To solve the difficulties in practice caused by the subjectivity, relativity and evidence combination focus element explosion during the process of solving the uncertain problems of fault diagnosis with evidence theory, this paper proposes a fault diagnosis inference strategy by integrating rough sets with evidence theory along with the theories of information fusion and mete-synthesis. By using rough sets, redundancy of characteristic data is removed and the unrelated essential characteristics are extracted, the objective way of basic probability assignment is proposed, and an evidence synthetic method is put forward to solve high conflict evidence. The method put forward in this paper can improve the accuracy rate of fault diagnosis with the redundant and complementary information of various faults by synthesizing all evidences with the rule of the composition of evidence theory. Besides, this paper proves the feasibility and validity of experiments and the efficiency in improving fault diagnosis.

#### Keywords

- fault diagnosis
- information fusion
- rough sets
- D-S evidence theory

Information fusion is applied widely during fault diagnosis and there are many ways. Evidence theory or D-S theory (DST) [1, 2] can compound the uncertain information from several information sources and is a very effective uncertainty reasoning way in information fusion technology. However, the evidence theory itself has some problems [3], such as dependency on the evidence provided by expert knowledge, focus element explosion caused by evidence combination, rigorous demands to combination conditions (mutual independence among evidences), inefficiency to evidential conflicts and the subjectivity in the distribution of credibility and so on. Many scholars have improved the theory to overcome these limitations, but still there are some deficiencies [4,5,6]. Rough set theory is a new mathematical tool to deal with vagueness and uncertain knowledge and is very practical. It can extract unrelated essential characteristics to eliminate redundancies without any initial or additional data [7, 8]. Both the rough set and evidence theories are math tools for uncertain information and they focus on the group ability. There is a strong complementary relationship between these two theories [9,10,11]. This paper provides a new research thought for intelligent fault diagnosis by combining the evidence theory with rough set theory to solve the subjectivity, relativity, integrated focus element explosion and conflict problems of evidences, improve the accuracy of fault diagnosis and at last carry out online diagnostics.

The paper just gives two definitions about the importance of attribute and the basic ideas about rough set theory, which are referred in the literature [12].

The importance of any attribute

To

Here the paper just gives the D-S evidence theory synthesis formula and the related knowledge, which are referred in the literature [1, 2].

If _{1},_{2},...,_{n}_{1},_{2},...,_{n}

Among them:

From the D-S evidence theory synthesis formula: when

Among them:

Eq. (4) has some subjective factors, while the physical meaning of Eqs (2) and (3) is not clear. Since

Among them:
_{i}

Suppose the conflict sum of evidence _{i}_{i}_{i}_{j}_{j}

Definition 4 shows the greater the amount of self-conflict for evidence, the smaller its weight factor is. Otherwise, the smaller the amount of self-conflict for evidence, the greater its weight factor is. The weight factor shows the important degree of evidences provided by the information source in the synthesis process and the incidence of the synthetic results.

Here is an example to prove the validity of the way. Suppose Θ = {

_{1}(_{1}(_{1}(

_{2}(_{2}(_{2}(

_{3}(_{3}(_{3}(

Three evidences are synthesised by applying the synthetic methods of DST, Yager [14], Sun Quan, Li Bicheng [15] and the synthesis method proposed in this paper, respectively (evidence weight _{1} = 0.37414, _{2} = 0.21902, _{3} = 0.40684) and the results are shown in Table 1. Table 1 shows the evidence synthetic method with the weight factor raised in this paper that can enhance the reasonability and reliability in the evidence combination. It can avoid the subjectivity and randomness from the weight factor by determining the evidence weight coefficient with the amount of evidence self-conflict, reflect the importance of evidence during synthesis and get a better result than other methods.

Comparison of results of various evidence synthesis methods.

DST | 0.99901 | 0 | 0 | 1 | 0 |

Yager | 0.99901 | 0 | 0 | 0.00099 | 0.99901 |

Sun Quan | 0.99901 | 0.3210 | 0.0030 | 0.1880 | 0.4880 |

Li Bicheng | 0.99901 | 0.6260 | 0.0067 | 0.3673 | 0 |

This paper | 0.99901 | 0.7321 | 0.0059 | 0.2620 | 0 |

Application of the attribute reduction of the rough set can effectively optimise the key features as the evidence for diagnosis decisions. Attributes significance of rough set can evaluate the weight of each evidence objectively. And then there comes the formula

This paper creates a fault diagnosis model based on inference strategy to fuse rough set and evidence theories, as shown in Figure 1. First, the basic thought is building an information decision table by discretising the fault sample data sets with continuous attributes. Then, optimising feature parameters suitable for fault diagnosis as theory body by applying the rough set theory for attribute reduction of the decision table. In practice, for the discretised sample set to diagnose, basic probability assignment of related evidences can be calculated based on the reduction and the diagnosis result can be obtained with the inductive decision by the use of the D-S combination rule.

