加速寿命数据的贝叶斯建模与分析

doi:10.12305/j.issn.1001-506X.2021.05.32

摘要/Abstract

摘要：

在加速寿命试验的可靠性设计中, 随机化设计的限制以及删失数据不可避免地导致低分位数估计出现较大的偏差。针对上述的问题, 结合贝叶斯抽样技术以及非线性混合模型(nonlinear mixed model, NLMM)提出了一种可靠性改进的分析方法。首先, 需要检验所收集的数据是否服从威布尔分布以及验证形状参数是否是恒定常数。其次, 考虑随机效应对尺度参数和形状参数的影响, 运用NLMM构建了尺度参数和形状参数与试验因子之间的函数关系。然后, 利用贝叶斯方法估计低分位数的可靠性寿命。最后, 实际案例研究表明, 在考虑删失问题和未完全随机设计的影响时, 所提方法能够获得更为稳健和可靠的估计结果。

关键词: 删失数据, 贝叶斯理论, 分位数估计, 随机效应, 非线性混合模型

Abstract:

In the reliability design of accelerated life tests, the limitations of randomized design and censored data inevitably lead to significant deviations in low percentile estimation. Given the above problems, a reliability-improved analysis method is proposed combining Bayesian sampling technology and nonlinear mixed model. Firstly, it is necessary to check whether the collected data follow the Weibull distribution and verify whether the shape parameters are constant. Secondly, considering the effects of random effects on scale parameters and shape parameters, a nonlinear mixed model is used to construct the functional relationship between scale parameters, shape parameters and test factors. Thirdly, a Bayesian method is used to estimate the reliability life of low percentiles. Finally, a practical example reveals that the proposed method can obtain more robust and reliable estimation results under considering the impact of censored data and non-randomization.

Key words: censored data, Bayesian theory, percentile estimate, random effect, nonlinear mixed model(NLMM)

中图分类号:

TB472

汪建均, 杨桂康, 冯泽彪. 加速寿命数据的贝叶斯建模与分析[J]. 系统工程与电子技术, 2021, 43(5): 1420-1429.

Jianjun WANG, Guikang YANG, Zebiao FENG. Bayesian modeling and analysis of accelerated life data[J]. Systems Engineering and Electronics, 2021, 43(5): 1420-1429.

图/表 11

表1

图1

表2

图2

图3

图4

表3

表4

表5

模型Ⅰ~Ⅴ的分位数估计以及置信区间"

