# Simultaneous-Fault Diagnosis of Satellite Power System Based on Fuzzy Neighborhood ζ-Decision-Theoretic Rough Set

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## Abstract

**:**

## 1. Introduction

- (1)
- A novel and concise data-driven loss function matrix is designed for DTRS.
- (2)
- A fuzzy neighborhood ζ-decision-theoretic rough set model is proposed with the help of the fuzzy neighborhood relationship and the proposed loss function matrix, which can deal with hybrid data common in engineering.
- (3)
- The proposed FNζDTRS model, used for attribute reduction, has a significant advantage in classification accuracy compared with other existing rough sets. This proves that it is more suitable for real fault diagnosis.
- (4)
- A diagnosis strategy of simultaneous-fault is put forward based on a coupling mapping relationship between single-fault and its associated simultaneous-fault. This ensures that our strategy can handle both single-fault and simultaneous-fault.
- (5)
- The proposed strategy is successfully applied to the simultaneous-fault diagnosis of the satellite power system and only requires single-fault samples in the training phase, which is highly feasible for practical applications.

## 2. Preliminaries and Related Work

**Definition**

**1.**

## 3. Fuzzy Neighborhood ζ-Decision-Theoretic Rough Set

#### 3.1. Granular Computing Based on Fuzzy Neighborhood Relationship

**Definition**

**2.**

#### 3.2. Determination of the Two Threshold Parameters

**Case**

**1:**

**Case**

**2:**

**Theorem**

**1.**

_{1}) ${\lambda}_{PP}\le {\lambda}_{BP}<{\lambda}_{NP}$,

_{2}) ${\lambda}_{NN}\le {\lambda}_{BN}<{\lambda}_{PN}$.

_{1}) ${\lambda}_{PP}\le {\lambda}_{BP}<{\lambda}_{NP}$

_{2}) ${\lambda}_{NN}={\lambda}_{BN}={\lambda}_{PN}=0$

**Proof.**

_{1}) If ${[x]}^{\delta}\u2288X$, then $\tilde{S}\left(X|{[x]}^{\delta}\right)>0$, $0<\tilde{P}\left(X|{[x]}^{\delta}\right)<1$. Since $0\le \zeta <1$, ${\lambda}_{BP}=\tilde{S}\left(X|{[x]}^{\delta}\right)\tilde{P}\left(X|{[x]}^{\delta}\right)\zeta $, then $0\le {\lambda}_{BP}<\tilde{S}\left(X|{[x]}^{\delta}\right)$. Due to ${\lambda}_{PP}=0$ and ${\lambda}_{NP}=\tilde{S}\left(X|{[x]}^{\delta}\right)$, hence ${\lambda}_{PP}\le {\lambda}_{BP}<{\lambda}_{NP}$.

_{2}) If ${[x]}^{\delta}\u2288X$, then ${\tilde{S}}^{c}\left(X|{[x]}^{\delta}\right)>0$, $0<\tilde{P}\left(X|{[x]}^{\delta}\right)<1$. Since $0\le \zeta <1$, ${\lambda}_{BN}={\tilde{S}}^{c}\left(X|{[x]}^{\delta}\right)\left(1-\tilde{P}\left(X|{[x]}^{\delta}\right)\right)\zeta $, then $0\le {\lambda}_{BN}<{\tilde{S}}^{c}\left(X|{[x]}^{\delta}\right)$. Due to ${\lambda}_{NN}=0$ and ${\lambda}_{PN}={\tilde{S}}^{c}\left(X|{[x]}^{\delta}\right)$, hence ${\lambda}_{NN}\le {\lambda}_{BN}<{\lambda}_{PN}$.

_{1}) If ${[x]}^{\delta}\subseteq X$, then $\tilde{S}\left(X|{[x]}^{\delta}\right)>0$, $\tilde{P}\left(X|{[x]}^{\delta}\right)=1$. Since $0\le \zeta <1$, ${\lambda}_{BP}=\tilde{S}\left(X|{[x]}^{\delta}\right)\tilde{P}\left(X|{[x]}^{\delta}\right)\zeta $, then $0\le {\lambda}_{BP}<\tilde{S}\left(X|{[x]}^{\delta}\right)$. Due to ${\lambda}_{PP}=0$ and ${\lambda}_{NP}=\tilde{S}\left(X|{[x]}^{\delta}\right)$, hence ${\lambda}_{PP}\le {\lambda}_{BP}<{\lambda}_{NP}$.