A rough set theory can deal with discretised data, while original sample data are always continuous, so continuous data should be discretised first. There are many ways to discretise and each has its advantage. In practice, each field seeks a proper algorithm according to the characteristic [16, 17].

Symptom attribute reduction can reduce the relativity among evidence and decide the fault with attributes as less as possible, remaining the sorting quality invariant and avoiding focus element explosion. On the other hand, the weight of each attribute can be obtained from the information in the decision table and the subjectivity from experts can be avoided, overcoming the difficulties in practical application efficiently caused by the subjectivity and relativity of evidence during the fault diagnosis with evidence theory. About the reduction algorithm of the rough set theory, still there is no recognised and high-efficient algorithm. With the completion of the reduction algorithm considered, attribute reduction is applied in this paper by combining discernibility matrix, dependability of attribute and the heuristic reduction algorithm of information entropy by improving the importance of attribute. The detailed process of this algorithm is referred in the literature [18, 19].

Basic probability assignment of evidence is realised with the decision attribute

_{1},_{2},...,_{n}, then the measure of lower approximation of X_{i}_{i}

The proof of Theorems 1 and 2 can be referred in the literature [10]. From Theorems 1 and 2, it is possible to calculate belief function based on a decision table with rough set theory. It provides the theoretical basis for the fusion reasoning of evidence theory and rough set theory.

The basic probability assignment

Here

Next, a practical algorism of basic probability assignment is given based on a rough set decision table: {Input: decision table after reduction; samples to be diagnosed}; {Output: basic probability assignments of all evidences}.

Quantification of samples to be diagnosed: to discretise the samples to be diagnosed according to the discretisation criterion of original sample data;

Recognition framework Θ is decision attribute set and the reasoning evidence is condition attributes

To determine the division

To determine the division _{i}

To determine equivalence class in _{i}

To get the basic probability assignment of evidence _{i}

If there is only 1 −

The diagnosis can be concluded based on the decision of basic probability assignment after a combination of the D-S rule [20].

The engine is an important equipment of a ship, concerning the survivability and battle effectiveness of a ship. The performance of the engine can influence and restrict the performance of the technique and tactics of a ship. The structure of the engine is complex, while the fault analysis to it is difficult. To ensure the accuracy of fault diagnosis, more characteristic parameters should be used. The number of characteristics to be extracted gets larger with the increase of various kinds of parameters and this makes the amount of information to be processed too large and cannot meet the needs of online diagnosis. Only a few key characteristics are sensitive to a fault. They are independent and provide complementary information for each other to improve the accuracy of diagnosis. While the redundant characteristics are not sensible to fault or they are related to other characteristics but useless. The rough set theory can eliminate redundant information effectively, select the key characteristics, get probability assignment and make reasoning decisions with D-S combination rule to solve the problems such as subjectivity in evidence obtaining, relativity of evidence and focus element explosion of evidence combination.

After the experiments of five fault phenomena appearing in a kind of ship engine and the management to the diagnostic knowledge, this paper selects 10 sets of data samples to build. The common fault information is shown in Figure 2. In Figure 2, _{1}, airflow _{2}, fuel pressure _{3}, revolution speed _{4}, torque _{5}} stands for the five characteristic (symptom) parameters to describe the state of equipment, _{1}, high-temperature _{2}, airflow meter damage _{3}, fuel injector fault _{4}, ignition fault _{5}} shows the fault symptoms to each state of the equipment. Select three sets of data randomly as the test samples (to be diagnosed) to verify the diagnostic effects from the method proposed in this paper.

The attribute value of the sample data in Table 2 is continuous and needs to be discretised. Here a discretisation for continuous attribute value based on SOFM network classification is used and the details and the computing process of this algorithm can be referred in the literature [21]. The decision table after discretisation is shown in Table 3. After attribute reduction with the method proposed in this paper, only three condition attributes, _{1}, _{2}, _{5}, are kept in the decision table. They are the diagnostic proofs for five fault symptoms: cooling water temperature (_{1}), air flow (_{2}), torque (_{3}).

Table of fault information.