次序	模型	t_.01	t_.05	t_.1	t_.5
	Ⅰ	280.76[179.5, 475.3]	508.00[367.1, 728.4]	660.08[498.4, 882.5]	1 309.92[1 026.5, 1 576.8]
	Ⅱ	284.51[183.2, 516.9]	511.90[367.7, 768.7]	663.50[491.0, 926.4]	1 308.16[972.5, 1 625.1]
1	Ⅲ	375.80[200.1, 675.40]	609.45[405.9, 882.8]	754.52[550.8, 1 003.7]	1 319.33[1 080.2, 1 531.2]
	Ⅳ	427.31[243.2, 779.3]	655.91[452.0, 965.4]	792.55[589.8, 1 067.3]	1 300.51[1 081.9, 1 492.2]
	Ⅴ	381.80[119.9, 635.9]	591.50[287.4, 859.5]	722.60[416.9, 994.9]	1 250.00[951.3, 1 703.0]
	Ⅰ	208.92[137.1, 344.4]	378.02[285.5, 520.7]	491.18[389.9, 629.3]	974.750[821.5, 1 100.1]
	Ⅱ	211.94[142.2, 372.9]	381.33[284.3, 550.1]	494.26[383.8, 659.9]	974.490[772.0, 1 138.9]
2	Ⅲ	224.37[149.5, 374.1]	394.17[302.7, 544.1]	505.54[408.7, 645.6]	969.600[840.4, 1 074.5]
	Ⅳ	235.33[158.5, 420.6]	407.01[310.8, 587.7]	518.42[417.0, 682.6]	976.520[835.1, 1 098.4]
	Ⅴ	288.00[91.98, 487.9]	475.50[234.6, 706.3]	596.60[347.7, 843.8]	1 101.00[833.9, 1 475.0]
	Ⅰ	155.46[102.3, 258.1]	281.29[211.9, 389.1]	365.50[289.9, 470.4]	725.340[603.9, 829.80]
	Ⅱ	157.88[106.1, 275.5]	284.07[213.7, 407.7]	368.19[287.7, 488.5]	725.930[576.9, 847.60]
3	Ⅲ	129.05[73.20, 239.5]	248.86[171.6, 368.1]	332.59[248.6, 449.2]	710.460[582.4, 826.90]
	Ⅳ	117.99[66.60, 249.3]	237.72[163.6, 379.7]	323.90[242.1, 459.4]	727.790[594.2, 846.80]
	Ⅴ	131.80[33.08, 238.7]	246.30[104.9, 378.9]	327.20[168.6, 481.1]	708.500[493.8, 997.90]
	Ⅰ	115.68[74.20, 197.6]	209.32[150.7, 302.8]	271.98[203.7, 366.0]	539.740[416.0, 666.00]
	Ⅱ	117.61[77.10, 211.6]	211.61[152.6, 317.2]	274.28[203.9, 382.3]	540.770[403.4, 668.30]
4	Ⅲ	71.060[19.20, 190.3]	152.77[66.50, 285.3]	214.20[113.5, 345.9]	518.770[381.6, 660.30]
	Ⅳ	52.470[11.80, 181.9]	128.49[49.30, 278.2]	190.83[92.50, 341.3]	537.250[400.6, 664.50]
	Ⅴ	72.860[2.948, 183.1]	152.80[19.28, 304.5]	216.20[49.85, 387.1]	584.000[371.1, 925.40]
	Ⅰ	210.28[132.6, 362.9]	380.48[270.6, 553.1]	494.38[367.4, 674.3]	981.100[750.7, 1 210.8]
	Ⅱ	211.60[135.6, 382.9]	380.71[271.1, 572.8]	493.46[363.7, 687.2]	972.910[723.7, 1 208.5]
5	Ⅲ	316.52[153.4, 571.1]	493.95[311.5, 723.4]	601.23[421.2, 809.3]	1 005.63[798.2, 1 202.1]
	Ⅳ	437.26[284.6, 654.3]	571.27[431.2, 745.7]	642.87[514.2, 794.1]	875.620[730.3, 997.60]
	Ⅴ	481.30[203.9, 713.6]	658.80[373.1, 877.8]	759.00[483.1, 961.8]	1 110.00[923.6, 1 353.0]
	Ⅰ	156.48[102.4, 261.0]	283.13[211.3, 394.2]	367.89[289.6, 475.7]	730.060[605.7, 836.80]
	Ⅱ	157.63[104.6, 277.4]	283.61[210.3, 409.4]	367.59[284.7, 490.6]	724.750[574.3, 846.80]
6	Ⅲ	192.40[122.1, 325.4]	323.19[239.6, 449.7]	406.39[320.9, 522.5]	740.110[633.5, 827.00]
	Ⅳ	273.33[203.4, 404.7]	384.71[316.7, 4912.0]	447.40[382.0, 539.1]	664.150[582.0, 723.10]
	Ⅴ	194.60[87.61, 283.3]	287.70[174.5, 388.3]	343.30[235.7, 453.1]	553.300[424.6, 798.40]
	Ⅰ	116.44[76.80, 193.3]	210.68[159.8, 290.7]	273.75[218.2, 351.1]	543.260[454.8, 615.10]
	Ⅱ	117.42[78.50, 205.3]	211.27[157.0, 304.0]	273.83[213.6, 365.2]	539.890[427.0, 630.20]
7	Ⅲ	113.00[70.70, 196.0]	206.82[151.5, 293.9]	270.11[211.2, 353.7]	543.210[453.8, 622.80]
	Ⅳ	161.14[117.7, 251.7]	249.46[200.1, 331.0]	302.57[251.9, 376.4]	501.410[422.2, 565.50]
	Ⅴ	139.70[58.35, 215.1]	214.60[127.6, 292.4]	260.60[172.0, 343.6]	440.000[334.6, 598.90]
	Ⅰ	86.640[56.00, 147.0]	156.77[114.6, 224.4]	203.71[154.6, 271.2]	404.260[317.0, 489.10]
	Ⅱ	87.470[57.30, 157.5]	157.38[113.6, 235.9]	203.99[151.6, 283.7]	402.190[298.4, 497.30]
8	Ⅲ	63.760[23.80, 148.3]	128.97[68.50, 219.8]	176.05[108.0, 265.6]	397.420[301.1, 493.50]
	Ⅳ	88.150[41.70, 179.6]	154.13[94.00, 243.2]	197.27[134.4, 280.3]	376.300[288.6, 454.50]
	Ⅴ	118.10[27.33, 216.2]	192.90[75.76, 296.6]	241.30[110.4, 352.8]	444.800[325.4, 602.80]

