SCIENCE CHINA Information Sciences, Volume 64 , Issue 7 : 172214(2021) https://doi.org/10.1007/s11432-020-3001-7

## New health-state assessment model based on belief rule base with interpretability

• AcceptedJun 29, 2020
• PublishedMay 20, 2021
Share
Rating

### Acknowledgment

This work was supported in part by National Natural Science Foundation of China (Grant Nos. 61773388, 61702142, 61751304, 61833016, 61867001), China Postdoctoral Science Foundation (Grant Nos. 2015M570847, 2016T90938), Key Research and Development Plan of Hainan (Grant No. ZDYF2019007), Natural Science Foundation of Hainan (Grant No. 2019CXTD405), and Guangxi Key Laboratory of Trusted Software (Grant No. KX202050).

### References

[1] Zhao F J, Zhou Z J, Hu C H. A New Evidential Reasoning-Based Method for Online Safety Assessment of Complex Systems. IEEE Trans Syst Man Cybern Syst, 2018, 48: 954-966 CrossRef Google Scholar

[2] Narmada S, Jacob M. Reliability analysis of a complex system with a deteriorating standby unit under common-cause failure and critical human error. MicroElectron Reliability, 1996, 36: 1287-1290 CrossRef Google Scholar

[3] Dao C D, Zuo M J. Optimal selective maintenance for multi-state systems in variable loading conditions. Reliability Eng Syst Saf, 2017, 166: 171-180 CrossRef Google Scholar

[4] Yin X, Wang Z, Zhang B. A Double Layer BRB Model for Health Prognostics in Complex Electromechanical System. IEEE Access, 2017, 5: 23833-23847 CrossRef Google Scholar

[5] Chen Y W, Yang J B, Xu D L. On the inference and approximation properties of belief rule based systems. Inf Sci, 2013, 234: 121-135 CrossRef Google Scholar

[6] Hinton G E. Reducing the Dimensionality of Data with Neural Networks. Science, 2006, 313: 504-507 CrossRef ADS Google Scholar

[7] Mulder Y D, Wyseur B, Preneel B. Cryptanalysis of a perturbated white-box AES implementation. In: Proceedings of International Conference on Cryptology in India, 2010. 292--310. Google Scholar

[8] Jian-Bo Yang , Jun Liu , Jin Wang . Belief rule-base inference methodology using the evidential reasoning Approach-RIMER. IEEE Trans Syst Man Cybern A, 2006, 36: 266-285 CrossRef Google Scholar

[9] Zhou Z J, Hu G Y, Hu C H. A Survey of Belief Rule-Base Expert System. IEEE Trans Syst Man Cybern Syst, 2019, : 1-15 CrossRef Google Scholar

[10] Zhou Z J, Hu C H, Yang J B. Online updating belief rule based system for pipeline leak detection under expert intervention. Expert Syst Appl, 2009, 36: 7700-7709 CrossRef Google Scholar

[11] Chang L, Dong W, Yang J. Hybrid belief rule base for regional railway safety assessment with data and knowledge under uncertainty. Inf Sci, 2020, 518: 376-395 CrossRef Google Scholar

[12] Cheng C, Qiao X, Teng W. Principal component analysis and belief-rule-base aided health monitoring method for running gears of high-speed train. Sci China Inf Sci, 2020, 63: 199202 CrossRef Google Scholar

[13] Zhou Z, Feng Z, Hu C. A hidden fault prediction model based on the belief rule base with power set and considering attribute reliability. Sci China Inf Sci, 2019, 62: 202202 CrossRef Google Scholar

[14] Yang J B, Liu J, Xu D L. Optimization Models for Training Belief-Rule-Based Systems. IEEE Trans Syst Man Cybern A, 2007, 37: 569-585 CrossRef Google Scholar

[15] Chen Y W, Yang J B, Xu D L. Inference analysis and adaptive training for belief rule based systems. Expert Syst Appl, 2011, 38: 12845-12860 CrossRef Google Scholar

[16] Chang L, Zhou Y, Jiang J. Structure learning for belief rule base expert system: A comparative study. Knowledge-Based Syst, 2013, 39: 159-172 CrossRef Google Scholar

[17] Chang L L, Zhou Z J, Chen Y W. Belief Rule Base Structure and Parameter Joint Optimization Under Disjunctive Assumption for Nonlinear Complex System Modeling. IEEE Trans Syst Man Cybern Syst, 2018, 48: 1542-1554 CrossRef Google Scholar

