Planetary gearbox fault diagnosis based on data-driven valued characteristic multigranulation model with incomplete diagnostic information

Journal of Sound and Vibration 429 (2018) 63e77

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Planetary gearbox fault diagnosis based on data-driven valued characteristic multigranulation model with incomplete diagnostic information Jun Yu a, Yongjun He b, * a b

School of Automation, Harbin University of Science and Technology, Harbin 150080, China School of Computer Science and Technology, Harbin University of Science and Technology, Harbin 150080, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 31 December 2017 Received in revised form 11 May 2018 Accepted 12 May 2018 Available online 24 May 2018 Handling Editor: K. Shin

There are many uncertain factors that may result in incomplete diagnostic information of planetary gearboxes, such as sensor malfunctions, communication lags, and data discretization, etc. Therefore, incomplete diagnostic information of planetary gearboxes may simultaneously contain two categories of unknown attribute values. However, existing fault diagnosis methods of planetary gearboxes are hard to realize fault diagnosis using incomplete diagnostic information that simultaneously contains two categories of unknown attribute values. To overcome this issue, a fault diagnosis method of planetary gearboxes based on data-driven valued characteristic multigranulation model with incomplete diagnostic information is proposed. First, a calculation method of characteristic similarity degrees among cases is introduced, and a data-driven valued characteristic relation is defined. The data-driven valued characteristic relation is used to analyze and process incomplete diagnostic information that simultaneously contains two categories of unknown attribute values. Then, a data-driven valued characteristic multigranulation model is defined according to multigranulation model. An attribute reduction algorithm based on pessimistic data-driven valued characteristic multigranulation model is employed to extract fault diagnosis decision rules. Finally, naive Bayesian classifier is constructed to identify planetary gearbox conditions. The effectiveness of this method is validated and the advantages are investigated using a fault diagnosis experiment of planetary gearbox. Experimental results demonstrate that this method can accurately determine indiscernibility relation among cases, reduce computational complexity, and enhance fault diagnosis accuracy. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Planetary gearbox Data-driven Valued characteristic relation Multigranulation model Fault diagnosis

1. Introduction Planetary gearboxes have been extensively applied in complex mechanical equipment, such as wind turbines, helicopters, and power plants to derive benefits from their merits of compact structure, large transmission ratio, smooth operation, and high transmission efficiency [1e3]. Because of heavy load, high speed, and harsh working environment, planetary gearboxes

* Corresponding author. E-mail address: [email protected] (Y. He). https://doi.org/10.1016/j.jsv.2018.05.020 0022-460X/© 2018 Elsevier Ltd. All rights reserved.

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are prone to various faults [4,5]. Such faults may result in the entire system shutdown, huge economic losses and even casualties. Consequently, fault diagnosis of planetary gearboxes has played an important role to avoid catastrophic accidents. In recent years, major efforts have been made in fault diagnosis of planetary gearboxes. Researchers have proposed many fault diagnosis methods. Among these fault diagnosis methods, data-driven fault diagnosis methods remove the requirement of the prior knowledge and accurate dynamic models. Therefore, they have been extensively used in planetary gearbox fault diagnosis. In the data-driven fault diagnosis methods, feature space is mapped into decision space, and functional relation is constructed between the two spaces to identify fault modes. Typical data-driven fault diagnosis methods of planetary gearboxes consist of deep neural networks (DNNs), fuzzy logic (FL) scheme, and support vector machine (SVM), etc. DNNs with deep architectures have been widely applied in machine fault diagnosis owing to their strong representation ability and simple structure [6e9]. Jia et al. [7] proposed a novel DNN-based method for rotating machinery fault diagnosis. A tangent function is regarded as the active function of DNNs. A five-layer DNN is constructed to identify fault types. Planetary gearbox dataset was used to assess the effectiveness of the proposed method. Compared to traditional methods, this method obtains the highest diagnosis accuracy. Chen et al. [8] presented an integrated fault diagnosis scheme of planetary gearboxes based on deep belief networks (DBNs) in order to enhance diagnosis accuracy. A dimensionality reduction algorithm is introduced to acquire a sensitive and lower dimensional feature set. Then these features are regarded as the input of DBNs to classify planetary gearbox fault modes. Experimental results demonstrate the presented scheme possesses high classification accuracy. Wang et al. [9] put forward a wind turbine gearbox fault diagnosis approach using DNNs. A DNN is used as a prediction model to identify impending faults. Compared to other data-driven fault diagnosis methods, the DNN model possesses the highest accuracy without high computational complexity. However, training of DNNs' structure and parameters requires a great of training samples. Slow convergence speed during training remains unsolved. FL scheme is a common inference strategy for fusing identification results, which has been successfully applied in mechanical system fault identification [10e12]. Chen et al. [11] proposed a remaining useful life (RUL) prediction approach based on neuro-fuzzy inference. A UH-60 helicopter planetary gearbox plate was used to evaluate the proposed approach. Experimental results indicate that the diagnosis accuracy of the proposed approach outperforms conventional RUL prediction method. Chen et al. [12] introduced a diagnosis method using fuzzy entropy and adaptive neuro-fuzzy inference system (ANFIS) to identify planetary gearbox faults. Fuzzy entropies are regarded as the input of ANFIS model. Although FL scheme can achieve diagnostic results by fuzzy inference rules, the process of acquiring fuzzy rules may be time-consuming. Furthermore, these rules from human's expertise are not reliable. SVM, as a powerful machine learning algorithm [13,14], has been used for planetary gearbox fault diagnosis owing to its high accuracy and excellent generalization ability. Liu et al. [15] introduced a feature ranking criterion of multi-class SVM. Feature effectiveness is estimated by its contribution to mode identification, which is evaluated by a kernel function. Lei et al. [16] presented a fault diagnosis method using mRVM to classify seven conditions of planetary gearboxes. Accumulative amplitudes and energy ratio are adopted as the input of the mRVM to identify the seven conditions. Experimental results demonstrate this method possesses high classification accuracy and good robustness. Li et al. [17] put forward a fault diagnosis scheme of planetary gearboxes combining modified multi-scale symbolic dynamic entropy and minimum redundancy maximum relevance (mRMR). Refined fault features by mRMR are considered as the input of least square support vector machine to identify fault patterns. Nevertheless, the procedure of optimizing kernel function parameters is very complicated and many optimization algorithms are not satisfactory. Data-driven fault diagnosis methods do not require additional prior knowledge or accurate analytical models, and classify fault modes by training samples. Therefore, data-driven fault diagnosis methods possess important application value in planetary gearbox fault diagnosis. However, there are many uncertain factors that may result in incomplete diagnostic information of planetary gearboxes, such as sensor malfunctions, communication lags, and data discretization, etc. It brings a great challenge to the application of data-driven fault diagnosis methods. In order to deal with this problem, Wang et al. [18] proposed a rule extraction method from incomplete decision table using granular computing for fault diagnosis of helicopter gearbox. Semantic analysis of missing attribute values is performed by characteristic relation. Then, information granules are built to obtain optimal decision rules. These decision rules can be utilized to recognize fault conditions. Wang et al. [19] presented a rule extraction method based on maximum characteristic granule (MCG) for fault diagnosis from incomplete historical test records. MCGs defined according to characteristic relation are employed to construct a resolution function matrix. General decision rules are acquired by equivalent forms of propositional logic. It can be noted that decision rule extraction is the key of planetary gearbox fault diagnosis with incomplete diagnostic information. However, most of the existing rule extraction methods can be utilized to process incomplete diagnostic information that only contains one category of unknown attribute values. Although some rule extraction methods can acquire diagnosis knowledge from incomplete diagnostic information that simultaneously contains two categories of unknown attribute values, the similarity degrees among cases cannot be quantitatively described. It results that the extraction decision rules are not reliable. To overcome this issue, a fault diagnosis method of planetary gearboxes based on data-driven valued characteristic multigranulation model with incomplete diagnostic information is proposed in this paper. First, a data-driven valued characteristic relation is defined to analyze and process incomplete diagnostic information that simultaneously contains two categories of unknown attribute values. Then, an attribute reduction algorithm based on pessimistic data-driven valued characteristic multigranulation model is employed to extract fault diagnosis decision rules. Finally, naive Bayesian classifier (NBC) is constructed to identify planetary gearbox conditions. Experimental results demonstrate that this method can