_{2}) If ${[x]}^{\delta}\subseteq X$, then ${\tilde{S}}^{c}\left(X|{[x]}^{\delta}\right)=0$, and $\tilde{P}\left(X|{[x]}^{\delta}\right)=1$. Since $0\le \zeta <1$, ${\lambda}_{BN}={\tilde{S}}^{c}\left(X|{[x]}^{\delta}\right)\left(1-\tilde{P}\left(X|{[x]}^{\delta}\right)\right)\zeta $, then ${\lambda}_{BN}=0$. Due to ${\lambda}_{NN}=0$ and ${\lambda}_{PN}={\tilde{S}}^{c}\left(X|{[x]}^{\delta}\right)$, hence ${\lambda}_{NN}={\lambda}_{BN}={\lambda}_{PN}=0$. QED. □

#### 3.3. Establishment of FNζDTRS

**Case**

**1:**

**Case**

**2:**

**Theorem**

**2.**

**Proof.**

**Part I:**For ${\zeta}_{1}\ge {\zeta}_{2}\Rightarrow {\alpha}^{fn}\left({\zeta}_{1}\right)\le {\alpha}^{fn}\left({\zeta}_{2}\right)$, two cases need to be considered.

**Case**

**1:**

**Part II:**For ${\zeta}_{1}\ge {\zeta}_{2}\Rightarrow {\beta}^{fn}\left({\zeta}_{1}\right)\ge {\beta}^{fn}\left({\zeta}_{2}\right)$, two cases need to be considered as well.

**Case**

**1:**

**Theorem**

**3.**

**Proof.**

**Part I:**Proof of $0\le {\beta}^{fn}$ under two cases.

**Case**

**1:**

**Case**

**2:**

**Part II:**Proof of the inequality ${\beta}^{fn}\le {\alpha}^{fn}$ in two cases.

**Case**

**1:**

**Case**

**2:**

**Part III:**Proof of the inequality ${\alpha}^{fn}\le 1$ in two cases.

**Case**

**1:**

**Case**

**2:**

**Theorem**

**4.**

**Proof.**

#### 3.4. FNζDTRS-Based Attribute Reduction Algorithm

## 4. Strategy of Simultaneous-Fault Diagnosis

## 5. Numerical Experiment of Attribute Reduction

#### 5.1. Parameters Test for FNζDTRS

#### 5.2. Comparison Experiments on Attribute Reduction

- (a)
- The analysis based on the classification accuracy indicates that the FNζDTRS model is superior to other rough set models. The main reason may lie in the different methods to describe spatial granules. Discretization methods such as EF and SMDNS are commonly introduced to process continuous data in the traditional DTRS models, which results in the destruction of the spatial structure of granules. Using special measures (such as fuzzy relationship, neighborhood relationship, etc.) can avoid the distortion of the discretization method, but it also has some disadvantages, such as simple measurement, insufficient description ability, etc. The proposed FNζDTRS model utilizes fuzzy neighborhood relationships to overcome the above shortcomings. Compared with other DTRS models, the description of spatial granules is more precise in our model and results in the higher classification accuracy.
- (b)
- With respect to the number of reduction attributes, the FDTRS model has the least number of reduction attributes, but it fails to achieve a desired classification accuracy, whereas the FNζDTRS model can maintain high classification accuracy while keeping the number of reduction attributes small. The results show that the classification ability can be maintained or improved only when the reduction attributes are accurately selected. The above conclusion also conforms to the basic principle of attribute reduction, that is, in the operation of reduction, we want to get a relatively concise set, which can ensure that the original classification accuracy is not reduced, and the purpose is to improve the operation efficiency.
- (c)
- The standard deviation is used to measure the robustness of models. It is obvious that the standard deviation of the FNζDTRS model is the smallest regardless of the classification accuracy or the number of reduced attributes, which directly proves that the robustness of the FNζDTRS model is the highest compared to other models. The above robustness characteristics also show that we have a large selection range when setting our two parameters, which is conducive to the wide application of the model in practical projects.

## 6. Simultaneous-Fault Diagnosis of Satellite Power System

#### 6.1. Simultaneous-Fault Diagnosis Based on the Fault Matching Strategy

_{3}, its corresponding shunt current data can directly distinguish F3 from F4. The shunt current value of F3 fluctuates between 6.42–8.67 and that of F4 is between 12.45–14.77. Therefore, F3 and F4 can be distinguished by setting the threshold value of the shunt current average.