[18] Zhou Z J, Hu C H, Yang J B. Online Updating Belief-Rule-Base Using the RIMER Approach. IEEE Trans Syst Man Cybern A, 2011, 41: 1225-1243 CrossRef Google Scholar

[19] Zhou Z J, Hu C H, Yang J B. A sequential learning algorithm for online constructing belief-rule-based systems. Expert Syst Appl, 2010, 37: 1790-1799 CrossRef Google Scholar

[20] Li G, Zhou Z, Hu C. A new safety assessment model for complex system based on the conditional generalized minimum variance and the belief rule base. Saf Sci, 2017, 93: 108-120 CrossRef Google Scholar

[21] Casillas J, Cordón O, Herrera F, et al. Accuracy Improvements in Linguistic Fuzzy Modeling. Berlin: Springer, 2013. Google Scholar

[22] Zadeh L A. A Fuzzy-Set-Theoretic Interpretation of Linguistic Hedges. J Cybernetics, 1972, 2: 4-34 CrossRef Google Scholar

[23] Zhou S M, Gan J Q. Low-level interpretability and high-level interpretability: a unified view of data-driven interpretable fuzzy system modelling. Fuzzy Sets Syst, 2008, 159: 3091-3131 CrossRef Google Scholar

[24] Wu W K, Yang L H, Fu Y G, et al. Parameter training approach for belief rule base using the accelerating of gradient algorithm. J Front Comput Sci Technol, 2009, 22: 5166--5170. Google Scholar

[25] Li G L, Zhou Z J, Hu C H, et al. An optimal safety assessment model for complex systems considering correlation and redundancy. Int J Approx Reason, 2019 104: 38--56. Google Scholar

[26] Chang L L, Zhou Z J, Liao H. Generic Disjunctive Belief-Rule-Base Modeling, Inferencing, and Optimization. IEEE Trans Fuzzy Syst, 2019, 27: 1866-1880 CrossRef Google Scholar

• Figure 1

(Color online) The relationship between $P({\bf{\Omega~}})$ and $D({\bf{\Omega~}},{{\bf{\Omega~}}^0})$.

• Figure 2

(Color online) 2000 points generated by using (9) with $\kappa~=~1,2$ and (7). (a) The point distribution when $\kappa~=~1$;protect łinebreak (b) the point distribution when $\kappa~=~2$; (c) the point distribution generated by (7).

• Figure 3

(Color online) Conflict belief distributions of (a) W-shape, (b) U-shape, and (c) V-shape in the health-state assessment.

• Figure 4

(Color online) The structure of the health-state assessment model based on interpretable BRB.

• Figure 5

(Color online) The replacement operation.

• Figure 6

(Color online) The original vibration data and features. (a) The original vibration data; (b) kurtosis; (c) skewness.

• Figure 7

(Color online) (a) The estimated health state and (b) the corresponding belief degrees.

• Figure 8

(Color online) The errors between model outputs (BRB1, BRB2, FRB, ELM) and the actual health state. (a) The errors of BRB1, FRB, and ELM; (b) the errors of BRB1 and BRB2.

• Figure 9

(Color online) The rule weights in BRB0, BRB1, and BRB2.

• Figure 10

(Color online) The belief distributions in BRB0, BRB1, and BRB2.

• Table 1

Table 1Referential values of attributes and output

 Attribute $\delta_i$ L M H VH Output N S M SE $K$ 1 0.1 2.4 5.2 9.2 Health state 0 0.25 0.75 1 $S$ 1 0 0.8 1.4 2.9
• Table 2