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accurately determine indiscernibility relation among cases, reduce computational complexity, and enhance fault diagnosis accuracy. The remainder of this paper is organized as follows. In Section 2, the preliminaries of characteristic relation and multigranulation model are briefly introduced. An attribute reduction algorithm based on pessimistic data-driven valued characteristic multigranulation model is presented in Section 3. Section 4 proposes a fault diagnosis method of planetary gearboxes based on data-driven valued characteristic multigranulation model with incomplete diagnostic information. Section 5 validates the proposed method. Finally, conclusions are given in Section 6.

2. Preliminaries of characteristic relation and multigranulation model 2.1. Characteristic relation An information system (IS) is defined as a quadruple ðU; A; V; f Þ, where U is a non-empty finite set of cases called the universe, A ¼ C∪D is a non-empty finite set of attributes, where C is the condition attribute set and D is the decision attribute. V is regarded as the finite set of attributes. Each attribute a2A is associated with a set Va , where Va is the set of values of a, called the domain of attribute a. f : U  A/Va is a total function such as f ðx; aÞ2Va for each a2A and x2U. If one or more attribute values in the IS are unknown, then it is regarded as an incomplete information system (IIS) [20]. Characteristic relation is a generalized indiscernibility relation that can be utilized to handle an IIS [21]. Characteristic relation has been widely applied in many domains [22e25]. Referring to Prof. Grzymala-Busse's study [21], unknown attribute values could be divided into two categories, namely lost value and “do not care” condition. Lost value: an unknown attribute value is called a lost value, when its corresponding value is forgotten to enter into an IIS or mistakenly erased. The original value exists but it is not accessible for some reasons. “Do not care” condition: an unknown attribute value is called a “do not care” condition, when its corresponding value exists and may be replaced by any typical value for that attribute. To distinguish between the two categories of unknown attribute values, lost value is denoted by “?”, and “do not care” condition is denoted by “*”, respectively in this paper. Suppose that ða; vÞ is an attribute-value pair. For a complete IS, where each attribute value is specified, a block of ða; vÞ, denoted by ½ða;vÞ, is the set of all cases x for which aðxÞ ¼ v, where aðxÞ denotes the value of the attribute a for the case x. For an IIS, the definition of a block of an attribute-value pair is modified as follows [21]. If for an attribute a there exists a case x such that aðxÞ ¼ ?, then the case x should not be included in any blocks ½ða; vÞ for all values v of attribute a. If for an attribute a there exists a case x such that aðxÞ ¼ *, then the case x should be included in blocks ½ða; vÞ for all specified values v of attribute a. For a case x2U, the characteristic set KB ðxÞ is defined as the intersection of the sets Kðx;aÞ, for all a2B, where the set Kðx; aÞ is defined as follows [21]. For a specified attribute value aðxÞ, the set Kðx; aÞ is the block ½ða; aðvÞÞ of attribute a and its value aðxÞ. If aðxÞ ¼ ? or aðxÞ ¼ *, then the set Kðx; aÞ ¼ U. Characteristic set KB ðxÞ may be regarded as the set of cases that are indistinguishable from x using all attributes from B and using a given interpretation of unknown attribute values. Characteristic relation RðBÞ is a relation about U defined as follows. ðx; yÞ2RðBÞ if and only if y2KB ðxÞ. The characteristic relation RðBÞ is known if the characteristic set KB ðxÞ is known for all x2U. 2.2. Multigranulation model In recent years, multigranulation model has attracted considerable attention for its multi-level view. It has extensively been applied in many domains, including attribute reduction, knowledge discovery, and fault diagnosis, etc. Multigranulation model was first introduced by Qian et al. [26,27]. It describes lower and upper approximations via multiple granulation relations to realize approximate approximation of a target concept. Different from traditional rough set models, multigranulation model is constructed according to a series of equivalence relations rather than a single one. Definition 1. [26]. Let ðU; A ¼ C∪D; V; f Þ be an IS in which B1 ; B2 ; :::; Bm ⊆C, then cX⊆U, the optimistic lower approximation and upper approximation of X are denoted by m X

O

Bi ðXÞ ¼

i¼1

m X i¼1

8 > < > :

Pm

O

i¼1 Bi

ðXÞ and

x2U : ½xB1 ∢X∨½xB2 ∢X∨…∨½xBm ∢X

O

Bi ðXÞ ¼

m X

Pm

i¼1 Bi

O

ðXÞ, respectively,

9 > = > ;

(1)

O

Bi ð XÞ

(2)

i¼1

where ½xBi ð1  i  mÞ are the equivalence classes of the case x with respect to the attribute subset Bi , and  X is the

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complement set of X. 〈

Pm

O

i¼1 Bi

ðXÞ;

Pm

i¼1 Bi

O

ðXÞ〉 is the optimistic multigranulation model.