_{1} | _{2} | _{3} | _{4} | _{5} | |||
---|---|---|---|---|---|---|---|

Original samples | 1 | 0.95 | 1.40 | 1.20 | 5100 | 0.39 | _{1} |

2 | 0.70 | 1.74 | 0.94 | 3450 | 0.98 | _{1} | |

4 | 0.06 | 1.73 | 0.98 | 3900 | 0.91 | _{2} | |

5 | 0.65 | 3.31 | 0.63 | 1950 | 0.55 | _{3} | |

7 | 0.30 | 1.31 | 1.32 | 4350 | 0.40 | _{4} | |

8 | 0.28 | 1.23 | 1.11 | 4300 | 0.44 | _{4} | |

10 | 0.05 | 2.47 | 1.01 | 3800 | 0.20 | _{5} | |

Test samples | 3 | 1.02 | 1.73 | 0.57 | 3500 | 0.98 | _{1} |

6 | 0.67 | 3.71 | 0.77 | 1900 | 0.45 | _{3} | |

9 | 0.18 | 0.95 | 1.14 | 3600 | 0.30 | _{5} |

Decision table after discretization.

_{1} | _{2} | _{3} | _{4} | _{5} | |||
---|---|---|---|---|---|---|---|

Original samples | 1 | 3 | 1 | 3 | 3 | 2 | _{1} |

2 | 2 | 2 | 2 | 2 | 3 | _{1} | |

4 | 1 | 2 | 2 | 2 | 3 | _{2} | |

5 | 2 | 3 | 1 | 1 | 2 | _{3} | |

7 | 2 | 1 | 3 | 3 | 2 | _{4} | |

8 | 1 | 1 | 2 | 3 | 2 | _{4} | |

10 | 1 | 2 | 2 | 2 | 1 | _{5} | |

Test samples | 3 | 3 | 2 | 1 | 2 | 3 | verify _{1} |

6 | 2 | 3 | 1 | 1 | 2 | verify _{3} | |

9 | 1 | 1 | 2 | 2 | 1 | verify _{5} |

From Definition 1, the importance of attribute _{1}, _{2}, _{5} about

It shows _{1} is the attribute that influences the accuracy mostly to diagnostic decision rule, followed by _{2} and _{5}. This agrees with the engineering practice. Then the weight of each evidence _{i}

For evidence _{1}, _{2}, _{3}, the basic probability assignment can be calculated with _{3} need not be verified anymore. For the two sets of test samples, _{1} and _{5} are fused with D-S evidence theory and the combining method by improving evidence proposed in the paper, the basic probability assignments are shown in Tables 4 and 5. Θ is the overall uncertainty.

Basic probability assignment about _{1}.

_{1} | _{1} | _{2} | _{3} | _{4} | _{5} | |||
---|---|---|---|---|---|---|---|---|

_{1} | 0 | 0.9500 | 0 | 0 | 0 | 0 | 0.05 | |

_{2} | 0 | 0.3167 | 0.3167 | 0 | 0 | 0.3167 | 0.05 | |

_{3} | 0 | 0.4750 | 0.4750 | 0 | 0 | 0 | 0.05 | |

_{1} ⊕_{2} | DS | 0.9144 | 0.0397 | 0 | 0 | 0.0397 | 0.0062 | |

Improve DS | 0.6017 | 0.8269 | 0.0703 | 0 | 0 | 0.0703 | 0.0325 | |

_{1} ⊕_{3} | DS | 0.9522 | 0.0433 | 0 | 0 | 0 | 0.0045 | |

Improve DS | 0.4512 | 0.8899 | 0.0850 | 0 | 0 | 0 | 0.0251 | |

_{2} ⊕_{3} | DS | 0.4771 | 0.4771 | 0 | 0 | 0.0397 | 0.0062 | |

Improve DS | 0.6017 | 0.4282 | 0.4282 | 0 | 0 | 0.1111 | 0.0325 | |

_{1} ⊕_{2} ⊕_{3} | DS | 0.9488 | 0.0469 | 0 | 0 | 0.0039 | 0.0006 | |

Improve DS | 0.7972 | 0.7534 | 0.1498 | 0 | 0 | 0.0569 | 0.0399 |

Basic probability assignment about _{5}.

