Cosine Similarity Measure between Vague Sets and Its Application of Fault Diagnosis

In order to propose a novel cosine similarity measure between vague sets and to apply it to the fault diagnosis of turbine, a new similarity measure value between a testing sample and the knowledge of system faults is evaluated in the vibration fault diagnosis of turbine. The testing sample is near to a type of fault knowledge if the measure value is big. Thus, the type of vibration fault is determined according to the maximum measure value (more than some threshold). The fault-diagnosis problems of the turbine are investigated by use of the proposed cosine similarity measure. The results demonstrate that the proposed method not only diagnoses the main fault types of the turbine, but also provides useful information for multi-fault analyses and future trends. Therefore, the proposed method is reasonable and effective and provides another useful tool for fault analyses.


INTRODUCTION
The technique of fault diagnosis has produced the huge economic benefits by scheduling preventive maintenance and preventing extensive downtime periods caused by extensive failure.Therefore, it becomes a research hotspot.In the past, various faultdiagnosis techniques have been proposed, including expert systems (Wang and Yang, 1996), neural networks (Chen et al., 1996).The expert system can take human expertise and has been successfully applied in this field.However, there are some intrinsic shortcomings for the expert system, such as the difficulty of acquiring knowledge and maintaining a database.
These may vary from utility to utility due to the heuristic nature of the method and no general mathematical formulation can be utilized.Neural networks can directly acquire experience from training data and exhibit highly nonlinear input-output relationships.This can overcome some of the shortcomings of the expert system.However, the training data must be sufficient and compatible to ensure proper training.A further limitation of the approach of neural networks is its inability to produce linguistic output, because it is difficult to understand the content of network memory.To overcome above shortcomings, Wang (2004) proposed a vibration fault diagnosis method of generator sets based on extension theory.Then Ye (2006Ye ( , 2009) ) proposed the fault diagnosis methods of turbine based on similarity measures between vague sets and fuzzy cross entropy of vague sets.
In fact, the degree of similarity or dissimilarity between the objects under study plays an important role.In vector space, especially, the cosine similarity measure (Salton and McGill, 1987) is often used in information retrieval, citation analysis and automatic classification.However, this similarity measure does not deal with the similarity measures for vague information.Therefore, Ye (2011) proposed a cosine similarity measure between intuitionistic fuzzy sets and applied it to pattern recognition and medical diagnosis.The main purposes of this study are to propose another cosine similarity measure between vague sets in vector space based on the extension of the cosine similarity measure (Salton and McGill, 1987) and to apply it to the fault diagnosis of turbine.Then the feasibility and rationality of the proposed fault diagnosis method is validated by the fault diagnosis example of the turbine.

COSINE SIMILARITY MEASURE
Let X = (x 1 , x 2 , ..., x n ) and Y = (y 1 , y 2 , ..., y n ) be the two vectors of length n where all the coordinates are positive.The cosine similarity measure between the two vectors (measuring the "similarity" of these vectors) (Salton and McGill, 1987) is defined as: where is the inner product of the vectors X and Y and where are the Euclidean norms of X and Y (also called the L 2 norms).
Cosine formula is then defined as the inner product of two vectors divided by the product of their lengths.This is nothing but the cosine of the angle between the vectors.The cosine similarity measure takes value in the interval [0, l].It is undefined if x i = 0 or/and y i = 0 (i = 1, 2, …, n).Then, let the cosine measure value be zero when x i = 0 or/and y i = 0 (i = 1, 2, …, n).

VAGUE SET AND ITS SIMILARITY MEASURE
Vague set: The vague set, which is a generalization of the concept of a fuzzy set, has been introduced by Gau and Buehrer (1993).Vague set theory is introduced as follows.
A vague set A in the universe of discourse X = {x 1 , x 2 , …, x n } is characterized by a truth-membership function t A (x i ) and a false-membership function f A (x i ) for an element , where t A (x i ) is a lower bound on the grade of membership of x i derived from the evidence for x i , f A (x i ) is a lower bound on the negation of x i derived from the evidence against x i and the functions t A (x i ) and f A (x i ) are constrained by the condition 0≤t indicates that the exact grade of membership µ A (x i ) of x i may be unknown.But it is bounded by 1 shows a vague set in the universe of discourse X.
In the sequel, we will omit the argument x i of t A (x i ) and f A (x i ) throughout unless they are needed for clarity.
Let X be the universe of discourse Let t i * = 1-f i , where 1≤i≤n.In this case, the vague set A can be rewritten as: Cosine similarity measure between vague sets: Assume that there are two vague sets The parameters (elements) in A and B can be considered as two pairs of vector representations with the length of n elements: ( )