Definition 2. [26]. Let ðU; A ¼ C∪D; V; f Þ be an IS in which B1 ; B2 ;:::; Bm ⊆C, then cX⊆U, the pessimistic lower approximation and upper approximation of X are denoted by m X

P

Bi ðXÞ ¼

i¼1

m X

8 > < > :

P

Bi ðXÞ ¼

i¼1

Pm

P

i¼1 Bi

ðXÞ and

x2U : ½xB1 ∢X∨½xB2 ∢X∨:::∨½xBm ∢X

m X

Pm

i¼1 Bi

P

ðXÞ, respectively,

9 > = (3)

> ;

P

Bi ð XÞ

(4)

i¼1

where ½xBi ð1  i  mÞ are the equivalence classes of the case x with respect to the attribute subset Bi , and  X is the comP P P Pm plement set of X. 〈 m i¼1 Bi ðXÞ; i¼1 Bi ðXÞ〉 is the pessimistic multigranulation model.

In the optimistic multigranulation model, the word “optimistic” means that in multiple independent granular structures; one needs only at least a granular structure to satisfy with the inclusion condition between knowledge granule and target concept. In the pessimistic multigranulation model, the word “pessimistic” means that in multiple independent granular structures; one needs all granular structures to satisfy with the inclusion condition between knowledge granule and target concept. 3. Attribute reduction of IIS with two categories of unknown attribute values 3.1. Data-driven valued characteristic relation Characteristic relation can be used to describe and analyze an IIS which simultaneously contains two categories of unknown attribute values. An IIS is given in Table 1. Take this table as an example, the set of all cases is denoted by U. Condition attribute set C ¼ fc1 ; c2 ; c3 g contains three condition attributes. Decision attribute d, Vd ¼ fY; Ng, which denotes fault condition and normal condition, respectively. For the cases in Table 1, the blocks of attribute-value pairs are as follows. ½ðc1 ; 1Þ ¼ fu2 ; u7 ; u8 g, ½ðc1 ; 2Þ ¼ fu1 ; u4 ; u5 ; u7 g, ½ðc1 ; 3Þ ¼ fu6 ; u7 g, ½ðc2 ; 1Þ ¼ fu2 ; u4 ; u6 ; u7 g, ½ðc2 ; 2Þ ¼ fu3 ; u8 g, ½ðc3 ; 1Þ ¼ fu1 ; u3 ; u6 ; u7 ; u8 g, ½ðc3 ; 2Þ ¼ fu2 ; u4 ; u5 ; u7 g. For the cases in Table 1, the characteristic sets are as follows. KC ðu1 Þ ¼ fu1 ;u4 ;u5 ;u7 g∩U∩fu1 ;u3 ;u6 ;u7 ;u8 g ¼ fu1 ;u7 g, KC ðu2 Þ ¼ fu2 ;u7 g, KC ðu3 Þ ¼ fu3 ;u8 g, KC ðu4 Þ ¼ fu4 ;u7 g, KC ðu5 Þ ¼ fu4 ; u5 ; u7 g, KC ðu6 Þ ¼ fu6 ; u7 g, KC ðu7 Þ ¼ fu2 ; u4 ; u6 ; u7 g, KC ðu8 Þ ¼ fu8 g. From these characteristic sets, it can be noted that the characteristic relation is only used to qualitatively determine whether two cases belong to a characteristic set. However, characteristic relation cannot be employed to quantitatively describe the similarity degree between two cases. For instance, with respect to the cases u4 ¼ f2; 1; 2g, u5 ¼ f2; ?; 2g, and u7 ¼ f*; 1; *g, the case u5 only possesses one unknown attribute value, and other specific attribute values are the same as the case u4 ; the case u7 possesses two unknown attribute values, and only one specific attribute value is the same as the case u4 . Obviously, the similarity degree between the cases u4 and u5 is much higher than between the cases u4 and u7 . According to the definition of characteristic set, the cases u4 and u5 belong to the characteristic set KC ðu5 Þ. However, with respect to the decision attribute d, Vd ðu4 Þ is Y, Vd ðu5 Þ is Y, and Vd ðu7 Þ is N. The cases u4 and u5 possess the same decision attribute value; nevertheless, the cases u4 and u7 possess different decision attribute values. Two cases in the same characteristic set possess different decision attribute values. Consequently, it is inaccurate that characteristic relation is employed to determine

Table 1 Incomplete information system. U

c1

c2

c3

D

u1 u2 u3 u4 u5 u6 u7 u8

2 1 ? 2 2 3 * 1

? 1 2 1 ? 1 1 2

1 2 1 2 2 1 * 1

Y Y Y Y Y Y N N

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indiscernibility relation among cases. The reason lies in that characteristic relation doesn't take the probabilities that an unknown attribute value is equal to its corresponding specific value into account. It means that characteristic relation cannot be used to quantify the similarity degrees among cases. Owing to this reason, the reliability of determining the indiscernibility relation among cases using characteristic relation may reduce. To address this issue, a data-driven valued characteristic relation is defined as follows. jV j

Definition 3. Let ðU; A ¼ C∪D; V; f Þ be an IIS, and B⊆C is a condition attribute set. For cb2B, Vb ¼ fk1b b1 ; k2b b2 ; /; kb b bjVb j g, where fb1 ; b2 ; /; bjVb j g denotes the set that all the unequal specific values of condition attribute b in the universe U, kib ði ¼ 1; 2; /; jVb jÞ denotes the number of all cases whose condition attribute values are bi in the universe U, and jXj denotes the cardinality of the set X. For cx; y2U, the characteristic similarity degree VRB ðx; yÞ in condition attribute set B can be calculated as

VRB ðx; yÞ ¼

Y

R ðx; yÞ$NB ðx; yÞ b2B b

(5)

where Rb ðx; yÞ denotes the similarity degree between the cases x and y in the condition attribute b, and NB ðx; yÞ denotes the percentage of non-lost attribute values in the cases x and y. Rb ðx; yÞ and NB ðx; yÞ can be respectively calculated as follows.