表5

图5

图6

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方法	尺度参数	形状参数
文献[5]	log η_i=θ₀+θ₁ Volt+θ₂ Temp	常数
文献[10]	log η_i=θ₀+θ₁ Volt+θ₂ Temp+μ_i	常数
文献[6]	log η_i=θ₀+θ₁ Volt+θ₂ Temp	log β_i=γ₀+γ₁ Volt+γ₂ Temp
文献[2]	log η_i=θ₀+θ₁ Volt+θ₂ Temp+μ_i	log β_i=γ₀+γ₁ Volt+γ₂ Temp
本文方法	log η_i=θ₀+θ₁ Volt+θ₂ Temp+μ_i	log β_i=γ₀+γ₁ Volt+γ₂ Temp+ε_i

试验次序	温度/℃	电压/V	寿命数据
1	170	200	439	904	1 092	1 105	1 105*	1 105*	1 105*	1 105*
2	170	250	572	690	904	1 090	1 090*	1 090*	1 090*	1 090*
3	170	300	315	315	439	628	628*	628*	628*	628*
4	170	350	258	258	347	588	588*	588*	588*	588*
5	180	200	959	1 065	1 065	1 087	1 087*	1 087*	1 087*	1 087*
6	180	250	216	315	455	473	473*	473*	473*	473*
7	180	300	241	315	332	380	380*	380*	380*	380*
8	180	350	241	241	435	455	455*	455*	455*	455*

参数	Ⅰ	Ⅱ	Ⅲ	Ⅳ	Ⅴ
θ₀	13.407 0	13.519 70	13.216 3	15.646 50	15.000 0
θ₁	-0.028 9	-0.029 60	-0.028 0	-0.043 20	-0.039 5
θ₂	-0.005 9	-0.005 90	-0.005 8	-0.005 20	-0.005 1
γ₀	-	-	0.419 7	-5.700 70	-12.190 0
γ₁	-	-	0.008 2	0.047 20	0.110 5
γ₂	-	-	-0.003 1	-0.004 90	-0.013 1
σ_μ	-	0.047 46	-	0.184 14	0.262 7
σ_ε	-	-	-	-	0.400 8

试验次序	Ⅰ		Ⅱ		Ⅲ		Ⅳ		Ⅴ
试验次序	尺度参数	形状参数	尺度参数	形状参数	尺度参数	形状参数	尺度参数	形状参数	尺度参数	形状参数
1	1 496.77	2.75	1 492.86	2.78	1 470.850	3.37	1 432.060	3.80	1 397.0	3.77
2	1 113.79	2.75	1 112.08	2.78	1 100.570	2.89	1 104.550	2.98	1 244.0	3.23
3	828.80	2.75	828.43	2.78	823.510	2.48	851.940	2.33	828.3	2.56
4	616.73	2.75	617.13	2.78	616.195	2.13	657.105	1.82	723.0	2.01
5	1 121.00	2.75	1 110.27	2.78	1 111.480	3.66	929.870	6.10	1 198.0	5.32
6	834.20	2.75	827.08	2.78	831.670	3.14	717.210	4.77	608.8	4.30
7	620.75	2.75	616.12	2.78	622.300	2.70	553.189	3.73	488.6	3.87
8	461.92	2.75	458.97	2.78	465.640	2.31	426.676	2.92	503.5	3.27

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