Reliability evaluation of multi-component system based on time-varying Copula function

doi:10.11772/j.issn.1001-9081.2023040459

Abstract

Abstract:

Aiming at the mechanical system related to multi-component failure， a reliability evaluation method of multi-component system based on time-varying Copula function was proposed. Firstly， the nonlinear Wiener process was introduced to characterize the performance degradation process， and the Copula function was used to characterize the correlation between multiple component failures. Secondly， based on the evolutionary equation of the Copula function approximation of the Fourier series， the fitting effects of the Fourier series on common time-varying forms were verified by Monte Carlo （MC） simulation. In addition， the likelihood ratio statistic was used to test the existence of time-varying correlation， indicating the necessity of time-varying correlation research. The example analysis shows that compared with the static correlation model， the time-varying correlation model has the log-likelihood function value increased by 4.36%， and the Akaike Information Criterion （AIC） decreased by 3.81%， achieving more accurate reliability evaluation results.

Key words: Wiener process, time-varying Copula function, Fourier series, likelihood ratio test, reliability evaluation

摘要：

针对多部件失效相关的机械系统，提出一种基于时变Copula函数的多部件系统可靠性评估方法。首先，引入非线性Wiener过程刻画性能退化过程，并采用Copula函数刻画多部件失效间的相关性；其次，基于傅里叶级数近似Copula函数的演化方程，并通过蒙特卡洛（MC）模拟验证傅里叶级数对常见时变形式的拟合效果；此外，采用似然比统计量检验时变相关性的存在性，指明时变相关性研究的必要性。算例分析结果表明，与静态相关性模型相比，时变相关性模型对数似然函数值提高4.36%、赤池信息量准则（AIC）减小3.81%，可靠性评估结果更加准确。

关键词: Wiener过程, 时变Copula函数, 傅里叶级数, 似然比检验, 可靠性评估

CLC Number:

TB114.3

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.

Figures/Tables 12

Fig. 1 Reliability evaluation flow of multi-component system

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 $Λ ˜ ρ t$
Frank	$- ∞, + ∞ \ 0$	$ρ t ρ t ≠ 0$
Plackett	$0, + ∞$	$ρ t 2$
Clayton
Rotated Clayton
Gumbel	$1, + ∞$	$1 + ρ t 2$
Rotated Gumbel	$1, + ∞$	$1 + ρ t 2$
SJC	$0,1$	$1 / 1 + e - ρ t$

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 $Λ ˜ ρ t$
Frank	$- ∞, + ∞ \ 0$	$ρ t ρ t ≠ 0$
Plackett	$0, + ∞$	$ρ t 2$
Clayton
Rotated Clayton
Gumbel	$1, + ∞$	$1 + ρ t 2$
Rotated Gumbel	$1, + ∞$	$1 + ρ t 2$
SJC	$0,1$	$1 / 1 + e - ρ t$

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	$ρ t = ϕ 1 ρ t - 1 + ε t$	$ϕ 1 = 0.9, ε t ~ N (0,1)$	0.843 8
连续时变模型	多项式函数	$ρ t = β 0 + β 1 t + β 2 t 2 + β 3 t 3$	$β 0 = β 1 = β 2 = β 3 = 1$	0.998 0
跳跃时变模型	符号函数	$ρ t = s g n (t + t 0)$	$t 0 = - 5$	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	$ρ t = ϕ 1 ρ t - 1 + ε t$	$ϕ 1 = 0.9, ε t ~ N (0,1)$	0.843 8
连续时变模型	多项式函数	$ρ t = β 0 + β 1 t + β 2 t 2 + β 3 t 3$	$β 0 = β 1 = β 2 = β 3 = 1$	0.998 0
跳跃时变模型	符号函数	$ρ t = s g n (t + t 0)$	$t 0 = - 5$	0.864 5

Fig. 2 Fitting effects of Fourier series approximating AR model， polynomial function and sign function

Fig. 3 Parameter estimation steps for static Copula function

Fig. 4 Form selection and parameter estimation steps for time-varying Copula function