8 i. bðxÞ ¼ * or bðyÞ ¼ * k > < b jUj jV j b  . 2 Rb ðx; yÞ ¼ X > : kib jUj bðxÞ ¼ bðyÞ ¼ *

(6)

i¼1

NB ðx; yÞ ¼ 1 

jfb2BjbðxÞ ¼ ?∨bðyÞ ¼ ?gj jBj

(7)

Let bðxÞ and bðyÞ be specific condition attribute values. If bðxÞ ¼ bðyÞ, then Rb ðx; yÞ ¼ 1; if bðxÞsbðyÞ, then Rb ðx; yÞ ¼ 0. From Definition 3, we can see that the frequency of a specific attribute value is calculated according to statistical theory. The frequency is regarded as the probability of the specific attribute value. Consequently, the probability distribution of the specific attribute value can be acquired. The similarity degree between two cases in the “do not care” condition attribute is calculated according to the probability distribution. Then, the percentage of non-lost attribute values is calculated. Finally, the characteristic similarity degree between two cases is acquired without extra priori knowledge. Compared to data-driven valued tolerance relation, this calculation method can be used to calculate the similarity degree between two cases that simultaneously contain two categories of unknown attribute values. Accordingly, applicable cases are extended by this calculation method of the characteristic similarity degree between two cases. According to Definition 3, the characteristic similarity degree between the cases u4 and u5 , and the one between the cases u4 and u7 are respectively calculated as follows.

VRB ðu4 ; u5 Þ ¼ VRB ðu4 ; u7 Þ ¼

Y

R ðx; yÞ$NB ðx; yÞ b2B b

Y

R ðx; yÞ$NB ðx; yÞ b2B b

¼ NB ðu4 ; u5 Þ ¼ 2=3; ¼ Rc1 ðu4 ; x7 Þ$Rc3 ðu4 ; u7 Þ ¼ 3=8  3=8 ¼ 9=64:

With respect to Table 1, there are two same condition attribute values in the cases u4 and u5 . Nevertheless, there is only one same condition attribute values in the cases u4 and u7 . From the calculation results, it is reasonable that the characteristic similarity degree between the cases u4 and u5 is higher than the one between the cases u4 and u7 . Consequently, characteristic similarity degree can be used to quantify the similarity degree among cases. Definition 4. Let ðU; A ¼ C∪D; V; f Þ be an IIS, and B⊆C is a condition attribute set. For cx; y2U, data-driven valued characteristic relation can be defined as

VRðBÞ ¼ fðx; yÞ2U  Ujy2KB ðxÞ; VRB ðx; yÞ  rg

(8)

where r denotes a threshold. If the threshold is too large, then the case number of the characteristic set that satisfies the datadriven valued characteristic relation is too small. If the threshold is too small, then the case number of the characteristic set that satisfies the data-driven valued characteristic relation is too large. In order to avoid this situation, the threshold in the data-driven valued characteristic relation can be calculated as jUj P

a ¼ i¼1

minðVRB ðxi ÞÞ jUj

(9)

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where minðVRB ðxÞÞ denotes the minimum of the characteristic similarity degrees between the cases x and the other cases in the universeU. From Definition 4, it can be noted that the threshold is equal to the average value of the minimum characteristic similarity degrees of all cases in the universe U. Therefore, the threshold obtained by Eq. (9) is neither too large nor too small. Moreover, the threshold can be used to reasonably control information granular boundary. Compared to characteristic relation, the accuracy of determining indiscernibility relation among cases is improved by the data-driven valued characteristic relation. The threshold a among cases in Table 1 is equal to 0.14 by Eq. (9). According to Definition 4, the characteristic sets of the cases u4 and u5 that satisfy the data-driven valued characteristic relation are VKC ðu4 Þ ¼ fu4 g and VKC ðu5 Þ ¼ fu4 ; u5 g, respectively. Nevertheless, the characteristic sets of the cases u4 and u5 that satisfy the characteristic relation are KC ðu4 Þ ¼ fu4 ; u7 g and KC ðu5 Þ ¼ fu4 ; u5 ; u7 g, respectively. In fact, the decision attribute values of the cases u4 , u5 and u7 are Vd ðu4 Þ ¼ fYg, Vd ðu5 Þ ¼ fYg, and Vd ðu7 Þ ¼ fNg, respectively. From calculation results, it can be noted that the cases that possess the same decision attribute value are retained; nevertheless, the cases that possess different decision attribute values are removed for the limit of the characteristic similarity degree. Therefore, it is more accurate to determine the indiscernibility relation among cases using the data-driven valued characteristic relation than using the characteristic relation.

3.2. Attribute reduction algorithm based on pessimistic data-driven valued characteristic multigranulation model Attribute reduction is a vital topic in data-driven fault diagnosis methods. It removes irrelevant or redundant condition attribute values and preserves the decision ability of original information. However, many uncertain factors may result in the appearance of incomplete diagnostic information. The incomplete diagnostic information may simultaneously contain two categories of unknown attribute values. Therefore, attribute reduction of an IIS that simultaneously contains two categories of unknown attribute values has been a challenging problem. To address this issue, a data-driven valued characteristic multigranulation model is defined according to multigranulation model. Definition 5. Let ðU; A ¼ C∪D; V; f Þ be an IIS in which B1 ; B2 ; :::; Bm ⊆C, then cX⊆U, the optimistic lower approximation and upper approximation of X are denoted by m X

Pm

O

i¼1 VBi

ðXÞ and

Pm

i¼1 VBi

O

ðXÞ, respectively,

O

n o VBi ðXÞ ¼ x2U : VKðxÞB1 ∢X∨VKðxÞB2 ∢X∨:::∨VKðxÞBm ∢X

(10)

i¼1

m X

O

VBi ðXÞ ¼

i¼1

m X

O

VBi ð XÞ

(11)

i¼1

where VKðxÞBi ð1  i  mÞ are the characteristic sets of the case x that satisfy the data-driven valued characteristic relation O P O Pm with respect to the attribute subset Bi , and  X is the complement set of X. 〈 m i¼1 VBi ðXÞ; i¼1 VBi ðXÞ〉 is the optimistic data-

driven valued characteristic multigranulation model with respect to the attribute subset Bi . Let ðU; A ¼ C∪D; V; f Þ be an IIS in which B1 ; B2 ;:::; Bm ⊆C, then cX⊆U, the pessimistic lower approximation and

Definition 6.

upper approximation of X are denoted by m X

Pm

P

i¼1 VBi

ðXÞ and

Pm

i¼1 VBi

P

ðXÞ, respectively,

P

n o VBi ðXÞ ¼ x2U : VKðxÞB1 ∢X∨VKðxÞB2 ∢X∨:::∨VKðxÞBm ∢X

(12)

i¼1

m X i¼1

P

VBi ðXÞ ¼

m X

P

VBi ð XÞ

(13)

i¼1

where VKðxÞBi ð1  i  mÞ are the characteristic sets of the case x that satisfy the data-driven valued characteristic relation with respect to the attribute subset Bi , and  X is the complement set of X. 〈

Pm

P

i¼1 VBi

ðXÞ;

Pm

i¼1 VBi

driven valued characteristic multigranulation model with respect to the attribute subset Bi .

