis rule extraction results based on the confidence-improint framework.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1471/0.73196/0.909554/2.9278/3.6382k = 2471/0.84746/0.929844/3.3898/3.7194k = 3821/0.84746/0.927024/3.3898/3.7081k = 4831/0.84746/0.934374/3.3898/3.7375k = 5481/0.85124/0.936274/3.405/3.7451k = 6151/0.85124/0.928934/3.405/3.7157k = 740.98113/0.86555/0.952053.9245/3.4622/3.8082Table 15
Fault diagnosis rules extraction results based on the confidence-ACI framework.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1221/0.9557/0.986584/3.8228/3.9463k = 2251/0.98052/0.990424/3.9221/3.9617k = 3421/0.98052/0.991154/3.9221/3.9646k = 4471/0.98052/0.990844/3.9221/3.9633k = 5281/0.98077/0.988544/3.9231/3.9542k = 681/0.98077/0.985764/3.9231/3.943k = 730.98113/0.98077/0.980893.9245/3.9231/3.9236In order to show the performance difference of different methods more clearly, the key data in the table are shown in Figures23 and 24.Figure 23
Average confidence of different fault diagnosis rule discriminant criterion.Figure 24
Average lift of different fault diagnosis rule discriminant criterion.According to the above tables and figures, the following conclusions can be drawn:(1)
The confidence-lift framework extracts a total of 326 fault diagnosis rules, andk is at most 7. The confidence-improint framework extracts a total of 326 fault diagnosis rules, and k is at most 7. The confidence-ACI framework extracts a total of 175 frequent feature patterns, and k is at most 7.(2)
The rules extracted based on the confidence-lift framework are exactly the same as those extracted based on the confidence-improint framework. It means that the improint-based discriminant criterion does not actually play any role. When the number of transactions in the transactional datasets is large, the lift of the inference will generally be very large. If a small value is set as the interestingness threshold in the confidence-improint framework, the improint of almost all the inferences will be greater than the threshold. Under this condition, the improint-based criterion degenerates to the lift-based criterion and loses its meaning.(3)
Regardless of the value ofk, the maximum confidence and maximum lift of the fault diagnosis rules extracted based on the confidence-ACI framework are not less than those of the other two frameworks. Moreover, the minimum confidence, the average confidence, the minimum lift, and the average lift of the fault diagnosis rules extracted based on the confidence-ACI framework are all greater than those of the other two frameworks. This result proves that the positive correlation between the feature states and the fault type of the fault diagnosis rules extracted based on the confidence-ACI framework is generally stronger.