Table 2The initial rules

 Number ${\theta~_k}$ $K~\wedge~S$ Health state levels Health Number ${\theta~_k}$ $K~\wedge~S$ Health state levels Health ${\textstyle{{\{~\rm~N,~~~S,~~M,~~SE\}~}~\over~{\{~{\beta~_1},{\beta~_2},{\beta~_3},{\beta~_4}\}~}}}$ state ${\textstyle{{\{\rm~N,~~~S,~~M,~~SE\}~}~\over~{\{~{\beta~_1},{\beta~_2},{\beta~_3},{\beta~_4}\}~}}}$ state 1 0.7 L $\wedge$ L 0.00, 0.10, 0.53, 0.37 0.7925 9 1 H $\wedge$ L 0.00, 0.40, 0.60, 0.00 0.55 2 0.8 L $\wedge$ M 0.00, 0.00, 0.00, 1.00 1 10 0.7 H $\wedge$ M 0.00, 0.75, 0.25, 0.00 0.375 3 0.7 L $\wedge$ H 0.00, 0.00, 0.30, 0.70 0.925 11 1 H $\wedge$ H 0.15, 0.85, 0.00, 0.00 0.2125 4 0.8 L $\wedge$ VH 0.20, 0.60, 0.20, 0.00 0.3 12 1 H $\wedge$ VH 0.80, 0.20, 0.00, 0.00 0.05 5 1 M $\wedge$ L 0.00, 0.00, 0.90, 0.10 0.775 13 0.8 VH $\wedge$ L 0.85, 0.15, 0.00, 0.00 0.0375 6 0.7 M $\wedge$ M 0.00, 0.00, 0.00, 1.00 1 14 1 VH $\wedge$ M 0.40, 0.60, 0.00, 0.00 0.15 7 0.7 M $\wedge$ H 0.00, 0.00, 0.75, 0.25 0.8125 15 1 VH $\wedge$ H 0.90, 0.10, 0.00, 0.00 0.025 8 1 M $\wedge$ VH 0.10, 0.70, 0.20, 0.00 0.325 16 0.7 VH $\wedge$ VH 1.00, 0.00, 0.00, 0.00 0
• Table 3

Table 3The initial referential values of health state

 The parameter Value The parameter Value The parameter Value Maximum generation $G$ 100 The zoom factor $F$ $[0.2,05]$ Population size $\lambda$ 50 Covariance matrix ${\boldsymbol{\sigma~}}~=~\kappa~*~{\boldsymbol~I}$ $\kappa~=~0.1$ The crossover rate $Cr$ 0.2
• Table 4

Table 4The optimized rules

 Number ${\theta~_k}$ $K~\wedge~S$ Health state levels ${\textstyle{{\{\rm~N,~S,~M,~SE\}~}~\over~{\{~{\beta~_1},{\beta~_2},{\beta~_3},{\beta~_4}\}~}}}$ Health state $[0,1]$ 1 0.69 L $\wedge$ L 0.020, 0.068, 0.505, 0.407 0.80275 2 0.96 L $\wedge$ M 0.000, 0.007, 0.031, 0.962 0.987 3 0.62 L $\wedge$ H 0.027, 0.081, 0.234, 0.658 0.85375 4 0.8 L $\wedge$ VH 0.20, 0.60, 0.20, 0.00 0.3 5 1.00 M $\wedge$ L 0.016, 0.048, 0.868, 0.068 0.731 6 0.61 M $\wedge$ M 0.000, 0.000, 0.019, 0.981 0.99525 7 0.60 M $\wedge$ H 0.002, 0.009, 0.727, 0.262 0.8095 8 0.91 M $\wedge$ VH 0.104, 0.644, 0.141, 0.111 0.37775 9 1.00 H $\wedge$ L 0.000, 0.119, 0.639, 0.242 0.751 10 0.60 H $\wedge$ M 0.108, 0.770, 0.119, 0.003 0.28475 11 1.00 H $\wedge$ H 0.232, 0.768, 0.000, 0.000 0.192 12 0.98 H $\wedge$ VH 0.980, 0.017, 0.003, 0.000 0.0065 13 0.8 VH $\wedge$ L 0.85, 0.15, 0.00, 0.00 0.0375 14 1 VH $\wedge$ M 0.40, 0.60, 0.00, 0.00 0.15 15 0.99 VH $\wedge$ H 0.903, 0.086, 0.008, 0.003 0.0305 16 1.00 VH $\wedge$ VH 0.943, 0.045, 0.007, 0.005 0.0215
• Table 5

Table 5MSEs generated by five models

 BRB0 BRB1 BRB2 FRB ELM MSE 0.0114 0.0054 0.0051 0.0092 0.0064
• Table 6

Table 6Average MSEs generated by using three optimization algorithms

 Algorithm MSE Satisfy interpretability constraint in Eq. (12) Be consistent with initial judgement PSO-I 0.0049 YES YES GA-I 0.0057 YES YES SA-I 0.0052 YES YES

Citations

Altmetric