P

ðXÞ〉 is the pessimistic data-

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Definition 7. Let ðU; A ¼ C∪D; V; f Þ be an IIS in which B1 ; B2 ;:::; Bm ⊆C, and D1 ; D2 ; :::; Dm is a partition of the universe U. The attribute dependency of the decision classes that satisfy the data-driven valued characteristic relation with respect to the attribute subset Bi is denoted by gVBi ðDÞ,

   , X   . m gVBi ðDÞ ¼ POSVBi ðDÞ jUj ¼  VBi ðXÞ jUj    i¼1 

(14)

where POSVBi ðDÞ denotes the union of the lower approximations of X with respect to the attribute subset Bi . The classical multigranulation model is extended to the data-driven valued characteristic multigranulation model by datadriven valued characteristic relation instead of equivalence relation. Data-driven valued characteristic relation can be utilized to analyze and process incomplete diagnostic information that simultaneously contains two categories of unknown attribute values. Consequently, the data-driven valued characteristic multigranulation model largely extends the theory and application of the classical multigranulation model. The optimistic data-driven valued characteristic multigranulation model belongs to a risk-seeking decision strategy; nevertheless the pessimistic data-driven valued characteristic multigranulation model belongs to a risk-averse decision strategy. So as to extract reliable decision rules from an IIS that simultaneously contains two categories of unknown attribute values, an attribute reduction algorithm based on pessimistic data-driven valued characteristic multigranulation model is introduced. The attribute reduction algorithm is described as follows. Input: an IIS ðU; A ¼ C∪D; V; f Þ, and an attribute subset B1 ; B2 ; :::; Bm . Output: decision rules. Step 1: Calculate the characteristic similarity degrees VRC ðx; yÞ among cases, and the thresholds aðxi Þ; i ¼ 1; 2; :::; n; Step 2: Determine the characteristic sets VKC ðxi Þ that satisfies data-driven valued characteristic relation; Step 3: Calculate the attribute dependency gVBi of the decision classes that satisfy data-driven valued characteristic relation with respect to the attribute subset B1 ; B2 ; :::; Bm ; Step 4: Remove condition attribute value c1 , and calculate a new attribute dependency g0 ; Step 5: If the attribute dependency gVBi  g0, then the condition attribute value c1 is redundant, or else the condition attribute value c1 is indispensable; Step 6: Repeat step 4 and 5 to the other condition attribute values, till the last one; Step 7: Remove all the redundant condition attribute values, and generate decision rules. Different from the traditional dimension reduction algorithms, the proposed attribute reduction algorithm can remove redundant condition attribute values of an IIS that simultaneously contains two categories of unknown attribute values, namely lost value and “do not care” condition. Furthermore, the proposed attribute reduction algorithm belongs to a riskaverse decision strategy. The similarity degree among cases can be described via characteristic similarity degree. The threshold can reasonably control information granular boundary. Therefore, extracted decision rules by the proposed attribute reduction algorithm are much more reliable than the traditional dimension reduction algorithms. 4. Method framework of planetary gearbox fault diagnosis with incomplete diagnostic information NBC is a simplest fundamental form of Bayesian networks [28,29]. It is an effective mode identification tool for its low computational complexity and less memory requirements [30,31]. Therefore, it has been successfully applied in fault diagnosis of planetary gearbox. Moreover, the pessimistic data-driven valued characteristic multigranulation model can be applied in attribute reduction to extract reliable decision rules from an IIS that simultaneously contains two categories of unknown attribute values. Accordingly, a fault diagnosis method of planetary gearboxes based on data-driven valued characteristic multigranulation model with incomplete diagnostic information is proposed, and the method framework is shown in Fig. 1. First, fault diagnosis features of planetary gearbox with typical gear faults are extracted and construct an IIS for fault diagnosis. Data-driven valued characteristic relation is utilized to analyze and process the IIS to obtain characteristic sets. Then, an attribute reduction algorithm based on pessimistic data-driven valued characteristic multigranulation model is employed to extract fault diagnosis decision rules. Finally, NBC is constructed to identify planetary gearbox conditions. 5. Experimental validation 5.1. Experimental settings To validate the effectiveness of the proposed method, a fault diagnosis experiment was carried out. The advantages of this method were investigated according to the experimental results. An experimental test rig of planetary gearbox is presented in Fig. 2. The experimental test rig consists of a drive motor, a planetary gearbox, a load motor, and a data collection system. The drive motor is connected with the planetary gearbox through a shaft and two couplings. The planetary gearbox is attached to the load motor through another coupling. The planetary gearbox is in a lubrication semistate. SAE 40 oil is used as a lubricant.

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Fig. 1. Method framework of planetary gearbox fault diagnosis based on data-driven valued characteristic multigranulation model with incomplete diagnostic information.

Fig. 2. Experimental test rig of planetary gearbox.

In this experiment, four planetary gearbox conditions were considered, including normal condition (NC), face wear on sun gear (FWSG), a missing tooth on planet gear (MTPG), and a chipped tooth on ring gear (CTRG), respectively. The planetary gearbox is illustrated in Fig. 3. Two loads generated by the load motor controlled through a circuit were employed to simulate two operation states of the planetary gearbox. Three rotation speeds of the drive motor were adjusted to 75 rpm, 150 rpm, and 300 rpm, respectively. A piezoelectric acceleration sensor was mounted on the gearbox casing to acquire vibration signals. Sampling frequency was 5120 Hz and each sample contained 20480 data points. A current sensor was employed to obtain current signals of the load motor stator. 60 samples in each gearbox operation state were acquired and there were a total of 240 samples. The time-domain waveforms and their corresponding frequency spectra of four planetary gearbox conditions are illustrated in Figs. 4 and 5. The lubrication oil samples were collected and wear debris analysis technique was utilized to

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Fig. 3. Planetary gearbox.

Fig. 4. Time-domain waveforms of four planetary gearbox conditions: (a) NC, (b) FWSG, (c) MTPG, and (d) CTRG.

obtain a fault feature. The collected lubrication oil samples were diluted with unused lubrication oil. Wear debris was separated from the diluted oil samples and fixed to glass slides through the filtergram method. Visual inspection of the glass slides was carried out by use of a scanning electron microscope. Wear debris pictures of three planetary gearbox conditions are shown in Fig. 6.

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Fig. 5. Frequency spectra of four planetary gearbox conditions: (a) NC, (b) FWSG, (c) MTPG, and (d) CTRG.

Fig. 6. Wear debris pictures of three planetary gearbox conditions: (a) NC, (b) FWSG, and (c) MTPG.