Tab. 3 Crack degradation data for terminal 1 and terminal 2

终端	设备
终端	设备	10⁴	2×10⁴	3×10⁴	4×10⁴	5×10⁴	6×10⁴	7×10⁴	8×10⁴	9×10⁴
终端1	1	0.014	0.018	0.016	0.021	0.089	0.090	0.020	0.060	0.014
	2	0.031	0.017	0.075	0.011	0.024	0.025	0.080	0.010	0.043
	3	0.011	0.069	0.070	0.030	0.010	0.010	0.010	0.012	0.073
	4	0.030	0.020	0.080	0.030	0.050	0.060	0.090	0.020	0.055
	5	0.010	0.012	0.080	0.031	0.050	0.050	0.010	0.035	0.015
	6	0.011	0.050	0.090	0.026	0.084	0.085	0.022	0.036	0.016
	7	0.017	0.012	0.070	0.010	0.015	0.016	0.010	0.099	0.030
	8	0.026	0.016	0.010	0.010	0.012	0.010	0.010	0.021	0.016
	9	0.030	0.080	0.051	0.072	0.090	0.090	0.030	0.080	0.033
	10	0.080	0.012	0.016	0.032	0.010	0.010	0.020	0.013	0.034
终端2	1	0.010	0.020	0.025	0.052	0.058	0.018	0.017	0.060	0.042
	2	0.090	0.071	0.011	0.075	0.012	0.022	0.090	0.030	0.028
	3	0.010	0.050	0.021	0.037	0.024	0.016	0.011	0.063	0.030
	4	0.016	0.060	0.011	0.017	0.023	0.071	0.010	0.010	0.040
	5	0.036	0.060	0.080	0.028	0.038	0.039	0.044	0.090	0.080
	6	0.014	0.088	0.010	0.082	0.083	0.012	0.016	0.030	0.056
	7	0.037	0.027	0.014	0.018	0.028	0.040	0.070	0.020	0.072
	8	0.035	0.051	0.019	0.069	0.093	0.010	0.070	0.014	0.023
	9	0.067	0.081	0.013	0.012	0.011	0.034	0.011	0.010	0.046
	10	0.025	0.027	0.012	0.012	0.075	0.036	0.018	0.017	0.040

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	$μ 1$	0.405 4	0.025 7	0.405 2	（0.356 7，0.458 1）
	$σ 12$	0.012 1	0.002 2	0.041 8	（0.008 5，0.017 1）
	$c 1$	1.216 0	0.093 3	1.874 0	（1.032 0，1.407 0）
终端2	$μ 2$	0.373 4	0.018 8	0.565 2	（0.337 5，0.411 5）
	$σ 22$	0.007 0	0.001 2	0.035 1	（0.005 2，0.009 8）
	$c 2$	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	$μ 1$	0.405 4	0.025 7	0.405 2	（0.356 7，0.458 1）
	$σ 12$	0.012 1	0.002 2	0.041 8	（0.008 5，0.017 1）
	$c 1$	1.216 0	0.093 3	1.874 0	（1.032 0，1.407 0）
终端2	$μ 2$	0.373 4	0.018 8	0.565 2	（0.337 5，0.411 5）
	$σ 22$	0.007 0	0.001 2	0.035 1	（0.005 2，0.009 8）
	$c 2$	0.989 9	0.068 3	2.391 0	（0.858 5，1.127 0）

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

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	$τ U, τ L$	（0.14，0.18）	-1 109.081	2 222.162
Time-varying SJC（傅里叶）	$γ U, a U, b U, k U$	（-1.73，0.23，0.06，0.47）	-1 060.716	2 137.432
Time-varying SJC（傅里叶）	$γ L, a L, b L, k L$	（-1.3，-0.02，-0.72，0.06）	-1 060.716	2 137.432
文献［4］模型	$ω U, α U, β U$	（-1.333，-0.016，0.03）	-1 107.744	2 227.488
文献［4］模型	$ω L, α L, β L$	（-1.331，0.022，0.364）	-1 107.744	2 227.488

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	$τ U, τ L$	（0.14，0.18）	-1 109.081	2 222.162
Time-varying SJC（傅里叶）	$γ U, a U, b U, k U$	（-1.73，0.23，0.06，0.47）	-1 060.716	2 137.432
Time-varying SJC（傅里叶）	$γ L, a L, b L, k L$	（-1.3，-0.02，-0.72，0.06）	-1 060.716	2 137.432
文献［4］模型	$ω U, α U, β U$	（-1.333，-0.016，0.03）	-1 107.744	2 227.488
文献［4］模型	$ω L, α L, β L$	（-1.331，0.022，0.364）	-1 107.744	2 227.488

Fig. 5 Single-component system reliability curves

Fig. 6 Multi-component system reliability curves

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