Single versus mixture Weibull distributions for nonparametric satellite reliability

doi:10.1016/j.ress.2009.10.001

Reliability Engineering & System Safety

Volume 95, Issue 3, March 2010, Pages 295-300

https://doi.org/10.1016/j.ress.2009.10.001 Get rights and content

Abstract

Long recognized as a critical design attribute for space systems, satellite reliability has not yet received the proper attention as limited on-orbit failure data and statistical analyses can be found in the technical literature. To fill this gap, we recently conducted a nonparametric analysis of satellite reliability for 1584 Earth-orbiting satellites launched between January 1990 and October 2008. In this paper, we provide an advanced parametric fit, based on mixture of Weibull distributions, and compare it with the single Weibull distribution model obtained with the Maximum Likelihood Estimation (MLE) method. We demonstrate that both parametric fits are good approximations of the nonparametric satellite reliability, but that the mixture Weibull distribution provides significant accuracy in capturing all the failure trends in the failure data, as evidenced by the analysis of the residuals and their quasi-normal dispersion.

Introduction

The present work expands on satellite reliability models previously derived by the authors and based on recent on-orbit failure data [1]. Given the previously derived nonparametric satellite reliability, we provide in this work an advanced parametric fit, based on mixture of Weibull distributions, and compare it with the single Weibull distribution model obtained with the Maximum Likelihood Estimation (MLE) method. We assess the quality of the two parametric models (single MLE Weibull versus the mixture Weibull distribution), and conclude with recommendations for researchers and industry professionals should they wish to use these satellite reliability results.

Reliability has long been recognized as a critical attribute for space systems. Spacecraft operate in remote environments, and since for the majority of them maintenance is not an option, designing high-reliability into these high-value assets is an essential engineering and financial imperative. Unfortunately, limited on-orbit failure data and statistical analyses of satellite reliability exist in the literature. To fill this gap, we recently conducted a nonparametric analysis of satellite reliability for 1584 Earth-orbiting satellites launched between January 1990 and October 2008 [1], [2]. Because our dataset is censored, we made extensive use of the Kaplan–Meier estimator [3] for calculating the satellite reliability function. We also derived confidence intervals for the nonparametric reliability result [4], [5], [6]. Fig. 1 shows the nonparametric satellite reliability with 95% confidence intervals.

Fig. 1 reads as follows. For example, after a successful launch, satellite reliability drops to approximately 96% after 2 years on-orbit. More precisely, we have $\overset{⌢}{R} (t) = 0.964 for 1 . 982 years (724 days) \leq t < 2 . 155 years (787 days)$

The complete tabular data for Fig. 1 are provided in Appendix A. Vertical cuts across Fig. 1 read as follows. For example, after one year on-orbit (t=1 year), satellite reliability will fall between 96.1% and 97.8% with a 95% likelihood—these values constitute the lower and upper bounds of the 95% confidence interval at t=1 year. In addition, the most likely estimate of satellite reliability at this point in time is $\hat{R} (t = 1 year)$ =96.9%.

Nonparametric analysis provides powerful results (as shown in Fig. 1) since the reliability calculation is unconstrained to fit any particular pre-defined lifetime distribution. However, this flexibility makes nonparametric results inconvenient to use for practical purposes often encountered in engineering design (e.g., reliability-based design optimization). In addition, some failure trends and patterns such as infant mortality or wear-out are more clearly identified and recognizable with parametric analysis. Several parametric forms can be fitted to the nonparametric reliability, such as single distributions or mixture distributions. Single distributions are simpler than the mixture distributions, whose added complexity must be balanced by an increased accuracy in the parametric fit to justify their use. In the following, we provide a brief overview of these two parametric fits and calculate the parameters for a single Weibull distribution, as well as a mixture of 2-Weibull distributions, fitted to the nonparametric results. We then analyze the quality of fit of the two resulting fits with respect to the benchmark that is the nonparametric satellite reliability. The justification for the use of Weibull distribution for satellite reliability can be found in Ref. [1].

Section snippets

Single distribution: Weibull distribution and maximum likelihood estimation

Several methods are available to fit a parametric distribution to the nonparametric function, such as graphical procedures (also known as probability plotting) and the Maximum Likelihood Estimation (MLE) method. Two Weibull fits have already been fitted to the satellite nonparametric reliability in previous works [1], [7]. Only the MLE fit is presented in this paper due to the better accuracy of the parametric model it provides. Recall that the Weibull reliability function can be expressed as $R (t$

Comparative analysis of the single versus the mixture distributions Weibull fits

Both parametric models provide relatively precise approximation of the nonparametric reliability as can been seen from Fig. 2a and b. However, a visual inspection of Fig. 2b indicates that the 2-Weibull mixture distribution follows with a higher precision the trends present in the nonparametric satellite reliability. To quantitatively gauge the improvement between the single Weibull and the 2-Weibull mixture distribution, we first calculate both the maximum and the average error between the

Conclusion

In this work, we fitted 2-Weibull mixture distribution to the previously derived nonparametric satellite reliability. We analyzed and compared the quality of fit of the mixture distribution with that of the single MLE Weibull fit. While both parametric models provide relatively accurate approximation of the nonparametric satellite reliability, the use of mixtures of Weibull distributions demonstrated a significant improvement in the quality of the fit compared with the single distribution model.

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Short communicationSingle versus mixture Weibull distributions for nonparametric satellite reliability

Abstract

Introduction

Section snippets

Single distribution: Weibull distribution and maximum likelihood estimation

Comparative analysis of the single versus the mixture distributions Weibull fits

Conclusion

Satellite reliability: statistical data analysis and modeling

Journal of Spacecraft and Rockets

Nonparametric estimation from incomplete observations

Journal of the American Statistical Association

Practical methods for reliability data analysis

Short communication
Single versus mixture Weibull distributions for nonparametric satellite reliability