融合专家先验知识和单调性约束的贝叶斯网络参数学习方法

doi:10.3969/j.issn.1001-506X.2020.03.019

摘要/Abstract

摘要：

针对小样本集条件下的贝叶斯网络参数学习问题,提出一种融合专家先验知识和单调性约束的贝叶斯网络参数学习方法。该方法通过将专家先验知识以正态分布形式融入单调性约束的贝叶斯网络参数学习过程,进一步提高了小样本集条件下贝叶斯网络参数学习的精度和稳定性。在小样本集条件下进行仿真实验,结果表明,与其他3种主要方法相比,所提方法平均(Kullback-Leibler, KL)散度大幅降低,运行时间高于其余3种方法。综合考虑学习精度和运行时间,所提方法优于其他3种方法。将所提方法应用于燃气轮机健康状态评估,评估结果与实际状态一致,验证了方法的有效性。

关键词: 贝叶斯网络, 参数学习, 小样本集, 单调性约束, 正态分布

Abstract:

Aiming at the Bayesian network parameter learning problem under the condition of the small sample set, a Bayesian network parameter learning method combining expert prior knowledge and monotonic constraints is proposed. This method integrates the expert priori knowledge into the process of parameter learning of Bayesian network with monotonic constraints in the form of normal distribution and further improves the accuracy and stability of parameter learning of Bayesian network under the condition of the small sample set. Simulation experiments are conducted under the condition of small sample sets, and the results show that compared with the other three main methods, the average Kullback-Leibler (KL) divergence is significantly reduced, and the running time is higher than the other three methods. Considering the learning accuracy and running time comprehensively, this method is superior to the other three methods mentioned above. The method is applied to the gas turbine health condition evaluation, and the result is consistent with the actual condition, which verifies the effectiveness of the method.

Key words: Bayesian network, parametric learning, small sample set, monotonic constraint, normal distribution

中图分类号:

TP18

曾强, 黄政, 魏曙寰. 融合专家先验知识和单调性约束的贝叶斯网络参数学习方法[J]. 系统工程与电子技术, 2020, 42(3): 646-652.

Qiang ZENG, Zheng HUANG, Shuhuan WEI. Bayesian network parameter learning method based on expert priori knowledge and monotonic constraints[J]. Systems Engineering and Electronics, 2020, 42(3): 646-652.

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节点	参数值
S	P_S\|C(0\|0)=0.2
	P_S\|C(1\|0)=0.8
	P_S\|C(0\|1)=0.6
	P_S\|C(1\|1)=0.4
R	P_R\|C(0\|0)=0.9
	P_R\|C(1\|0)=0.1
	P_R\|C(0\|1)=0.2
	P_R\|C(1\|1)=0.8
W	P_{W\|S, R}(0\|0, 0)=0.9
	P_{W\|S, R}(1\|0, 0)=0.1
	P_{W\|S, R}(0\|0, 1)=0.2
	P_{W\|S, R}(1\|0, 1)=0.8
	P_{W\|S, R}(0\|1, 0)=0.1
	P_{W\|S, R}(1\|1, 0)=0.9
	P_{W\|S, R}(0\|1, 1)=0.01
	P_{W\|S, R}(1\|1, 1)=0.99

编号	约束
1	P_S\|C(1\|0)>P_S\|C(0\|0)
2	P_S\|C(0\|1)>P_S\|C(1\|1)
3	P_R\|C(0\|0)>P_R\|C(1\|0)
4	P_R\|C(1\|1)>P_R\|C(1\|0)
5	P_{W\|S, R}(1\|0, 1)>P_{W\|S, R}(0\|0, 1)
6	P_{W\|S, R}(1\|1, 1)>P_{W\|S, R}(0\|1, 1)
7	P_{W\|S, R}(0\|0, 0)>P_{W\|S, R}(1\|0, 0)
8	P_{W\|S, R}(1\|1, 0)>P_{W\|S, R}(0\|1, 0)