_{1} | _{1} | _{2} | _{3} | _{4} | _{5} | |||
---|---|---|---|---|---|---|---|---|

_{1} | 0 | 0 | 0.3167 | 0 | 0.3167 | 0.3167 | 0.05 | |

_{2} | 0 | 0.3167 | 0 | 0 | 0.6334 | 0 | 0.05 | |

_{3} | 0 | 0 | 0 | 0 | 0 | 0.9500 | 0.05 | |

_{1} ⊕_{2} | DS | 0.0531 | 0.0531 | 0 | 0.8326 | 0.0531 | 0.0081 | |

Improve DS | 0.7020 | 0.0794 | 0.1746 | 0 | 0.5339 | 0.1746 | 0.0375 | |

_{1} ⊕_{3} | DS | 0 | 0.0398 | 0 | 0.0398 | 0.9145 | 0.0060 | |

Improve DS | 0.6017 | 0 | 0.1520 | 0 | 0.1520 | 0.6636 | 0.0324 | |

_{2} ⊕_{3} | DS | 0.1624 | 0 | 0 | 0.3248 | 0.4872 | 0.0256 | |

Improve DS | 0.9025 | 0.1587 | 0 | 0 | 0.3175 | 0.4762 | 0.0476 | |

_{1} ⊕_{2} ⊕_{3} | DS | 0.0245 | 0.0245 | 0 | 0.3840 | 0.5634 | 0.0036 | |

Improve DS | 0.9677 | 0.0688 | 0.1707 | 0 | 0.3191 | 0.3932 | 0.0482 |

From Tables 4 and 5, the case cannot be diagnosed and a diagnostic error will occur if the diagnosis is with single evidence. For example, _{2} and _{3} in Figure 4 cannot be diagnosed, _{2} in Table 5 is a diagnostic error. The diagnostic capability of single evidence is limited. So it is necessary to combine all the evidence to get a better diagnosis. The diagnostic capability can be improved because the overall uncertainty reduces in comparison with a single evidence diagnosis after two evidences are combined. But it is possible for the case that cannot be diagnosed or diagnostic errors because of the limitation of self-capability of single evidence and original training samples, for example, _{2} ⊕ _{3} in Table 4 cannot be diagnosed, and _{1} ⊕ _{2} in Table 5 is a diagnostic error. For some evidences, the diagnosis is not certain after fusion, the uncertainty is reduced gradually. After the fusion of all evidences _{1}, _{2}, _{3}, certainty strengthens. With one evidence or two evidences, the case cannot be diagnosed or diagnostic errors may occur, while with three evidences fused the correct diagnosis can be obtained finally.

It shows that the diagnostic capability can be improved by reducing the uncertainty efficiently by fusion reasoning for all the evidences that are not redundant.

The fusion results of the method proposed in this text are not as good as those from D-S evidence theory, according to Tables 4 and 5. Because the former is based on solving the fusion of high conflict evidences (

An integrated diagnostic method with evidence theory and rough set fusion reasoning is put forward and it can overcome the problems of subjectivity, dependency and focus element explosion of traditional evidences. It has a significant theoretical significance and practical value.

This paper proposed a synthetic method by considering the weight factor. The method can solve the synthetic problems of the evidences with high conflict and is effective for synthesising normal evidences.

The application effect of an integrated diagnosis of rough set and evidence theories is proved to be good with diagnosis examples. With the increase of original sample data, the accuracy of diagnosis will improve. The method proposed in this text can be promoted and applied in the fault diagnosis to other devices if the sample data are large enough.

The research of this paper will be further deepened. It will focus on developing artificial intelligence diagnostic system for better application in industrial practice to achieve maximum production benefits in the direction of our subsequent research.

#### Basic probability assignment about D5.

_{1} | _{1} | _{2} | _{3} | _{4} | _{5} | |||
---|---|---|---|---|---|---|---|---|