## 3.4.1. Performance Comparison of Different Feature State Coding Methods

This paper proposes a feature state coding method based onK-means clustering. The other two commonly used coding methods are equal-width discretization coding and equal-density discretization coding. In order to test the performance differences, using the same collected signal data, three different transactional datasets are generated according to the above three methods. Then, three transactional datasets will be used to mine the frequent feature pattern and extract fault diagnosis rules, respectively. The experiment uses the SCL framework, and the threshold is the same as those in Section 3.2. The frequent feature pattern mining results are shown in Tables 5–7.Table 5
Frequent feature pattern mining results based on the equal-density discretization coding method.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 173160160160k = 219160108140.9474k = 311160108137.2727k = 45154127133.6k = 51127127127Table 6
Frequent feature pattern mining results based on the equal-width discretization coding method.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 178532118293.0256k = 215449106199.1333k = 36116106112.1667k = 41108108108Table 7
Frequent feature pattern mining results based on theK-means clustering coding method.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 179401118201.7848k = 276267102161.1711k = 3150263102141.54k = 4165259102129.1152k = 5105232102118.981k = 640222102111.225k = 79119104106k = 81104104104To show the performance difference of different methods more clearly, the key data in the table are shown in Figure19.Figure 19
Average support of different feature state coding methods.According to the above tables and figures, the following conclusions can be drawn:(1)
Based on the equal-density discretization coding method, a total of 109 frequent feature patterns are mined, andk is at most 5. Based on the equal-width discretization coding method, a total of 100 frequent feature patterns are mined, and k is at most 4. Based on the K-means clustering coding method, a total of 625 frequent feature patterns are mined, and k is at most 8. It can be seen that the coding method based on K-means clustering can mine more frequent feature patterns that meet the requirements under the same constraint conditions.(2)
Whenk > 5, the other two discretization coding methods can no longer extract frequent feature patterns, while the K-means clustering method can still mine frequent feature patterns. In addition, the maximum support, minimum support, and average support of the mined frequent feature patterns are maintained at a relatively high numerical level.(3)
Overall, the number of frequent feature patterns mined based onK-means clustering coding method is about 6 times those of the other two coding methods. However, most of the metrics values are maintained at a similar level with the other two discretization coding methods and even have obvious advantages. It can be seen that the frequent feature patterns mined by the coding method based on K-means clustering ensure the accuracy and reliability of the conclusions under the premise of a significant increase in the results’ number. This phenomenon also proves that the transaction generated by this method is more accurate than the other two methods, and the coding method based on K-means clustering is more in line with the matching relationship between the feature value and the fault type.The fault diagnosis rule extraction results are shown in Tables8–10.Table 8
Fault diagnosis rules extraction results based on the equal-density discretization coding method.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1661/0.725/0.868374/2.9/3.4735k = 2191/0.7375/0.851894/2.95/3.4076k = 3110.98148/0.7375/0.785053.9259/2.95/3.1402k = 450.7874/0.75188/0.776063.1496/3.0075/3.1043k = 510.7874/0.7874/0.78743.1496/3.1496/3.1496Table 9
Fault diagnosis rule extraction results based on the equal-width discretization coding method.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1141/0.86286/0.960554/3.4514/3.8422k = 281/0.98148/0.997694/3.9259/3.9907k = 361/1/14/4/4k = 411/1/14/4/4Table 10
Fault diagnosis rule extraction results based on theK-means clustering coding method.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1471/0.73196/0.909554/2.9278/3.6382k = 2491/0.84746/0.931914/3.3898/3.7276k = 31121/0.84746/0.932454/3.3898/3.7298k = 41371/0.84746/0.935284/3.3898/3.7411k = 5961/0.84874/0.942254/3.395/3.769k = 6391/0.85124/0.954614/3.405/3.8185k = 791/0.86555/0.970144/3.4622/3.8806k = 810.98077/0.98077/0.980773.9231/3.9231/3.9231In order to show the performance difference of different methods more clearly, the key data in the table are shown in Figures20 and 21.Figure 20
Average confidence of different feature state coding methods.Figure 21
Average lift of different feature state coding methods.According to the above tables and figures, the following conclusions can be drawn:(1)
Based on the equal-density discretization coding method, a total of 102 rules are extracted, andk is at most 5. Based on the equal-width discretization coding method, a total of 29 rules are extracted, and k is at most 4. Based on the K-means clustering coding method, a total of 490 rules are extracted, and k is at most 8. The K-means clustering coding method can extract the largest number of fault diagnosis rules, and the number of feature states contained in the rules is also the largest. This result shows that the K-means clustering coding method can extract more available fault diagnosis rules while ensuring that the extracted rules meet the threshold constraints.(2)
The maximum confidence and the maximum lift of the fault diagnosis rules extracted by theK-means clustering coding method are not lower than the results of the equal-density discretization method. The minimum confidence, the average confidence, the minimum lift, and the average lift are all significantly greater than the results of the equal-density-based coding method. It can be seen that the transactional datasets generated by the K-means clustering coding method are more accurate than the transactional datasets generated by the coding method based on equal-density discretization, so more reliable and credible fault diagnosis rules can be extracted.(3)
Whenk > 5, the other two discretization coding methods can no longer extract the fault diagnosis rules, and the K-means clustering coding method can still extract the fault diagnosis rules. In addition, when k < 8, the maximum confidence of the extracted fault diagnosis rule can reach 1, and the maximum lift can reach 4. When k = 8, the confidence of the extracted fault diagnosis rule can reach 0.98077, and the lift can reach 3.9231. It can be seen that the transaction generated based on the K-means clustering coding method can not only ensure that the number of the extracted fault diagnosis rules is larger, but also that the mined rules have the higher metrics level. It proves that the K-means clustering coding method has better usability and reliability.(4)
In general, the number of fault diagnosis rules extracted by theK-means clustering coding method is roughly equivalent to 5 times the number of rules extracted based on the equal-density discretization coding method, and 17 times the number of rules extracted based on the equal-width discretization coding method. However, most of the metrics values are maintained at a similar level with the other two discretization coding methods and even have obvious advantages. It can be seen that the fault diagnosis rules extracted by the K-means clustering coding method ensure the accuracy and reliability of the conclusions under the premise of a significant increase in the results’ number. This result also proves that the transactions generated by this coding method are more accurate than the other two methods, and the K-means clustering coding method is more in line with the matching relationship between the feature value and the fault type.