5.2. Diagnosis results Time-domain analysis methods are much simpler and faster fault feature extraction methods than frequency-domain and time-frequency-domain methods. Accordingly, they have been widely adopted in fault diagnosis of planetary gearboxes [32e34]. In the experiment, three time-domain parameters are utilized to characterize the obtained samples. The three timedomain parameters can be calculated as follows.

Waveform indicator:K ¼ xrms =x0  Peak indicator:C ¼ xp xrms  Pulse indicator:P ¼ xp x0 where xrms is mean square amplitude, x0 is average amplitude, and xp is peak value.

(15) (16) (17)

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Table 2 Wear particle density of three planetary gearbox conditions. Condition

Density dilution ratio

Dl

Ds

W

NC FWSG MTPG

10:1 10:1 10:1

4 21 5

11 31 9

15 52 14

Table 3 IIS for fault diagnosis of planetary gearbox. U

Kv

Cv

Pv

KI

CI

W

D

u1 u2 … u60 u61 u62 … u120 u121 u122 … u180 u181 u182 … u240

1 1 … 2 1 2 … 2 2 * … 3 2 3 … 3

1 * … 2 1 * … 2 3 2 … 3 2 * … 2

1 1 … 3 3 2 … 2 2 * … 3 * 2 … 1

* 1 … * * 1 … 2 2 * … 3 3 4 … 3

1 * … 2 2 1 … 2 3 2 … * 3 * … 3

1 1 … 1 2 2 … 2 2 2 … 2 ? ? … ?

NC

FWSG

MTPG

CTRG

The three time-domain parameters of vibration signals can be regarded as three fault features, represented by Kv ; Cv ; Pv. The waveform indicator and peak indicator of the current signals can be considered as two fault features, represented by KI ; CI . Because of the five parameters are continuous variables, it is necessary to carry out data discretization. Take waveform indicator of vibration signals for example, waveform indicator values are distributed within three intervals, i.e., Kv ð1Þ ¼ ½1; 1:2Þ, Kv ð2Þ ¼ ½1:2;1:4Þ, and Kv ð3Þ ¼ ½1:4;3:5. According to this discretization principle, the values of the other parameters can be divided into three or four intervals so as to reduce computational complexity. Nevertheless, during discretization, some contradictory attribute values were removed to simulate “do not care” conditions. Wear particle density of the lubrication oil is regarded as a fault feature, represented by W. The wear particle density of three planetary gearbox conditions is given in Table 2. The density of larger wear particle is denoted by Dl, and the diameter of larger wear particle is more than 5 mm. The density of smaller wear particle is denoted by Ds, and the diameter of larger wear particle is less than 5 mm. Wear particle density is the amount of wear particles accumulating on unit area of the glass slide. The lubrication oil of planetary gearbox with CTRG was not collected in order to simulate lost values. After discretization, the whole 6 fault features are employed to form condition attributes. An IIS for fault diagnosis of planetary gearbox is listed in Table 3. As listed in Table 3, there exist two categories of unknown attribute values. The existing methods are not suitable for fault diagnosis. Thus, the proposed method is adopted to diagnose planetary gearbox faults. The entire 240 cases can be formed from the 240 samples. 200 cases are regarded as training cases and the remains are utilized for test. First, the data-driven valued characteristic relation is utilized to analyze and process the training cases to obtain the characteristic sets. Then, the attribute reduction algorithm based on pessimistic data-driven valued characteristic multigranulation model is used to extract fault diagnosis decision rules from the training cases. These decision rules extracted are listed in Table 4. Finally, NBC is constructed to identify planetary gearbox conditions of the test cases. Diagnosis results of four planetary gearbox conditions are listed in Table 5, where average accuracy is calculated according to all the cases of each

Table 4 Decision rules extracted from the training cases. No.

Decision rules

k

Case

1 2 3 4 5 6 7 … 91

ðKv ; 1Þ∧ðPv ; 2Þ∧ðW; 1Þ/NC ðPv ; 3Þ∧ðW; 1Þ/NC ðKv ; 1Þ∧ðCv ; 1Þ/FWSG ðKI ; 1Þ∧ðCI ; 1Þ/FWSG ðKv ; 2Þ∧ðKI ; 2Þ/MTPG ðKv ; 3Þ∧ðPv ; 3Þ∧ðCv ; 3Þ/MTPG ðKI ; 3Þ∧ðCI ; 3Þ/CTRG … ðKv ; 3Þ∧ðKI ; 4Þ/CTRG

2 3 2 3 2 3 1 … 3

u1, u11 u2, u17, u33 u61, u71 u62, u81, u97 u121, u122 u128, u129, u161 u181 … u182, u199, u230

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Table 5 Diagnosis results of four planetary gearbox conditions. Condition

Ratio of training and test cases

Training accuracy (%)

Test accuracy (%)

Average accuracy (%)

NC FWSG MTPG CTRG

5:1 5:1 5:1 5:1

100 100 100 100

80 90 100 100

96.67 98.33 100 100

planetary gearbox condition. As listed in Table 5, test accuracy of each condition exceeds 80%, and average accuracy of each condition is more than 96%. Accordingly, the proposed method can accurately determine planetary gearbox conditions. 5.3. Result analyses and discussions To further investigate the influence of the threshold on average accuracy, the training cases in Table 3 are processed by tolerance relation (TR), non-symmetric similarity relation (NSR), and data-driven valued characteristic relation (DVCR), respectively. Under the three indiscernibility relations, average accuracy varies with the threshold. Relation curve between average accuracy and the threshold is depicted in Fig. 7, where average accuracy increases first and then decreases along with the increase of the threshold. The maximum average accuracy appears when the threshold is about 0.3. This indicates that there is an optimum threshold and the best diagnostic effectiveness can be achieved by reasonable threshold selection. Furthermore, the average accuracy under data-driven valued characteristic relation obviously exceeds the other two indiscernibility relations. This is because the data-driven valued characteristic relation can be utilized to analyze and process the IIS that simultaneously contains two categories of unknown attribute values. Nevertheless, all unknown attribute values are regarded as one category by the other two indiscernibility relations. Therefore, the proposed method can acquire satisfactory diagnosis effectiveness. In order to further analyze the sensitivity of average accuracy, a series of training case sets with randomly generated unknown attribute values according to the 200 training cases are used to extract decision rules. Average accuracy under three indiscernibility relations is calculated. Relation curve between average accuracy and the percentages of unknown attribute values is depicted in Fig. 8, where average accuracy decreases along with the percentage increase. Average accuracy rapidly decreases when the percentage is more than 15%. This is because specific attribute values play a crucial role in decision rule extraction. Satisfactory diagnosis effectiveness is hard to obtain, when the percentage of specific attribute values is too low. Although average accuracy is influenced by the percentages of unknown attribute values, the proposed method still possesses the highest average accuracy in the three indiscernibility relations. The diagnostic effectiveness of the proposed method is still satisfactory, when the percentage of unknown attribute values is less than 20%. In order to further validate the effectiveness of the proposed method, the proposed method is compared with FL, SVM and DNNs methods using the 240 samples in Table 3. First, the unknown attribute values of these samples are completed by regression algorithm. Then, these samples after completion are used to verify the diagnosis accuracy of FL, SVM and DNNs methods. FL is a rule-based system, which has successfully combined fuzzy theory with human's inference capability. It makes use of experts' knowledge through linguistic transformation. Membership functions give scaled value of numerical values defined by linguistic labels. Linguistic rules are represented by ‘if-then’ rules. In this study, linguistic rules and membership functions are generated by J48 algorithm. Fuzzy inference engine is constructed by the fuzzy toolbox available in MATLAB 2016a. SVM based on statistical learning theory possesses high accuracy and excellent generalization ability for a

