Journal of Computer Applications ›› 2024, Vol. 44 ›› Issue (3): 953-959.DOI: 10.11772/j.issn.1001-9081.2023040459
Special Issue: 前沿与综合应用
• Frontier and comprehensive applications • Previous Articles Next Articles
Lei WANG, Shijuan CHENG(), Yu HAN
Received:
2023-04-21
Revised:
2023-08-14
Accepted:
2023-08-18
Online:
2024-03-12
Published:
2024-03-10
Contact:
Shijuan CHENG
About author:
WANG Lei, born in 1998, M. S. candidate. Her research interests include reliability theory and engineering, information fusion.Supported by:
通讯作者:
程世娟
作者简介:
王蕾(1998—),女,山东潍坊人,硕士研究生,主要研究方向:可靠性理论与工程、信息融合基金资助:
CLC Number:
Lei WANG, Shijuan CHENG, Yu HAN. Reliability evaluation of multi-component system based on time-varying Copula function[J]. Journal of Computer Applications, 2024, 44(3): 953-959.
王蕾, 程世娟, 韩雨. 基于时变Copula函数的多部件系统可靠性评估[J]. 《计算机应用》唯一官方网站, 2024, 44(3): 953-959.
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URL: https://www.joca.cn/EN/10.11772/j.issn.1001-9081.2023040459
Copula函数 | 参数取值范围 | 变换函数 |
---|---|---|
Frank | ||
Plackett | ||
Clayton | ||
Rotated Clayton | ||
Gumbel | ||
Rotated Gumbel | ||
SJC |
Tab. 1 Parameter value ranges and transformation functions of common Copula functions
Copula函数 | 参数取值范围 | 变换函数 |
---|---|---|
Frank | ||
Plackett | ||
Clayton | ||
Rotated Clayton | ||
Gumbel | ||
Rotated Gumbel | ||
SJC |
时变类型 | 时变模式 | 数据生成形式 | 模拟参数设定 | 拟合优度 |
---|---|---|---|---|
随机时变模型 | AR模型 | 0.843 8 | ||
连续时变模型 | 多项式函数 | 0.998 0 | ||
跳跃时变模型 | 符号函数 | 0.864 5 |
Tab. 2 Analysis of Monte Carlo simulation results
时变类型 | 时变模式 | 数据生成形式 | 模拟参数设定 | 拟合优度 |
---|---|---|---|---|
随机时变模型 | AR模型 | 0.843 8 | ||
连续时变模型 | 多项式函数 | 0.998 0 | ||
跳跃时变模型 | 符号函数 | 0.864 5 |
终端 | 设备 | 不同循环次数下的裂纹扩展数据/mm | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
0 | 104 | 2×104 | 3×104 | 4×104 | 5×104 | 6×104 | 7×104 | 8×104 | 9×104 | ||
终端1 | 1 | 0 | 0.014 | 0.018 | 0.016 | 0.021 | 0.089 | 0.090 | 0.020 | 0.060 | 0.014 |
2 | 0 | 0.031 | 0.017 | 0.075 | 0.011 | 0.024 | 0.025 | 0.080 | 0.010 | 0.043 | |
3 | 0 | 0.011 | 0.069 | 0.070 | 0.030 | 0.010 | 0.010 | 0.010 | 0.012 | 0.073 | |
4 | 0 | 0.030 | 0.020 | 0.080 | 0.030 | 0.050 | 0.060 | 0.090 | 0.020 | 0.055 | |
5 | 0 | 0.010 | 0.012 | 0.080 | 0.031 | 0.050 | 0.050 | 0.010 | 0.035 | 0.015 | |
6 | 0 | 0.011 | 0.050 | 0.090 | 0.026 | 0.084 | 0.085 | 0.022 | 0.036 | 0.016 | |
7 | 0 | 0.017 | 0.012 | 0.070 | 0.010 | 0.015 | 0.016 | 0.010 | 0.099 | 0.030 | |
8 | 0 | 0.026 | 0.016 | 0.010 | 0.010 | 0.012 | 0.010 | 0.010 | 0.021 | 0.016 | |
