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Reliability evaluation of components with multiple failure modes based on mixture Weibull distribution using expectation maximization algorithm

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Abstract

The mixture Weibull distribution is widely used in the reliability analysis of lifetime data of components, especially for those caused by multiple failure modes. In order to accurately estimate the parameters of mixture Weibull distributions, an optimized expectation maximization (EM) algorithm is proposed based on the particle swarm optimization (PSO) algorithm. The Weibull probability plot (WPP) method is introduced to determine the approximate values and rough limits of unknown parameters, and the approximate values will be used as the initial points in the optimization of the EM procedure. By taking the maximum of the Q function as the optimization goal and the rough limits of each parameter as the constraints, a parameter estimation model of the mixture Weibull distribution is built. The PSO algorithm has been introduced into the maximization step of the EM algorithm to deal with the transcendental equations. Numerical cases are used to illustrate the accuracy of our method in the parameter estimation for the mixture Weibull distribution, and an engineering case is used to show the application of our method in the reliability evaluation of components with multiple failure modes. Compared with the least squares estimation and the maximum likelihood estimation, our method shows advantages in both accuracy and efficiency.

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Abbreviations

AIC :

Akaike information criterion

CDF :

Cumulative distribution function

EM :

Expectation maximization

E-step :

Expectation step

GA :

Genetic algorithm

LF :

Likelihood function

LSE :

Least squares estimation

MLE :

Maximum likelihood estimation

M-step :

Maximization step

PDF :

Probability density function

PSO :

Particle swarm optimization

RMSE :

Root mean squared error

WPP :

Weibull probability plot

T :

Vector of the lifetime sample

t i :

The ith observed lifetime data

N :

Total number of the lifetime sample

K :

Number of sub-populations in the mixture distribution

f(t):

PDF of the mixture Weibull distributions

F(t):

CDF of the mixture Weibull distributions

R(t):

Reliability function of the mixture Weibull distributions

α k :

Scale parameters of the kth sub-population in the mixture Weibull distributions

β k :

Shape parameters of the kth sub-population in the mixture Weibull distributions

π k :

Weight parameters of the kth sub-population in the mixture Weibull distributions

θ :

Vector of all the unknown parameters of the mixture Weibull distributions

η E :

Number of failure lifetime data

η R :

Number of censoring lifetime data

L(θ):

Likelihood function of the mixture Weibull distributions

Z :

Vector of the hidden variables in the EM algorithm

Z ik :

Variable represents if ti belongs to the kth subpopulation in the mixture model

g(t i ∣α k,β k):

Density function of the kth sub-population in the mixture model

Q(θ, θ (j)):

The expectation function or the Q function in the EM algorithm

θ (0) :

The initial values of parameters in the EM algorithm

θ (j) :

Parameters’ estimation values of the j-th iteration in the EM algorithm

\(\hat R(t)\) :

Empirical reliability function of the mixture Weibull distributions

J t :

Set of all indices j of the failure time data in the empirical reliability function for censoring lifetime data

η j :

Number of lifetime data that function and can be observed before time tj

η k :

Number of points on the kth fitting straight line in the WPP

j (α) :

Ordinal number of the lifetime sample whose value is roughly equal to αk

π k max :

Upper bound of the weight factor for the kth subpopulation

π k min :

Lower bound of the weight factor for the kth subpopulation

π k (0) :

Initial value of the weight factor for the kth sub-population

X k max :

Upper bound of the scale or shape factor for the kth subpopulation

X k min :

Lower bound of the scale or shape factor for the kth subpopulation

X (0)k :

Initial value of the scale or shape factor for the kth subpopulation

popsize :

Total size of the particles in the PSO algorithm

maxiter :

Total number of swarm iterations in the PSO algorithm

D :

Dimensions of each particle in the PSO algorithm, which equal to the total number of estimation parameters (D = 2K)

X :

Vector of all the particle positions

x d :

Particle position value of the dth dimension

v d :

Particle velocity value of the dth dimension

x d s+1 :

The updated particle position value of the dth dimension at the s+1th iteration in the PSO algorithm

v d s+1 :

The updated particle velocity n value of the dth dimension at the s+1th iteration in the PSO algorithm

w :

Adjustable inertial factor in the PSO algorithm

c 1, c 2 :

Acceleration factors in the PSO algorithm

pbest :

Individual particle optimal solution in the PSO algorithm

gbest :

Global particle optimal solution in the PSO algorithm

ε :

An arbitrarily small positive real which is set as the convergence condition of the EM algorithm

GE :

Total number of generations in the EM algorithm

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Acknowledgements

The authors wish to appreciate the support from the National Natural Science Foundation of China (U1733124), Funds for Civil Aviation Safety Capacity Building (2021-196), and the Aeronautical Science Foundation of China (20180252002).

Author information

Authors and Affiliations

  1. College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

    Yuting Wu, Zhong Lu & Jiayu Wu

  2. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Zhong Lu

  3. Department of Mechanical Engineering, University of Manitoba, Winnipeg, R3T5V6, Canada

    Xihui Liang

Authors
  1. Yuting Wu
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Corresponding author

Correspondence to Zhong Lu.

Additional information

Yuting Wu serves as an Assistant Engineer at the Xi’an Aeronautics Computing Technique Research Institute, Shaanxi, China. She received her master’s degree in Transportation Engineering from the Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China, in 2023. And this work was finished by her when she was pursuing her master’s degree at NUAA. Her research interests include system safety assessment and data-driven risk analysis.

Zhong Lu serves as a Professor at the College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing, China. He received his Ph.D. in Vehicle Operation Engineering from Nanjing University of Aeronautics and Astronautics. His research interests include reliability engineering, system safety assessment, and risk analysis.

Jiayu Wu is currently working toward her Master’s degree at the College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing, China. Her research interests include system safety assessment, reliability engineering, and risk analysis.

Xihui Liang serves as an Assistant Professor at the Department of Mechanical Engineering, University of Manitoba. His research interests include reliability and maintainability engineering, nonlinear dynamics, and signal processing.

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Wu, Y., Lu, Z., Wu, J. et al. Reliability evaluation of components with multiple failure modes based on mixture Weibull distribution using expectation maximization algorithm. J Mech Sci Technol 38, 649–660 (2024). https://doi.org/10.1007/s12206-024-0113-1

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  • DOI: https://doi.org/10.1007/s12206-024-0113-1

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