., ,
Based on the extension of cosine similarity measure (Salton and McGill, 1987), a similarity measure between T A and T B is proposed in the vector space as follows: A cosine similarity measure between * A T and *

B
T is proposed in the vector space as follows: Thus, the cosine similarity measure between A and B is proposed in the vector space as follows: where, α∈[0, 1] is the weight of similarity measures.
The cosine similarity measure between vague sets A and B satisfies the following properties:

Proof:
(P1) It is obvious that the property is true according to cosine values for Eq. ( 2) and ( 3).(P2) It is obvious that the property is true.(P3) When A = B, there are t Ai = t and (P4) It is obvious that the property is true.

FAULT DIAGNOSIS BASED ON THE COSINE SIMILARITY MEASURE
In this section, we will apply the cosine similarity measure between vague sets to fault diagnosis.Essentially, the technique of equipment diagnosis is a pattern recognition problem.In other word, the operating status of the machine is divided into normal and abnormal statuses.Further speaking, the signal sample of the abnormality belongs to which type on earth; this is a pattern recognition problem again.

Fault diagnosis principle:
Assume that there exist m fault patterns (knowledge of fault samples), which are represented by a vague set F i (i = 1, 2, …, m) and there is a testing sample to be recognized which is represented by a vague set F t .Then diagnosing result F k should be the nearest one to F t , i.e: where C vs (F t , F i ) expresses the similarity measure between the vague sets F t and F i .The value of C vs (F t , F i ) is calculated by using Eq.(2-4).Then, we decide that the testing sample F t should belong to the fault pattern F k , where, The cosine similarity measure of vague sets provides the strong means for the fault diagnosis.It can realize the classification and identification of the fault, i.e., we can compare the similarity measure values by calculating the similarity measures between a diagnosing sample and knowledge of system faults and then confirm the fault type according to the maximum similarity degree (more than some threshold).The fault diagnosis process using the cosine similarity measure of vague sets is shown in Fig. 2 (Ye, 2009).

Fault diagnosis of turbine:
An example with the steam turbine-generator set demonstrates the effectiveness of the fault diagnosis of turbine by using the proposed fault diagnosis method.The vibration of huge steam turbine-generator sets suffers the influence of a lot of factors, such as mechanical structure, load, vacuum degree, hot inflation of cylinder body and rotor, fluctuation of network load, temperature of lubricant oil and ground.In generator sets, interaction effects in these factors show the vibration of the generator sets.In the vibration fault diagnosis of the generator sets, we have established the relation between the cause and the fault symptom of the turbine adopted from (Ye, 2006(Ye, , 2009)).Now, we investigate the fault diagnosis problems by means of the cosine similarity measure of vague sets to demonstrate the effectiveness of the fault diagnosis problems of turbine.
In the diagnosis process, at first, we establish the knowledge database of fault types and then calculate the cosine similarity measures between a fault-testing sample and fault knowledge samples.
From the above fault diagnostic results, we can see that the proposed diagnosis method is effectiveness.The diagnosis results of the turbine show that the proposed method not only diagnoses the main fault types of the turbine, but also provides useful information for multi-fault analysis and future trends.Therefore, the method can offer effective and reasonable diagnosis information for multi-fault analyses.

CONCLUSION
In this study, we proposed a novel cosine similarity measure between vague sets and its fault diagnosis and investigated the cosine similarity measure for the fault diagnoses of turbine.The diagnosis results of the turbine show that the proposed method not only diagnoses the main fault types of the turbine, but also provides useful information for multi-fault analyses and future trends.The proposed methods in this study are effective and reasonable in fault diagnoses.This fault diagnosis method is easier and more practical than other traditional artificial intelligence methods.Furthermore, the proposed method will provide another useful tool for fault analyses.
Fig. 2: Block diagram of fault diagnosis using the cosine similarity measure of vague sets

Table 1 :
Knowledge of system faultsFrequency range (f: operating frequency)