9 | 0 | 0.030 | 0.080 | 0.051 | 0.072 | 0.090 | 0.090 | 0.030 | 0.080 | 0.033 | |
10 | 0 | 0.080 | 0.012 | 0.016 | 0.032 | 0.010 | 0.010 | 0.020 | 0.013 | 0.034 | |
终端2 | 1 | 0 | 0.010 | 0.020 | 0.025 | 0.052 | 0.058 | 0.018 | 0.017 | 0.060 | 0.042 |
2 | 0 | 0.090 | 0.071 | 0.011 | 0.075 | 0.012 | 0.022 | 0.090 | 0.030 | 0.028 | |
3 | 0 | 0.010 | 0.050 | 0.021 | 0.037 | 0.024 | 0.016 | 0.011 | 0.063 | 0.030 | |
4 | 0 | 0.016 | 0.060 | 0.011 | 0.017 | 0.023 | 0.071 | 0.010 | 0.010 | 0.040 | |
5 | 0 | 0.036 | 0.060 | 0.080 | 0.028 | 0.038 | 0.039 | 0.044 | 0.090 | 0.080 | |
6 | 0 | 0.014 | 0.088 | 0.010 | 0.082 | 0.083 | 0.012 | 0.016 | 0.030 | 0.056 | |
7 | 0 | 0.037 | 0.027 | 0.014 | 0.018 | 0.028 | 0.040 | 0.070 | 0.020 | 0.072 | |
8 | 0 | 0.035 | 0.051 | 0.019 | 0.069 | 0.093 | 0.010 | 0.070 | 0.014 | 0.023 | |
9 | 0 | 0.067 | 0.081 | 0.013 | 0.012 | 0.011 | 0.034 | 0.011 | 0.010 | 0.046 | |
10 | 0 | 0.025 | 0.027 | 0.012 | 0.012 | 0.075 | 0.036 | 0.018 | 0.017 | 0.040 |
Tab. 3 Crack degradation data for terminal 1 and terminal 2
终端 | 设备 | 不同循环次数下的裂纹扩展数据/mm | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
0 | 104 | 2×104 | 3×104 | 4×104 | 5×104 | 6×104 | 7×104 | 8×104 | 9×104 | ||
终端1 | 1 | 0 | 0.014 | 0.018 | 0.016 | 0.021 | 0.089 | 0.090 | 0.020 | 0.060 | 0.014 |
2 | 0 | 0.031 | 0.017 | 0.075 | 0.011 | 0.024 | 0.025 | 0.080 | 0.010 | 0.043 | |
3 | 0 | 0.011 | 0.069 | 0.070 | 0.030 | 0.010 | 0.010 | 0.010 | 0.012 | 0.073 | |
4 | 0 | 0.030 | 0.020 | 0.080 | 0.030 | 0.050 | 0.060 | 0.090 | 0.020 | 0.055 | |
5 | 0 | 0.010 | 0.012 | 0.080 | 0.031 | 0.050 | 0.050 | 0.010 | 0.035 | 0.015 | |
6 | 0 | 0.011 | 0.050 | 0.090 | 0.026 | 0.084 | 0.085 | 0.022 | 0.036 | 0.016 | |
7 | 0 | 0.017 | 0.012 | 0.070 | 0.010 | 0.015 | 0.016 | 0.010 | 0.099 | 0.030 | |
8 | 0 | 0.026 | 0.016 | 0.010 | 0.010 | 0.012 | 0.010 | 0.010 | 0.021 | 0.016 | |
9 | 0 | 0.030 | 0.080 | 0.051 | 0.072 | 0.090 | 0.090 | 0.030 | 0.080 | 0.033 | |
10 | 0 | 0.080 | 0.012 | 0.016 | 0.032 | 0.010 | 0.010 | 0.020 | 0.013 | 0.034 | |
终端2 | 1 | 0 | 0.010 | 0.020 | 0.025 | 0.052 | 0.058 | 0.018 | 0.017 | 0.060 | 0.042 |
2 | 0 | 0.090 | 0.071 | 0.011 | 0.075 | 0.012 | 0.022 | 0.090 | 0.030 | 0.028 | |
3 | 0 | 0.010 | 0.050 | 0.021 | 0.037 | 0.024 | 0.016 | 0.011 | 0.063 | 0.030 | |
4 | 0 | 0.016 | 0.060 | 0.011 | 0.017 | 0.023 | 0.071 | 0.010 | 0.010 | 0.040 | |
5 | 0 | 0.036 | 0.060 | 0.080 | 0.028 | 0.038 | 0.039 | 0.044 | 0.090 | 0.080 | |
6 | 0 | 0.014 | 0.088 | 0.010 | 0.082 | 0.083 | 0.012 | 0.016 | 0.030 | 0.056 | |
7 | 0 | 0.037 | 0.027 | 0.014 | 0.018 | 0.028 | 0.040 | 0.070 | 0.020 | 0.072 | |