#### 6.2. Comparison Experiment on Simultaneous-Fault Diagnosis

#### 6.2.1. Experimental Setup

#### 6.2.2. Experimental Results and Analysis

## 7. Conclusions

- (1)
- The proposed FNζDTRS model performs attribute reduction more effectively compared with other models, and it has strong generalization ability. This benefits from the concise loss functions and the introduction of the fuzzy neighborhood relationships. The advantages of our model can greatly promote the smooth implementation of the model in simultaneous-fault diagnosis, which reflects the effectiveness and superiority of our selection of this model.
- (2)
- The proposed FNζDTRS–FMS does not require simultaneous-fault samples to accomplish training and performs excellently in simultaneous-fault diagnosis compared to classic multi-label classification algorithms. This is completely consistent with the real situation, that is, the existing data cannot completely cover all the imagined failure modes. Therefore, the diagnostic strategy proposed in this paper has stronger application value.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$U=\left\{{x}_{1},{x}_{2},\cdots ,{x}_{m}\right\}$ | the universe, which is a finite and nonempty set. |

$D$ | the set of decision attributes that is a nonempty set. |

$C$ | the collection of conditional attributes. |

$X$ | the subset of samples with the same label ${d}_{k}$. |

${a}_{P}$, ${a}_{B}$, and ${a}_{N}$ | the classification of x into three regions, which are $x\in POS\left(X\right)$, $x\in BND\left(X\right)$, $x\in NEG\left(X\right)$. |

$POS\left(X\right)$ | the acceptance of the event $x\in X$. |

$BND\left(X\right)$ | the non-commitment of the event $x\in X$, denotes the deferment of the event $x\in X$. |

$NEG\left(X\right)$ | the rejection of $x\in X$. |

${\lambda}_{\u2022P}$ | the loss caused by taking actions (${a}_{P}$, ${a}_{B}$, ${a}_{N}$) while $x\in X$. |

${\lambda}_{\u2022N}$ | the loss caused by taking actions (${a}_{P}$, ${a}_{B}$, ${a}_{N}$) while $x\notin X$. |

