A Degradation Model Based on the Wiener Process Assuming Non-Normal Distributed Measurement Errors

Abstract

For highly reliable products whose failure times are scarce, traditional methods for lifetime time analysis are no longer effective and efficient. Instead, degradation data analysis that investigates degradation processes of products becomes a useful tool in evaluating reliability. It focuses on the inherent randomness of products, and investigates the lifetime properties by developing degradation models and extrapolating to lifetime variables. But degradation data are often subject to measurement errors, which may have tails and better be described by non-normal distribution. In this chapter, we consider a Wiener-based model and assume logistic distributed measurement errors. For parameter estimation of the model, the Monte-Carlo expectation-maximization method is adopted together with the Gibbs sampling. Also an efficient algorithm is proposed for a quick approximation of maximum likelihood value. Moreover the remaining useful lifetime is estimated and discussed. From the simulation results, we find that the proposed model is more robust than the model based on the Wiener process assuming normal-distributed errors. Finally, an example is given to illustrate the application of the proposed model.

Appendices

Appendix 1

This appendix presents the technical details for the inference in Sect. 3.4. Recall the parameter θ = (μ, σ, s)′. The first and second partial derivatives of the loglikelihood function in (8) with respect to θ are as follows. Let κ _l,i = x _l,i − x _l,i−1 − μλ _l,i.

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\partial\ell}{\partial \mu}&\displaystyle =&\displaystyle \frac{1}{\sigma^2}\sum_{l=1}^m\sum_{i=1}^{n_l}\kappa_{l,i}\\ \frac{\partial\ell}{\partial \sigma}&\displaystyle =&\displaystyle -\frac{1}{\sigma}\sum_{l=1}^m n_l+\frac{1}{\sigma^3}\sum_{l=1}^m\sum_{i=1}^{n_l}\frac{\kappa_{l,i}^2}{\lambda_{l,i}}\\ \frac{\partial\ell}{\partial s}&\displaystyle =&\displaystyle \frac{1}{s^2}\sum_{l=1}^m\sum_{i=1}^{n_l}(y_{l,i}-x_{l,i})-\frac{1}{s}\sum_{i=1}^m n_l-\frac{2}{s^2}\sum_{l=1}^m\sum_{i=1}^{n_l}\frac{y_{l,i}-x_{l,i}}{1+e^{(y_{l,i}-x_{l,i})/s}}\\ \frac{\partial^2\ell}{\partial\mu^2}&\displaystyle =&\displaystyle -\frac{1}{\sigma^2}\sum_{l=1}^{m}\varLambda_{l,n_l}\\ \frac{\partial^2\ell}{\partial\sigma^2}&\displaystyle =&\displaystyle \frac{1}{\sigma^2}\sum_{l=1}^m n_l-\frac{3}{\sigma^4}\sum_{l=1}^m\sum_{i=1}^{n_l}\frac{\kappa_{l,i}^2}{\lambda_{l,i}}\\ \frac{\partial^2\ell}{\partial s^2}&\displaystyle =&\displaystyle -\frac{2}{s^3}\sum_{l=1}^m\sum_{i=1}^{n_l}(y_{l,i}-x_{l,i})+\frac{1}{s^2}\sum_{i=1}^m n_l+\frac{4}{s^3}\sum_{l=1}^m\sum_{i=1}^{n_l}\frac{y_{l,i}-x_{l,i}}{1+e^{(y_{l,i}-x_{l,i})/s}}\\ &\displaystyle &\displaystyle -\frac{2}{s^4}\sum_{l=1}^m\sum_{i=1}^{n_l}\frac{(y_{l,i}-x_{l,i})^2e^{(y_{l,i}-x_{l,i})/s}}{[1+e^{(y_{l,i}-x_{l,i})/s}]^2}\\ \frac{\partial^2\ell}{\partial\mu\partial\sigma}&\displaystyle =&\displaystyle -\frac{2}{\sigma^3}\sum_{l=1}^m\sum_{i=1}^{n_l}\kappa_{l,i}\\ \frac{\partial^2\ell}{\partial\mu\partial s}&\displaystyle =&\displaystyle \frac{\partial^2\ell}{\partial\sigma\partial s}=0. \end{array} \end{aligned} $$

If the time transformation function Λ(t) contains an extra parameters b, i.e. Λ(t) = Λ(t;b), we also need the first and second partial derivatives of the loglikelihood function (8) with respect to b.

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\partial\ell}{\partial b}&\displaystyle =&\displaystyle -\frac{1}{2}\sum_{l=1}^m\sum_{i=1}^{n_l}\frac{1}{\lambda_{l,i}}\frac{\partial\lambda_{l,i}}{\partial b}+\frac{1}{\sigma^2}\sum_{l=1}^m\sum_{i=1}^{n_l}\left(\frac{\kappa_{l,i}\mu}{\lambda_{l,i}}+\frac{\kappa_{l,i}^2}{2\lambda^2_{l,i}}\right) \frac{\partial\lambda_{l,i}}{\partial b}\\ \frac{\partial^2\ell}{\partial b^2}&\displaystyle =&\displaystyle \frac{1}{2}\sum_{l=1}^m\sum_{i=1}^{n_l}\frac{1}{\lambda^2_{l,i}}\left(\frac{\partial\lambda_{l,i}}{\partial b}\right)^2-\frac{1}{2}\sum_{l=1}^m\sum_{i=1}^{n_l}\frac{1}{\lambda_{l,i}}\frac{\partial^2\lambda_{l,i}}{\partial b^2}\\ &\displaystyle &\displaystyle +\frac{1}{\sigma^2}\sum_{l=1}^m\sum_{i=1}^{n_l}\left(\frac{\kappa_{l,i}\mu}{\lambda_{l,i}}+\frac{\kappa_{l,i}^2}{2\lambda^2_{l,i}}\right) \frac{\partial^2\lambda_{l,i}}{\partial b^2}\\ &\displaystyle &\displaystyle -\frac{1}{\sigma^2}\sum_{l=1}^m\sum_{i=1}^{n_l}\left(\frac{\mu^2}{\lambda_{l,i}}+\frac{2\kappa_{l,i}\mu}{\lambda_{l,i}^2}+ \frac{\kappa_{l,i}^2}{\lambda_{l,i}^3}\right)\left(\frac{\partial\lambda_{l,i}} {\partial b}\right)^2\\ \frac{\partial^2\ell}{\partial \mu\partial b}&\displaystyle =&\displaystyle -\sum_{l=1}^m\sum_{i=1}^{n_l}\frac{\mu}{\sigma^2}\frac{\partial \lambda_{l,i}}{\partial b}\\ \frac{\partial^2\ell}{\partial \sigma\partial b}&\displaystyle =&\displaystyle -\frac{2}{\sigma^3}\sum_{l=1}^m\sum_{i=1}^{n_l}\left(\frac{\kappa_{l,i}\mu}{\lambda_{l,i}}+\frac{\kappa_{l,i}^2}{2\lambda^2_{l,i}}\right)\frac{\partial \lambda_{l,i}}{\partial b}\\ \frac{\partial^2\ell}{\partial s\partial b}&\displaystyle =&\displaystyle 0. \end{array} \end{aligned} $$

Appendix 2

This appendix presents part of codes for Monte Carlo simulations. All codes were compiled based Matlab software, and the related files are available upon request.