8 | 0 | 0.035 | 0.051 | 0.019 | 0.069 | 0.093 | 0.010 | 0.070 | 0.014 | 0.023 | |
9 | 0 | 0.067 | 0.081 | 0.013 | 0.012 | 0.011 | 0.034 | 0.011 | 0.010 | 0.046 | |
10 | 0 | 0.025 | 0.027 | 0.012 | 0.012 | 0.075 | 0.036 | 0.018 | 0.017 | 0.040 |
终端 | 参数 | 均值 | 标准差 | MC误差/10-3 | 95%HPD区间 |
---|---|---|---|---|---|
终端1 | 0.405 4 | 0.025 7 | 0.405 2 | (0.356 7,0.458 1) | |
0.012 1 | 0.002 2 | 0.041 8 | (0.008 5,0.017 1) | ||
1.216 0 | 0.093 3 | 1.874 0 | (1.032 0,1.407 0) | ||
终端2 | 0.373 4 | 0.018 8 | 0.565 2 | (0.337 5,0.411 5) | |
0.007 0 | 0.001 2 | 0.035 1 | (0.005 2,0.009 8) | ||
0.989 9 | 0.068 3 | 2.391 0 | (0.858 5,1.127 0) |
Tab. 4 Estimation results of crack degradation parameters for terminal 1 and terminal 2
终端 | 参数 | 均值 | 标准差 | MC误差/10-3 | 95%HPD区间 |
---|---|---|---|---|---|
终端1 | 0.405 4 | 0.025 7 | 0.405 2 | (0.356 7,0.458 1) | |
0.012 1 | 0.002 2 | 0.041 8 | (0.008 5,0.017 1) | ||
1.216 0 | 0.093 3 | 1.874 0 | (1.032 0,1.407 0) | ||
终端2 | 0.373 4 | 0.018 8 | 0.565 2 | (0.337 5,0.411 5) | |
0.007 0 | 0.001 2 | 0.035 1 | (0.005 2,0.009 8) | ||
0.989 9 | 0.068 3 | 2.391 0 | (0.858 5,1.127 0) |
Copula函数 | 估计值 | AIC | 排序 |
---|---|---|---|
Clayton | 0.55 | 2 518.597 | 3 |
Frank | 8.91 | 3 343.365 | 6 |
Gumbel | 1.66 | 2 924.685 | 4 |
SJC | (0.14,0.18) | 2 222.162 | 1 |
Plackett | 39.12 | 3 188.068 | 5 |
Rotated Clayton | 0.37 | 3 409.889 | 7 |
Rotated Gumbel | 1.59 | 2 313.342 | 2 |
Tab. 5 Comparison of parameter estimation and fitting goodness for static Copula function
Copula函数 | 估计值 | AIC | 排序 |
---|---|---|---|
Clayton | 0.55 | 2 518.597 | 3 |
Frank | 8.91 | 3 343.365 | 6 |
Gumbel | 1.66 | 2 924.685 | 4 |
SJC | (0.14,0.18) | 2 222.162 | 1 |
Plackett | 39.12 | 3 188.068 | 5 |
Rotated Clayton | 0.37 | 3 409.889 | 7 |
Rotated Gumbel | 1.59 | 2 313.342 | 2 |
模型 | 参数 | 估计值 | 对数似然函数 | AIC |
---|---|---|---|---|
SJC | (0.14,0.18) | -1 109.081 | 2 222.162 | |
Time-varying SJC(傅里叶) | (-1.73,0.23,0.06,0.47) | -1 060.716 | 2 137.432 | |
(-1.3,-0.02,-0.72,0.06) | ||||
文献[ | (-1.333,-0.016,0.03) | -1 107.744 | 2 227.488 | |
(-1.331,0.022,0.364) |
Tab. 6 Parameter estimation and fitting goodness for static and time-varying Copula function
模型 | 参数 | 估计值 | 对数似然函数 | AIC |
---|---|---|---|---|
SJC | (0.14,0.18) | -1 109.081 | 2 222.162 | |
Time-varying SJC(傅里叶) | (-1.73,0.23,0.06,0.47) | -1 060.716 | 2 137.432 | |
(-1.3,-0.02,-0.72,0.06) | ||||
文献[ | (-1.333,-0.016,0.03) | -1 107.744 | 2 227.488 | |
(-1.331,0.022,0.364) |
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