$\alpha ,\beta $ | the threshold parameters of the DTRS model. |

$\delta $ | fuzzy neighborhood radius. |

$\zeta $ | the compensation coefficient. |

## References

- Li, S.; Cao, H.; Yang, Y. Data-driven simultaneous fault diagnosis for solid oxide fuel cell system using multi-label pattern identification. J. Power Sources
**2018**, 378, 646–659. [Google Scholar] [CrossRef] - Suo, M.; Tao, L.; Zhu, B.; Chen, Y.; Lu, C.; Ding, Y. Soft decision-making based on decision-theoretic rough set and Takagi-Sugeno fuzzy model with application to the autonomous fault diagnosis of satellite power system. Aerosp Sci. Technol.
**2020**, 106, 106108. [Google Scholar] [CrossRef] - Suo, M.; Zhu, B.; An, R.; Sun, H.; Xu, S.; Yu, Z. Data-driven fault diagnosis of satellite power system using fuzzy Bayes risk and SVM. Aerosp Sci. Technol.
**2019**, 84, 1092–1105. [Google Scholar] [CrossRef] - Asgari, S.; Gupta, R.; Puri, I.K.; Zheng, R. A data-driven approach to simultaneous fault detection and diagnosis in data centers. Appl. Soft Comput.
**2021**, 110, 107638. [Google Scholar] [CrossRef] - Liang, P.; Deng, C.; Wu, J.; Yang, Z.; Zhu, J.; Zhang, Z. Single and simultaneous fault diagnosis of gearbox via a semi-supervised and high-accuracy adversarial learning framework. Knowl.-Based Syst.
**2020**, 198, 105895. [Google Scholar] [CrossRef] - Zhang, Z.; Li, S.; Xiao, Y.; Yang, Y. Intelligent simultaneous fault diagnosis for solid oxide fuel cell system based on deep learning. Appl. Energ.
**2019**, 233, 930–942. [Google Scholar] [CrossRef] - Pooyan, N.; Shahbazian, M.; Salahshoor, K.; Hadian, M. Simultaneous Fault Diagnosis using multi class support vector machine in a Dew Point process. J. Nat. Gas. Sci. Eng.
**2015**, 23, 373–379. [Google Scholar] [CrossRef] - Vong, C.; Wong, P.; Ip, W. A New Framework of Simultaneous-Fault Diagnosis Using Pairwise Probabilistic Multi-Label Classification for Time-Dependent Patterns. Ieee T Ind. Electron.
**2013**, 60, 3372–3385. [Google Scholar] [CrossRef] - Wong, P.K.; Zhong, J.; Yang, Z.; Vong, C.M. Sparse Bayesian extreme learning committee machine for engine simultaneous fault diagnosis. Neurocomputing
**2016**, 174, 331–343. [Google Scholar] [CrossRef] - Wu, B.; Cai, W.; Chen, H.; Zhang, X. A hybrid data-driven simultaneous fault diagnosis model for air handling units. Energy Build.
**2021**, 245, 111069. [Google Scholar] [CrossRef] - Zhang, M.; Zhou, Z. A Review on Multi-Label Learning Algorithms. IEEE Trans. Knowl. Data Eng.
**2014**, 26, 1819–1837. [Google Scholar] [CrossRef] - Boutell, M.R.; Luo, J.B.; Shen, X.P.; Brown, C.M. Learning multi-label scene classification. Pattern Recognit.
**2004**, 37, 1757–1771. [Google Scholar] [CrossRef] - Read, J.; Pfahringer, B.; Holmes, G.; Frank, E. Classifier chains for multi-label classification. Mach. Learn.
**2011**, 85, 333–359. [Google Scholar] [CrossRef] - Fuernkranz, J.; Huellermeier, E.; Mencia, E.L.; Brinker, K. Multilabel classification via calibrated label ranking. Mach. Learn.
**2008**, 73, 133–153. [Google Scholar] [CrossRef] - Zhang, M.; Zhou, Z. ML-KNN: A lazy learning approach to multi-label leaming. Pattern Recognit.
**2007**, 40, 2038–2048. [Google Scholar] [CrossRef] - Clare, A.; King, R.D. Knowledge Discovery in Multi-label Phenotype Data. In European Conference on Principles of Data Mining and Knowledge Discovery; Springer: Berlin/Heidelberg, Germany, 2001; pp. 42–53. [Google Scholar]
- Pawlak, Z. Rough sets. Int. J. Comput. Inf. Sci.
**1982**, 11, 341–356. [Google Scholar] [CrossRef] - Dong, L.; Chen, D.; Wang, N.; Lu, Z. Key energy-consumption feature selection of thermal power systems based on robust attribute reduction with rough sets. Inf. Sci.
**2020**, 532, 61–71. [Google Scholar] [CrossRef] - Su, L.; Yu, F. Matrix approach to spanning matroids of rough sets and its application to attribute reduction. Theor. Comput. Sci.
**2021**, 893, 105–116. [Google Scholar] [CrossRef] - Sahu, R.; Dash, S.R.; Das, S. Career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory. Decis. Mak. Appl. Manag. Eng.
**2021**, 4, 104–126. [Google Scholar] [CrossRef] - Dash, S.R.; Dehuri, S.; Sahoo, U.K. Interactions and Applications of Fuzzy, Rough, and Soft Set in Data Mining. Int. J. Fuzzy Syst. Appl.
**2015**, 3, 37–50. [Google Scholar] [CrossRef] - Zhang, P.; Li, T.; Wang, G.; Luo, C.; Chen, H.; Zhang, J.; Wang, D.; Yu, Z. Multi-source information fusion based on rough set theory: A review. Inf. Fusion
**2021**, 68, 85–117. [Google Scholar] [CrossRef] - Bai, H.; Ge, Y.; Wang, J.; Li, D.; Liao, Y.; Zheng, X. A method for extracting rules from spatial data based on rough fuzzy sets. Knowl.-Based Syst.
**2014**, 57, 28–40. [Google Scholar] [CrossRef] - Landowski, M.; Landowska, A. Usage of the rough set theory for generating decision rules of number of traffic vehicles. Transp. Res. Procedia
**2019**, 39, 260–269. [Google Scholar] [CrossRef] - Sharma, H.K.; Kumari, K.; Kar, S. A rough set theory application in forecasting models. Decis. Mak. Appl. Manag. Eng.
**2020**, 3, 1–21. [Google Scholar] [CrossRef] - Guo, Y.; Tsang, E.C.C.; Xu, W.; Chen, D. Adaptive weighted generalized multi-granulation interval-valued decision-theoretic rough sets. Knowl.-Based Syst.
**2020**, 187, 104804. [Google Scholar] [CrossRef] - Wang, T.; Liu, W.; Zhao, J.; Guo, X.; Terzija, V. A rough set-based bio-inspired fault diagnosis method for electrical substations. Int. J. Electr. Power Energy Syst.
**2020**, 119, 105961. [Google Scholar] [CrossRef] - Sang, B.; Yang, L.; Chen, H.; Xu, W.; Guo, Y.; Yuan, Z. Generalized multi-granulation double-quantitative decision-theoretic rough set of multi-source information system. Int. J. Approx. Reason.
**2019**, 115, 157–179. [Google Scholar] [CrossRef] - Zhang, P.F.; Li, T.R.; Yuan, Z.; Luo, C.; Liu, K.Y.; Yang, X.L. Heterogeneous Feature Selection Based on Neighborhood Combination Entropy. IEEE Trans. Neural Netw. Learn. Syst.
**2022**, 1–14. [Google Scholar] [CrossRef] - Wang, L.; Shen, J.; Mei, X. Cost Sensitive Multi-Class Fuzzy Decision-theoretic Rough Set Based Fault Diagnosis. In Proceedings of the 2017 36th Chinese Control Conference (CCC), Dalian, China, 26–28 July 2017; pp. 6957–6961. [Google Scholar]
- Yu, J.; Ding, B.; He, Y. Rolling bearing fault diagnosis based on mean multigranulation decision-theoretic rough set and non-naive Bayesian classifier. J. Mech. Sci. Technol.
**2018**, 32, 5201–5211. [Google Scholar] [CrossRef] - Suo, M.; Tao, L.; Zhu, B.; Miao, X.; Liang, Z.; Ding, Y.; Zhang, X.; Zhang, T. Single-parameter decision-theoretic rough set. Inf. Sci.
**2020**, 539, 49–80. [Google Scholar] [CrossRef] - Yao, Y.Y.; Wong, S.K.M. A decision theoretic framework for approximating concepts. Int. J. Man Mach. Stud.
**1992**, 37, 793–809. [Google Scholar] [CrossRef] - Suo, M.; Cheng, Y.; Zhuang, C.; Ding, Y.; Lu, C.; Tao, L. Extension of labeled multiple attribute decision making based on fuzzy neighborhood three-way decision. Neural Comput. Appl.
**2020**, 32, 17731–17758. [Google Scholar] [CrossRef] - Jia, X.; Liao, W.; Tang, Z.; Shang, L. Minimum cost attribute reduction in decision-theoretic rough set models. Inf. Sci.
**2013**, 219, 151–167. [Google Scholar] [CrossRef] - Yao, Y. Three-way decisions with probabilistic rough sets. Inf. Sci.
**2010**, 180, 341–353. [Google Scholar] [CrossRef] - Jiang, F.; Sui, Y. A novel approach for discretization of continuous attributes in rough set theory. Knowl.-Based Syst.
**2015**, 73, 324–334. [Google Scholar] [CrossRef] - Li, W.; Huang, Z.; Jia, X.; Cai, X. Neighborhood based decision-theoretic rough set models. Int. J. Approx. Reason.
**2016**, 69, 1–17. [Google Scholar] [CrossRef] - Song, J.; Tsang, E.C.C.; Chen, D.; Yang, X. Minimal decision cost reduct in fuzzy decision-theoretic rough set model. Knowl.-Based Syst.
**2017**, 126, 104–112. [Google Scholar] [CrossRef] - Wang, C.; Qi, Y.; Shao, M.; Hu, Q.; Chen, D.; Qian, Y.; Lin, Y. A Fitting Model for Feature Selection with Fuzzy Rough Sets. IEEE Trans. Fuzzy Syst.
**2017**, 25, 741–753. [Google Scholar] [CrossRef] - Wang, C.; Shao, M.; He, Q.; Qian, Y.; Qi, Y. Feature subset selection based on fuzzy neighborhood rough sets. Knowl.-Based Syst.
**2016**, 111, 173–179. [Google Scholar] [CrossRef]