## 3.4.2. Performance Comparison of Different Frequent Feature Pattern Discriminant Criterion

In order to test the performance difference of different frequent feature pattern mining methods, the methods based on constant support threshold is used to mine the frequent feature patterns by the same collected signal dataset. The minimum support threshold is the same as in Section 7.2. The test results are shown in Tables11 and 12.Table 11
Frequent feature pattern mining results based on the constant support threshold discriminant criterion.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 179401118201.7848k = 276267102161.1711k = 3150263102141.54k = 4165259102129.1152k = 5105232102118.981k = 640222102111.225k = 79119104106k = 81104104104Table 12
Frequent feature pattern mining results based on the DST discriminant criterion.
Frequent featurek-patternsNumberMaximum supportMinimum supportAverage supportk = 179401118201.7848k = 274267106162.7432k = 3120263104149.4083k = 4111259104138.2613k = 557232104127.6316k = 616222104119.0625k = 74119104108.2500To show the performance difference of different methods more clearly, the key data in the table are shown in Figure22.Figure 22
Average support of different frequent feature pattern discriminant criterion.According to the above tables and figure, the following conclusions can be drawn:(1)
Based on the constant support threshold discriminant criterion, a total of 625 frequent feature patterns are mined, andk is at most 8. Based on the DST criterion, a total of 461 frequent feature patterns are mined, and k is at most 7.(2)
In general, under the same support threshold condition, the number of frequent feature patterns mined by the DST-based discriminant criterion is roughly equivalent to 2/3 of the number of frequent feature patterns mined based on the constant support threshold discriminant criterion. However, the minimum support and average support of the frequent feature patterns mined by the DST-based discriminant criterion are both greater than those of the constant support threshold discriminant criterion. The maximum support of frequent feature patterns mined by the method based on the DST criterion is the same as the maximum support of frequent feature patterns mined based on the constant support threshold criterion.It can be seen that, on the basis of testing the support of the pattern, the DST-based discriminant criterion further tests the “relatively frequent co-occurrence” between each feature states in the pattern. Therefore, the patterns with stronger co-occurrence of the feature state and higher support can be extracted.