Fig. 7. Relation curve between average accuracy and the threshold.

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Fig. 8. Relation curve between average accuracy and the percentages of unknown attribute values.

Fig. 9. Average accuracy of four methods.

smaller number of samples. Structural risk minimization principle in SVM is employed to minimize the upper bound based on the expected risk. In this study, the MATLAB SVM toolbox is used to diagnose planetary conditions. The penalty parameter C is set to 150, and the Gaussian kernel parameter g is set to 12. DNNs with deep architectures contain multiple hidden layers, and each hidden layer performs a non-linear transformation from the previous layer to the next layer. Two main steps are used to realize the DNNs' training. The first step is pre-training the DNNs through unsupervised techniques, such as auto-encoder. The other step is fine-tuning the DNNs with supervised algorithms, such as back propagation algorithm [7]. In this study, three-layer DNNs are applied in fault diagnosis. The number of hidden layer nodes is set as 100. Ratio of training and test samples is 3:1, 4:1, and 5:1, respectively. Average accuracy of four methods is illustrated in Fig. 9. It can be found that the average accuracy of four methods increases along with ratio enhancement. The proposed method always obtains the superior average accuracy compared with the other methods. We suggest using the proposed method under the following conditions. Incomplete diagnostic information of planetary gearboxes simultaneously contains two categories of unknown attribute values. Moreover, condition attributes containing unknown values are more than two-thirds of all condition attributes, and the percentage of unknown attribute values exceeds 7%. According to the experimental validation above, the principal contributions of the proposed method are as follows. 1. Different from the existing indiscernibility relations, the data-driven valued characteristic relation can be utilized to analyze and process an IIS that simultaneously contains two categories of unknown attribute values, namely lost value and “do not care” condition. The similarity degree between two cases is described via characteristic similarity degree. The threshold is used to reasonably control information granular boundary so as to accurately determine indiscernibility relation among cases.

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2. The data-driven valued characteristic multigranulation model is the extension of the classical multigranulation model by data-driven valued characteristic relation instead of equivalence relation. Consequently, it largely extends the theory and application of the classical multigranulation model. The pessimistic data-driven valued characteristic multigranulation model belongs to a risk-averse decision strategy. Accordingly, the attribute reduction algorithm based on pessimistic datadriven valued characteristic multigranulation model is introduced to remove irrelevant or redundant condition attribute values and extract reliable decision rules. The relationship between condition attributes and decision attribute is expressed in a simplest way to reduce computational complexity. 3. The proposed method is capable of accurately diagnosing planetary gearbox conditions without additional prior knowledge and analytical models. Although average accuracy is influenced by the percentages of unknown attribute values, the proposed method still achieves satisfactory performance. Compared with FL, SVM and DNNs methods using the same dataset, the proposed method always obtains the superior average accuracy. This method introduces multigranulation into decision rule extraction so as to provide a novel strategy for planetary gearbox fault diagnosis with incomplete diagnostic information. 6. Conclusions In this paper, a novel fault diagnosis method of planetary gearboxes based on data-driven valued characteristic multigranulation model with incomplete diagnostic information is proposed to determine planetary gearbox conditions. A datadriven valued characteristic relation is introduced to analyze and process incomplete diagnostic information that simultaneously contains two categories of unknown attribute values. Then, a data-driven valued characteristic multigranulation model is defined and an attribute reduction algorithm based on pessimistic data-driven valued characteristic multigranulation model is presented to extract fault diagnosis decision rules. Finally, naive Bayesian classifier is constructed to identify planetary gearbox conditions. The effectiveness of this method is validated and the advantages are investigated using a fault diagnosis experiment of planetary gearbox. The experimental results demonstrate that this method can accurately determine indiscernibility relation among cases. Besides, computational complexity is reduced by using the attribute reduction algorithm. Furthermore, this method still achieves satisfactory performance with incomplete diagnostic information. Acknowledgments This work is supported by the National Natural Science Foundation of China (61673142, and 51275136), and the University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province (2016034). The authors are also grateful to the editors and the anonymous reviewers for their helpful comments and constructive suggestions. References [1] J.M. Ha, B.D. Youn, Long Oh, B. Han, Y. Jung, J. Park, Autocorrelation-based time synchronous averaging for condition monitoring of planetary gearboxes in wind turbines, Mech. Syst. Signal Process. 13 (3) (2017) 1360e1368. [2] Y.G. Lei, Z.Y. Liu, L. Jing, F.B. Lu, Phenomenological models of vibration signals for condition monitoring and fault diagnosis of epicyclic gearbox, J. Sound Vib. 370 (2016) 372e393. [3] Y.G. Lei, J. Lin, M.J. Zuo, Z.J. He, Condition monitoring and fault diagnosis of planetary gearboxes: a review, Measurement 48 (2014) 292e305. [4] X.H. Liang, H.S. Zhang, L.B. Liu, M.J. Zuo, The influence of tooth pitting on the mesh stiffness of a pair of external spur gears, Mech. Mach. Theor. 106 (2016) 1e15. [5] Z. Cheng, N.Q. Hu, X.F. Zhang, Crack level estimation approach for planetary gearbox based on simulation signal and GRA, J. Sound Vib. 331 (26) (2012) 5853e5863. [6] W.J. Sun, S.Y. Shao, R. Zhao, R.Q. Yan, X.W. Zhang, X.F. Chen, A sparse auto-encoder-based deep neural network approach for induction motor faults classification, Measurement 89 (2016) 171e178. [7] F. Jia, Y.G. Lei, J. Lin, X. Zhou, N. Lu, Deep neural networks: a promising tool for fault characteristic mining and intelligent diagnosis of rotating machinery with massive data, Mech. Syst. Signal Process. 72e73 (2016) 303e315. [8] H.Z. Chen, J.X. Wang, B.P. Tang, K. Xiao, J.Y. Li, An integrated approach to planetary gearbox fault diagnosis using deep belief networks, Meas. Sci. Technol. 28 (2017), 025010. [9] L. Wang, Z.J. Zhang, H. Long, J. Xu, R.H. Liu, Wind turbine gearbox failure identification with deep neural networks, IEEE Trans. Ind. Inf. 13 (3) (2017) 1360e1368. [10] H.M. Zhao, M. Sun, W. Deng, X.H. Yang, A new feature extraction method based on EEMD and multi-scale fuzzy entropy for motor bearing, Entropy 19 (1) (2017) 14. [11] C.C. Chen, G. Vachtsevanos, M.E. Orchard, Machine remaining useful life prediction: an integrated adaptive neuro-fuzzy and high-order particle filtering approach, Mech. Syst. Signal Process. 28 (2012) 597e607. [12] X.H. Chen, G. Cheng, H.Y. Li, M. Zhang, Diagnosing planetary gear faults using the fuzzy entropy of LMD and ANFIS, J. Mech. Sci. Technol. 30 (6) (2016) 2453e2462. [13] B. Gu, V.S. Sheng, K.Y. Tay, W. Romano, S. Li, Incremental support sector learning for ordinal regression, IEEE Trans. Neural Network Learn. Syst. 26 (7) (2015) 1403e1416. [14] B. Gu, V.S. Sheng, A robust regularization path algorithm for n-support vector classification, IEEE Trans. Neural Network Learn. Syst. 28 (5) (2017) 1241e1248. [15] Z.L. Liu, M.J. Zuo, H.B. Xu, Feature ranking for support vector machine classification and its application to machinery fault diagnosis, Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 227 (9) (2013) 2077e2089. [16] Y.G. Lei, Z.Y. Liu, X.H. Wu, N.P. Li, W. Chen, J. Lin, Health condition identification of multi-stage planetary gearboxes using an mRVM-based method, Mech. Syst. Signal Process. 60e61 (2015) 289e300.