$\mathit{Q}$ | ${\mathit{Q}}^{\mathit{c}}$ | |
---|---|---|

${a}_{P}$ | ${\lambda}_{PP}$ | ${\lambda}_{PN}$ |

${a}_{B}$ | ${\lambda}_{BP}$ | ${\lambda}_{BN}$ |

${a}_{N}$ | ${\lambda}_{NP}$ | ${\lambda}_{NN}$ |

$\mathit{Q}$ | ${\mathit{Q}}^{\mathit{c}}$ | |
---|---|---|

${a}_{p}$ | ${\lambda}_{PP}=0$ | ${\lambda}_{PN}={\tilde{S}}^{c}\left(X|{[x]}^{\delta}\right)$ |

${a}_{B}$ | ${\lambda}_{BP}=\tilde{S}\left(X|{[x]}^{\delta}\right)\tilde{P}\left(X|{[x]}^{\delta}\right)\zeta $ | ${\lambda}_{BN}={\tilde{S}}^{c}\left(X|{[x]}^{\delta}\right)\left(1-\tilde{P}\left(X|{[x]}^{\delta}\right)\right)\zeta $ |

${a}_{N}$ | ${\lambda}_{NP}=\tilde{S}\left(X|{[x]}^{\delta}\right)$ | ${\lambda}_{NN}=0$ |

Input: | Raw data of each single fault and normal state |

Output: | Fault mode |

Part I | Prior Knowledge Acquisition |

For each DS regarding to single fault or state | |

Initialized: red = $\varnothing ,Cl$ = C, Rred = H, //H is a large positive number. | |