## 3.4.3. Performance Comparison of Different Fault Diagnosis Rule Discriminant Criterion

In order to test the performance of different fault diagnosis rule extraction frameworks, using the same collected signal dataset, the three frameworks are used to extract fault diagnosis rules, respectively.The methods used for comparison are confidence-lift-based method and confidence-improint-based method [50]. The threshold is the same as those in Section 3.2. The results are shown in Tables 13–15.Table 13
Fault diagnosis rule extraction results based on the confidence-lift framework.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1471/0.73196/0.909554/2.9278/3.6382k = 2471/0.84746/0.929844/3.3898/3.7194k = 3821/0.84746/0.927024/3.3898/3.7081k = 4831/0.84746/0.934374/3.3898/3.7375k = 5481/0.85124/0.936274/3.405/3.7451k = 6151/0.85124/0.928934/3.405/3.7157k = 740.98113/0.86555/0.952053.9245/3.4622/3.8082Table 14
Fault diagnosis rule extraction results based on the confidence-improint framework.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1471/0.73196/0.909554/2.9278/3.6382k = 2471/0.84746/0.929844/3.3898/3.7194k = 3821/0.84746/0.927024/3.3898/3.7081k = 4831/0.84746/0.934374/3.3898/3.7375k = 5481/0.85124/0.936274/3.405/3.7451k = 6151/0.85124/0.928934/3.405/3.7157k = 740.98113/0.86555/0.952053.9245/3.4622/3.8082Table 15
Fault diagnosis rules extraction results based on the confidence-ACI framework.
Feature numberNumberConfidence maximum/minimum/averageLift maximum/minimum/averagek = 1221/0.9557/0.986584/3.8228/3.9463k = 2251/0.98052/0.990424/3.9221/3.9617k = 3421/0.98052/0.991154/3.9221/3.9646k = 4471/0.98052/0.990844/3.9221/3.9633k = 5281/0.98077/0.988544/3.9231/3.9542k = 681/0.98077/0.985764/3.9231/3.943k = 730.98113/0.98077/0.980893.9245/3.9231/3.9236In order to show the performance difference of different methods more clearly, the key data in the table are shown in Figures23 and 24.Figure 23
Average confidence of different fault diagnosis rule discriminant criterion.Figure 24
Average lift of different fault diagnosis rule discriminant criterion.According to the above tables and figures, the following conclusions can be drawn:(1)
The confidence-lift framework extracts a total of 326 fault diagnosis rules, andk is at most 7. The confidence-improint framework extracts a total of 326 fault diagnosis rules, and k is at most 7. The confidence-ACI framework extracts a total of 175 frequent feature patterns, and k is at most 7.(2)
The rules extracted based on the confidence-lift framework are exactly the same as those extracted based on the confidence-improint framework. It means that the improint-based discriminant criterion does not actually play any role. When the number of transactions in the transactional datasets is large, the lift of the inference will generally be very large. If a small value is set as the interestingness threshold in the confidence-improint framework, the improint of almost all the inferences will be greater than the threshold. Under this condition, the improint-based criterion degenerates to the lift-based criterion and loses its meaning.(3)
Regardless of the value ofk, the maximum confidence and maximum lift of the fault diagnosis rules extracted based on the confidence-ACI framework are not less than those of the other two frameworks. Moreover, the minimum confidence, the average confidence, the minimum lift, and the average lift of the fault diagnosis rules extracted based on the confidence-ACI framework are all greater than those of the other two frameworks. This result proves that the positive correlation between the feature states and the fault type of the fault diagnosis rules extracted based on the confidence-ACI framework is generally stronger.


## 4. Conclusions and Future Works

This paper proposes a vibrational signal fault diagnosis rule extraction method based on the DST-ACI discriminant criterion for the machine foot vibration signal. The proposed method includes three innovations, namely, the feature state coding method based onK-means clustering, frequent feature pattern mining method based on dynamic support threshold (DST) discriminant criterion, and the fault diagnosis rules extraction method based on the association coefficient interestingness (ACI) discriminant criterion. The fault simulation experiment was carried out using a fault simulation experiment platform. The performance of the proposed method was tested using the collected data. The results show that the confidence of the fault diagnosis rules extracted based on the DST-ACI discriminant criteria are all higher than 0.98, up to 1. The overall metrics value level of the support, confidence, and ACI of the proposed method is higher than the existing six methods.At present, in the research of fault diagnosis based on vibration signals, sensors used to monitor the equipment are often installed at key locations sensitive to vibration signals or close to the fault source. In the literature involving the fault diagnosis under the strong noisy condition, there is no research on the noise interference caused by the sensor installation position. The fault diagnosis research on the machine foot vibration signal is still in its infancy. This topic lacks the sufficient literature support and research results. Therefore, there are a lot of research directions that can be explored in the future.First, the measuring point of the machine foot is far away from the fault source. As a result, the fault signal will be attenuated during the transmission process, and the noise interference will be more. Therefore, the structure and dynamic characteristics of mechanical equipment can be explored with the help of modeling techniques such as finite element [52]. On this basis, the fault mechanism and the signal transmission process can be explored, so as to further study the fault diagnosis method for the machine foot signal.Second, in the early stage of equipment fault, generally only a certain type of fault will occur first. As the equipment operating time increases, due to the coupling of faulty parts with other parts, secondary fault will occur. Compared with the signal generated by a single type of fault, the signal generated by the compound fault will be more complex [53–55]. In this case, the technical method of using the machine foot vibration signal to diagnose the compound fault should be studied in the future.Third, deep learning technology has been widely used in various fields, and the use of neural networks to distinguish equipment fault has also become a very valuable research direction [56, 57]. The neural network can classify the complex input data by constructing a high-dimensional classifier. The convolutional neural network that is often used for deep learning even has the ability to automatically extract high-dimensional features [58–60]. Therefore, the fault diagnosis of the machine foot vibration signal based on deep learning will also become an important research topic. [61–69].



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