_{1} | 0 | 0 | 0.3167 | 0 | 0.3167 | 0.3167 | 0.05 | |

_{2} | 0 | 0.3167 | 0 | 0 | 0.6334 | 0 | 0.05 | |

_{3} | 0 | 0 | 0 | 0 | 0 | 0.9500 | 0.05 | |

_{1} ⊕_{2} | DS | 0.0531 | 0.0531 | 0 | 0.8326 | 0.0531 | 0.0081 | |

Improve DS | 0.7020 | 0.0794 | 0.1746 | 0 | 0.5339 | 0.1746 | 0.0375 | |

_{1} ⊕_{3} | DS | 0 | 0.0398 | 0 | 0.0398 | 0.9145 | 0.0060 | |

Improve DS | 0.6017 | 0 | 0.1520 | 0 | 0.1520 | 0.6636 | 0.0324 | |

_{2} ⊕_{3} | DS | 0.1624 | 0 | 0 | 0.3248 | 0.4872 | 0.0256 | |

Improve DS | 0.9025 | 0.1587 | 0 | 0 | 0.3175 | 0.4762 | 0.0476 | |

_{1} ⊕_{2} ⊕_{3} | DS | 0.0245 | 0.0245 | 0 | 0.3840 | 0.5634 | 0.0036 | |

Improve DS | 0.9677 | 0.0688 | 0.1707 | 0 | 0.3191 | 0.3932 | 0.0482 |

#### Comparison of results of various evidence synthesis methods.

DST | 0.99901 | 0 | 0 | 1 | 0 |

Yager | 0.99901 | 0 | 0 | 0.00099 | 0.99901 |

Sun Quan | 0.99901 | 0.3210 | 0.0030 | 0.1880 | 0.4880 |

Li Bicheng | 0.99901 | 0.6260 | 0.0067 | 0.3673 | 0 |

This paper | 0.99901 | 0.7321 | 0.0059 | 0.2620 | 0 |

#### Table of fault information.

_{1} | _{2} | _{3} | _{4} | _{5} | |||
---|---|---|---|---|---|---|---|

Original samples | 1 | 0.95 | 1.40 | 1.20 | 5100 | 0.39 | _{1} |

2 | 0.70 | 1.74 | 0.94 | 3450 | 0.98 | _{1} | |

4 | 0.06 | 1.73 | 0.98 | 3900 | 0.91 | _{2} | |

5 | 0.65 | 3.31 | 0.63 | 1950 | 0.55 | _{3} | |

7 | 0.30 | 1.31 | 1.32 | 4350 | 0.40 | _{4} | |

8 | 0.28 | 1.23 | 1.11 | 4300 | 0.44 | _{4} | |

10 | 0.05 | 2.47 | 1.01 | 3800 | 0.20 | _{5} | |

Test samples | 3 | 1.02 | 1.73 | 0.57 | 3500 | 0.98 | _{1} |

6 | 0.67 | 3.71 | 0.77 | 1900 | 0.45 | _{3} | |

9 | 0.18 | 0.95 | 1.14 | 3600 | 0.30 | _{5} |

#### Decision table after discretization.

_{1} | _{2} | _{3} | _{4} | _{5} | |||
---|---|---|---|---|---|---|---|

Original samples | 1 | 3 | 1 | 3 | 3 | 2 | _{1} |

2 | 2 | 2 | 2 | 2 | 3 | _{1} | |

4 | 1 | 2 | 2 | 2 | 3 | _{2} | |

5 | 2 | 3 | 1 | 1 | 2 | _{3} | |

7 | 2 | 1 | 3 | 3 | 2 | _{4} | |

8 | 1 | 1 | 2 | 3 | 2 | _{4} | |

10 | 1 | 2 | 2 | 2 | 1 | _{5} | |

Test samples | 3 | 3 | 2 | 1 | 2 | 3 | verify _{1} |

6 | 2 | 3 | 1 | 1 | 2 | verify _{3} | |

9 | 1 | 1 | 2 | 2 | 1 | verify _{5} |

#### Basic probability assignment about D1.

_{1} | _{1} | _{2} | _{3} | _{4} | _{5} | |||
---|---|---|---|---|---|---|---|---|

_{1} | 0 | 0.9500 | 0 | 0 | 0 | 0 | 0.05 | |

_{2} | 0 | 0.3167 | 0.3167 | 0 | 0 | 0.3167 | 0.05 | |

_{3} | 0 | 0.4750 | 0.4750 | 0 | 0 | 0 | 0.05 | |

_{1} ⊕_{2} | DS | 0.9144 | 0.0397 | 0 | 0 | 0.0397 | 0.0062 | |

Improve DS | 0.6017 | 0.8269 | 0.0703 | 0 | 0 | 0.0703 | 0.0325 | |

_{1} ⊕_{3} | DS | 0.9522 | 0.0433 | 0 | 0 | 0 | 0.0045 | |

Improve DS | 0.4512 | 0.8899 | 0.0850 | 0 | 0 | 0 | 0.0251 | |

_{2} ⊕_{3} | DS | 0.4771 | 0.4771 | 0 | 0 | 0.0397 | 0.0062 | |

Improve DS | 0.6017 | 0.4282 | 0.4282 | 0 | 0 | 0.1111 | 0.0325 | |

_{1} ⊕_{2} ⊕_{3} | DS | 0.9488 | 0.0469 | 0 | 0 | 0.0039 | 0.0006 | |

Improve DS | 0.7972 | 0.7534 | 0.1498 | 0 | 0 | 0.0569 | 0.0399 |

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