J. Yu, Y. He / Journal of Sound and Vibration 429 (2018) 63e77

77

[17] Y.B. Li, Y.T. Liu, G.Y. Li, M.Q. Xu, W.H. Huang, J. Lin, A fault diagnosis scheme for planetary gearboxes using modified multi-scale symbolic dynamic entropy and mRMR feature selection, Mech. Syst. Signal Process. 91 (2017) 295e312. [18] M. Wang, N.Q. Hu, G.J. Qin, A method for rule extraction based on granular computing: application in the fault diagnosis of a helicopter transmission system, J. Intell. Rob. Syst. 71 (2013) 445e455. [19] M. Wang, N.Q. Hu, G.J. Qin, Rule extracting based on MCG with its application in helicopter power train fault diagnosis, in: Proceedings of the 9th International Conference on Damage Assessment of Structures, Oxford, July 2011, 012058. [20] B. Walczak, D.L. Massart, Rough set theory, Chemometr. Intell. Lab. Syst. 47 (1) (1999) 1e16. [21] J.W. Grzymala-Busse, P.G. Clark, M. Kuehnhausen, Generalized probabilistic approximations of incomplete data, Int. J. Approx. Reason. 55 (1) (2014) 180e196. [22] J.W. Grzymala-Busse, P.G. Clark, Definability in mining incomplete data, in: Proceedings of the 20th International Conference on Knowledge-based and Intelligent Information and Engineering System, York, September 2016, pp. 179e186. [23] D. Liu, T.R. Li, J.B. Zhang, A rough set-based incremental approach for learning knowledge in dynamic incomplete information systems, Int. J. Approx. Reason. 55 (8) (2014) 1764e1786. [24] M. Wang, N.Q. Hu, G.J. Qin, A method for rule extraction based on granular computing: application in the fault diagnosis of a helicopter transmission system, J. Intell. Rob. Syst. 71 (3e4) (2013) 1764e1786. [25] H.M. Chen, T.R. Li, Maintenance of approximations in incomplete ordered decision systems while attribute values coarsening or refining, Knowl. Base Syst. 31 (2012) 140e161. [26] Y.H. Qian, J.Y. Liang, Y.Y. Yao, C.Y. Dang, MGRS: a multi-granulation rough set, Inf. Sci. 180 (6) (2010) 949e970. [27] Y.H. Qian, J.Y. Liang, C.Y. Dang, Incomplete multigranulation rough set, IEEE Trans. Syst. Man Cybern. Syst. Hum. 40 (2) (2010) 420e431. [28] V. Muralidharan, V. Sugumaran, A comparative study of Naive Bayes classifier and Bayes net classifier for fault diagnosis of monoblock centrifugal pump using wavelet analysis, Appl. Soft Comput. 12 (8) (2012) 2023e2029. [29] Y.L. He, R. Wang, S. Kwong, X.Z. Wang, Bayesian classifiers based on probability density estimation and their applications to simultaneous fault diagnosis, Inf. Sci. 259 (2014) 252e268. [30] J. Yu, W.T. Huang, X.Z. Zhao, Combined flow graphs and normal naive Bayesian classifier for fault diagnosis of gear box, Proc. IME C J. Mech. Eng. Sci. 230 (2) (2016) 303e313. [31] M. Askarian, G. Escudero, M. Graells, R. Zarghami, F.J. Farahani, N. Mostoufi, Fault diagnosis of chemical processes with incomplete observations: a comparative study, Comput. Chem. Eng. 84 (2016) 104e116. [32] M.C. Noll, J.W. Godfrey, R. Schelenz, G. Jacobs, Analysis of time-domain signals of piezoelectric strain sensors on slow spinning planetary gearboxes, Mech. Syst. Signal Process. 72e73 (2016) 727e744. [33] Y.G. Lei, D.T. Kong, J. Lin, M.J. Zuo, Fault detection of planetary gearboxes using new diagnostic parameters, Meas. Sci. Technol. 23 (5) (2012), 055605. [34] K. Feng, K.S. Wang, Q. Ni, M.J. Zuo, D.D. Wei, A phase angle based diagnostic scheme to planetary gear faults diagnostics under non-stationary operational conditions, J. Sound Vib. 408 (2017) 190e209.