While Cl ≠ ∅ | |

For $c\in Cl$ | |

$a=c\cup $ red, //$a$ is a temporary set. | |

Compute the risk generated by $a$. | |

End For | |

Find such a subset $a=c\cup red$ with the minimum risk, i.e., $Ra$. | |

If $Ra<Rred$ | |

The subset $a$ is the selected set | |

End If | |

End While | |

End For, return the reduction $red$ set of each state. | |

Part II | Rule Matching |

Utilize the above code to obtain the reduction set $r$ of the given fault data to be diagnosed. | |

For each $red$ | |

Compute the similarity between $red$ and $r$. | |

End For | |

Find such a $red$ with the maximum similarity, which could be considered as the similar fault mode f. | |

Return f |

ID | Full Name | Name | Samples | Attribute | Discrete | Continuous | Class | Source |
---|---|---|---|---|---|---|---|---|

1 | Mutagenesis-Atoms | Atoms | 1618 | 10 | 8 | 2 | 2 | KEEL |

2 | Australian Credit Approval | Australian | 690 | 14 | 8 | 6 | 2 | UCI |

3 | Breast Cancer | Breast | 277 | 9 | 6 | 3 | 2 | UCI |

4 | Heart Disease Cleveland | Cleve | 296 | 13 | 7 | 6 | 2 | UCI |

5 | Statlog Heart | Heart | 270 | 13 | 6 | 7 | 2 | UCI |

6 | Iris | Iris | 150 | 4 | 0 | 4 | 3 | UCI |

7 | Website Phishing | Phishing | 1353 | 10 | 10 | 0 | 3 | UCI |

8 | South African Hearth | Saheart | 462 | 9 | 1 | 8 | 2 | UCI |

9 | Seismic-Bumps | Seismic | 2584 | 18 | 12 | 6 | 2 | UCI |

10 | Congressional Voting Records | Vote | 435 | 16 | 16 | 0 | 2 | UCI |

ID | Name | DTRS-EF | DTRS-SMDNS | SPDTRS-EF | SPDTRS-SMDNS | NDTRS | FDTRS | FN3WD | FNζDTRS |
---|---|---|---|---|---|---|---|---|---|

1 | Atoms | 69.65 ± 2.38 | 71.08 ± 1.66 | 70.94 ± 1.20 | 71.27 ± 1.14 | 70.42 ± 1.61 | 70.87 ± 2.07 | 71.71 ± 1.00 | 72.08 ± 1.13 |

2 | Australian | 81.34 ± 5.09 | 82.39 ± 5.64 | 82.27 ± 2.87 | 83.46 ± 0.85 | 83.31 ± 2.76 | 72.06 ± 12.31 | 84.56 ± 0.39 | 84.97 ± 0.42 |

3 | Breast | 72.31 ± 1.06 | 72.53 ± 0.82 | 72.67 ± 0.70 | 72.65 ± 0.72 | 72.72 ± 0.67 | 70.38 ± 0.75 | 73.21 ± 0.29 | 74.01 ± 0.81 |

4 | Cleve | 79.13 ± 1.09 | 78.34 ± 5.02 | 78.99 ± 1.01 | 79.37 ± 0.61 | 77.69 ± 6.68 | 66.64 ± 11.29 | 80.13 ± 0.78 | 81.34 ± 0.37 |

5 | Heart | 78.34 ± 3.38 | 78.72 ± 5.20 | 79.99 ± 1.87 | 79.95 ± 0.88 | 76.31 ± 7.16 | 68.37 ± 10.42 | 80.27 ± 2.25 | 80.99 ± 1.23 |

6 | Iris | 94.85 ± 0.50 | 94.85 ± 0.62 | 94.93 ± 0.47 | 94.87 ± 0.55 | 94.95 ± 0.41 | 62.96 ± 17.98 | 94.82 ± 0.40 | 95.27 ± 0.33 |

7 | Phishing | 84.23 ± 9.62 | 86.05 ± 6.11 | 87.08 ± 1.10 | 87.08 ± 1.10 | 86.40 ± 5.08 | 69.93 ± 14.06 | 87.15 ± 1.06 | 87.14 ± 1.07 |

8 | Saheart | 69.23 ± 0.99 | 69.01 ± 1.29 | 69.49 ± 0.38 | 69.51 ± 0.43 | 69.35 ± 0.40 | 67.93 ± 1.99 | 69.63 ± 1.61 | 70.35 ± 1.06 |

9 | Seismic | 92.64 ± 0.80 | 92.33 ± 0.90 | 92.53 ± 0.46 | 91.94 ± 0.69 | 91.91 ± 0.70 | 92.01 ± 0.76 | 92.56 ± 0.48 | 93.21 ± 0.15 |