编号	参数先验
1	P_S\|C(0\|0)≈0.2
2	P_S\|C(1\|1)≈0.4
3	P_R\|C(1\|0)≈0.1
4	P_R\|C(0\|1)≈0.2
5	P_{W\|S, R}(1\|0, 0)≈0.1
6	P_{W\|S, R}(0\|0, 1)≈0.2
7	P_{W\|S, R}(0\|1, 0)≈0.1
8	P_{W\|S, R}(0\|1, 1)≈0.01

W节点状态	5样本集				15样本集
W节点状态	MLE	无先验MAP	均匀分布方法	本文方法	MLE	无先验MAP	均匀分布方法	本文方法
P_{W\|S, R}(0\|0, 0)	1.000 0	0.666 7	0.850 4	0.902 6	1.000 0	0.800 0	0.851 9	0.907 3
P_{W\|S, R}(1\|0, 0)	0.000 0	0.333 3	0.149 6	0.097 4	0.000 0	0.200 0	0.148 1	0.092 7
P_{W\|S, R}(0\|0, 1)	0.000 0	0.333 3	0.241 5	0.195 3	0.200 0	0.285 7	0.242 5	0.200 0
P_{W\|S, R}(1\|0, 1)	1.000 0	0.666 7	0.758 5	0.804 7	0.800 0	0.714 3	0.757 5	0.800 0
P_{W\|S, R}(0\|1, 0)	0.000 0	0.200 0	0.173 6	0.092 7	0.000 0	0.166 7	0.167 9	0.090 5
P_{W\|S, R}(1\|1, 0)	1.000 0	0.800 0	0.826 4	0.907 3	1.000 0	0.833 3	0.832 1	0.909 5
P_{W\|S, R}(0\|1, 1)	0.000 0	0.333 3	0.020 0	0.010 0	0.000 0	0.200 0	0.019 5	0.009 8
P_{W\|S, R}(1\|1, 1)	1.000 0	0.666 7	0.980 0	0.990 0	1.000 0	0.800 0	0.980 5	0.990 2

W节点状态	25样本集				35样本集
W节点状态	MLE	无先验MAP	均匀分布方法	本文方法	MLE	无先验MAP	均匀分布方法	本文方法
P_{W\|S, R}(0\|0, 0)	0.750 0	0.666 7	0.847 7	0.897 0	0.888 9	0.818 2	0.851 4	0.899 5
P_{W\|S, R}(1\|0, 0)	0.250 0	0.333 3	0.152 3	0.103 0	0.111 1	0.181 8	0.148 6	0.100 5
P_{W\|S, R}(0\|1, 0)	0.125 0	0.200 0	0.222 6	0.193 9	0.100 0	0.166 7	0.211 0	0.190 0
P_{W\|S, R}(1\|1, 0)	0.875 0	0.800 0	0.777 4	0.806 1	0.900 0	0.833 3	0.789 0	0.810 0
P_{W\|S, R}(0\|0, 1)	0.250 0	0.300 0	0.154 2	0.105 9	0.300 0	0.300 0	0.206 5	0.105 9
P_{W\|S, R}(1\|0, 1)	0.750 0	0.700 0	0.845 8	0.894 1	0.700 0	0.700 0	0.750 0	0.894 1
P_{W\|S, R}(0\|1, 1)	0.000 0	0.166 7	0.019 3	0.009 7	0.000 0	0.125 0	0.019 0	0.009 5
P_{W\|S, R}(1\|1, 1)	1.000 0	0.833 3	0.980 7	0.990 3	1.000 0	0.875 0	0.981 0	0.990 5

样本集	MLE	无先验信息MAP	先验均匀分布方法	本文方法
5	0.021	0.014	0.417	0.424
15	0.022	0.014	0.430	0.433
25	0.023	0.017	0.440	0.437
35	0.025	0.017	0.445	0.443