10 | Vote | 94.66 ± 0.38 | 94.67 ± 0.38 | 94.63 ± 0.36 | 94.63 ± 0.36 | 94.65 ± 0.38 | 83.86 ± 15.36 | 94.63 ± 0.36 | 94.65 ± 0.31 |

Average | 81.64 ± 2.53 | 82.00 ± 2.76 | 82.35 ± 1.04 | 82.47 ± 0.73 | 81.77 ± 2.59 | 72.50 ± 8.70 | 82.87 ± 0.86 | 83.40 ± 0.69 |

ID | Name | DTRS-EF | DTRS-SMDNS | SPDTRS-EF | SPDTRS-SMDNS | NDTRS | FDTRS | FN3WD | FNζDTRS |
---|---|---|---|---|---|---|---|---|---|

1 | Atoms | 5.5 ± 1.8 | 6.4 ± 1.8 | 6.8 ± 0.4 | 7.8 ± 0.4 | 5.2 ± 1.2 | 4.5 ± 1.3 | 1.2 ± 0.4 | 2.0 ± 0.0 |

2 | Australian | 11.2 ± 1.9 | 12.2 ± 3.0 | 11.3 ± 0.6 | 13.0 ± 0.1 | 12.7 ± 1.4 | 4.7 ± 2.7 | 10.8 ± 0.6 | 8.4 ± 0.5 |

3 | Breast | 7.7 ± 2.7 | 8.5 ± 2.0 | 9.0 ± 0.0 | 9.0 ± 0.1 | 9.0 ± 0.2 | 3.6 ± 1.0 | 8.0 ± 0.0 | 6.9 ± 0.6 |

4 | Cleve | 9.8 ± 0.5 | 12.5 ± 2.4 | 9.9 ± 0.5 | 13.0 ± 0.2 | 10.2 ± 2.6 | 4.1 ± 2.7 | 7.1 ± 1.6 | 3.0 ± 0.0 |

5 | Heart | 7.1 ± 2.0 | 12.1 ± 2.6 | 7.9 ± 0.6 | 12.7 ± 0.4 | 10.0 ± 3.0 | 4.8 ± 3.1 | 6.8 ± 1.3 | 3.0 ± 0.0 |

6 | Iris | 3.8 ± 0.7 | 3.6 ± 0.8 | 3.6 ± 0.5 | 4.0 ± 0.1 | 4.0 ± 0.0 | 1.7 ± 1.3 | 2.6 ± 0.5 | 1.0 ± 0.0 |

7 | Phishing | 8.3 ± 2.3 | 8.8 ± 1.4 | 9.0 ± 0.0 | 9.0 ± 0.0 | 8.8 ± 1.4 | 1.1 ± 0.2 | 9.0 ± 0.0 | 9.0 ± 0.0 |

8 | Saheart | 8.7 ± 1.6 | 8.5 ± 1.9 | 9.0 ± 0.0 | 9.0 ± 0.0 | 9.0 ± 0.0 | 6.4 ± 2.8 | 4.8 ± 0.4 | 2.6 ± 0.3 |

9 | Seismic | 6.6 ± 4.8 | 10.9 ± 6.3 | 4.0 ± 0.2 | 13.4 ± 0.9 | 13.8 ± 0.5 | 8.3 ± 0.7 | 2.0 ± 0.0 | 1.0 ± 0.0 |

10 | Vote | 8.5 ± 0.6 | 8.5 ± 0.6 | 8.5 ± 0.5 | 8.5 ± 0.5 | 8.5 ± 0.8 | 1.1 ± 0.3 | 8.5 ± 0.5 | 8.1 ± 0.3 |

Average | 7.7 ± 1.9 | 9.2 ± 2.3 | 7.9 ± 0.3 | 9.9 ± 0.3 | 9.1 ± 1.1 | 4.0 ± 1.6 | 6.1 ± 0.5 | 4.5 ± 0.2 |

ID | Attribute | Rate Range | Data Type | Unit |
---|---|---|---|---|

a_{1} | Duty cycle | 0–1 | Continuous | - |

a_{2} | Bus current | 13.5–17.3 | Continuous | A |

a_{3} | Shunt current | 5.3–12.4 | Continuous | A |

a_{4} | Battery current | 3.6–19.4 | Continuous | A |

a_{5} | Output power | 1070–1090 | Continuous | W |

a_{6} | Battery pressure | 2.0–5.4 | Continuous | MPa |

a_{7} | Battery quantity | 54.3–71.2 | Continuous | Ah |

a_{8} | Status word | −1 0 1 | Discrete | - |

a_{9} | Bus voltage | 40.5–43.1 | Continuous | V |

a_{10} | Battery voltage | 33.0–40.5 | Continuous | V |

Scenario | Fault Name | Scenario | Fault Name |
---|---|---|---|

0 | ---- | 6 | F1-F3 |

1 | F1 | 7 | F1-F4 |

2 | F2 | 8 | F2-F3 |

3 | F3 | 9 | F2-F4 |

4 | F4 | 10 | F1-F2-F3 |

5 | F1-F2 | 11 | F1-F2-F4 |

Fault Name | The Output Attribute Subset | Average Value of the Data for Attribute a_{3} |
---|---|---|

F1 | a_{5} | - |

F2 | a_{7} | - |

F3 | a_{2}, a_{3}, a_{9} | 7.46 |

F4 | a_{2}, a_{3}, a_{9} | 13.47 |

Fault Name | The Output Attribute Subset | Jaccard Similarity Coefficient | Average Value of the Data for Attribute a_{3} | Matching Result | |||
---|---|---|---|---|---|---|---|

F1 | F2 | F3 | F4 | ||||

F1-F2 | a_{5}, a_{7} | 0.50 | 0.50 | 0 | 0 | - | F1-F2 |

F1-F3 | a_{2}, a_{3}, a_{5}, a_{9} | 0.25 | 0 | 0.75 | 0.75 | 6.63 | F1-F3 |

F1-F4 | a_{2}, a_{3}, a_{5}, a_{9} | 0.25 | 0 | 0.75 | 0.75 | 12.62 | F1-F4 |

F2-F3 | a_{2}, a_{3}, a_{7}, a_{9} | 0 | 0.25 | 0.75 | 0.75 | 7.47 | F2-F3 |

F2-F4 | a_{2}, a_{3}, a_{7}, a_{9} | 0 | 0.25 | 0.75 | 0.75 | 13.46 | F2-F4 |

F1-F2-F3 | a_{2}, a_{3}, a_{5}, a_{7}, a_{9} | 0.20 | 0.20 | 0.60 | 0.60 | 6.63 | F1-F2-F3 |

F1-F2-F4 | a_{2}, a_{3}, a_{5}, a_{6}, a_{7}, a_{9} | 0.17 | 0.17 | 0.50 | 0.50 | 12.63 | F1-F2-F4 |

Algorithm | Accuracy/Subset Accuracy | Hamming Loss | Precision | Recall | F1 |
---|---|---|---|---|---|

FNζDTRS–FMS | 100.0 ± 0.0 | - | 100.0 ± 0.0 | 100.0 ± 0.0 | 100.0 ± 0.0 |

Binary Relevance | 82.34 ± 5.39 | 4.43 ± 1.35 | 100.0 ± 0.0 | 93.88 ± 1.87 | 96.28 ± 1.14 |

Classifier Chain | 65.78 ± 8.93 | 9.30 ± 2.63 | 100.0 ± 0.0 | 86.42 ± 4.45 | 91.31 ± 3.07 |

Calibrated Label Ranking | 0.0 ± 0.0 | 32.14 ± 7.11 | 100.0 ± 0.0 | 45.24 ± 0.0 | 61.90 ± 0.0 |

ML-KNN | 0.0 ± 0.0 | 32.15 ± 0.0 | 99.98 ± 0.0 | 45.23 ± 7.11 | 61.89 ± 0.0 |

ML-DT | 0.0 ± 0.0 | 32.35 ± 0.31 | 99.59 ± 0.62 | 45.03 ± 0.31 | 61.63 ± 0.42 |

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## Share and Cite

**MDPI and ACS Style**

Tao, L.; Wang, C.; Jia, Y.; Zhou, R.; Zhang, T.; Chen, Y.; Lu, C.; Suo, M.
Simultaneous-Fault Diagnosis of Satellite Power System Based on Fuzzy Neighborhood *ζ*-Decision-Theoretic Rough Set. *Mathematics* **2022**, *10*, 3414.
https://doi.org/10.3390/math10193414

**AMA Style**

Tao L, Wang C, Jia Y, Zhou R, Zhang T, Chen Y, Lu C, Suo M.
Simultaneous-Fault Diagnosis of Satellite Power System Based on Fuzzy Neighborhood *ζ*-Decision-Theoretic Rough Set. *Mathematics*. 2022; 10(19):3414.
https://doi.org/10.3390/math10193414

**Chicago/Turabian Style**

Tao, Laifa, Chao Wang, Yuan Jia, Ruzhi Zhou, Tong Zhang, Yiling Chen, Chen Lu, and Mingliang Suo.
2022. "Simultaneous-Fault Diagnosis of Satellite Power System Based on Fuzzy Neighborhood *ζ*-Decision-Theoretic Rough Set" *Mathematics* 10, no. 19: 3414.
https://doi.org/10.3390/math10193414