Applications of the Bayesian-Weibull Distribution in Life Data Analysis
[Editor's Note: This article has been updated since its original publication to reflect a more recent version of the software interface.]
One of the new features of Weibull++ is support for Bayesian statistics. The premise of Bayesian statistics is to incorporate prior knowledge along with a given set of current observations in order to make statistical inferences. The prior information could come from operational or observational data, from previous comparable experiments or from engineering knowledge.
This type of analysis is particularly useful if there is a lack of current test data and when there is a strong prior understanding about the parameter of the assumed life model and a distribution can be used to model the parameter. By incorporating prior information about a parameter, a posterior distribution for a parameter can be produced and an adequate estimate of reliability can be obtained. Weibull++ introduces a new type of distribution, Bayesian-Weibull, that combines the concepts of Bayesian statistics with the properties of the Weibull distribution. The Reliability Basics section of this HotWire issue introduces the theory behind the new distribution and provides two examples to illustrate its application.
A manufacturer is testing prototypes of a modified product. The test is terminated at 2000 hrs, with only two failures observed from a sample size of eighteen.
Because of the lack of failure data in the prototype testing, the manufacturer decided to use information gathered from prior tests on this product to increase the confidence in the results of the prototype testing. Failure analysis on the prototypes indicated that the same failure mode as that experienced by the previous design is observed again. The manufacturer hopes, however, that longer life has been achieved in the revised product (i.e. that the overall behavior of the distribution is the same but shifted to the right). Prior tests have yielded the following values for β.
First, in order to fit the data to a Bayesian-Weibull model, a prior distribution for β needs to be determined. Based on the prior tests' β values, the prior distribution for β was determined to be a lognormal distribution with μ = 0.8235, σ = 0.0943 (obtained by entering the β data into a Weibull++ Standard Folio and analyzing it based on the RRX analysis method).
To use the Bayesian-Weibull distribution, the manufacturer enters the test data set into a Standard Folio, and then on the control panel, chooses Bayesian-Weibull > B-W Lognormal Prior. After clicking Calculate, the parameters of the prior β distribution are entered into the input window, as shown next.
After performing the calculations, the Folio looks like the one shown next.
The reliability at t = 1000 hours, along with the two-sided confidence bounds, is calculated using the QCP as follows:
The reliability plot of the model is shown next.
A comparison of the probability plots and confidence bounds for the model and the regular two-parameter model is shown next. The plot shows that the confidence bounds obtained using the model are tighter and therefore lead to a more precise analysis.
This example illustrates how Bayesian analysis can be performed using a prior β distribution that describes the uncertainty related to the β estimation based on a single data set rather than a history of β values from many previous analyses.
A manufacturer wants to estimate the warranty period to offer its customers for a new product. The manufacturer performed a single test on a previous model of the product and obtained the following months-to-failure data.
The above figure also shows the results of the parameter estimation using the two-parameter Weibull and the RRX analysis method.
From this data, a list of β values can be obtained based on different confidence level values. In Weibull++, this is done using the General Spreadsheet and the Function Wizard.
The β for different CL values is calculated and the following table is generated.
The distribution parameters of β can be obtained based on using the β values obtained above. In Weibull++, insert a Free-Form (Probit) data sheet and enter the data as follows:
The best fit distribution for β is found to be the lognormal distribution. The parameters of the distribution are found to be μ = 0.8473 and σ = 0.1854. This now constitutes the prior distribution of β, which can be used as the prior knowledge in subsequent analysis.
The manufacturer performed another test on a new version of the product. Because of cost considerations, the test was limited to six units. Therefore, the manufacturer wants to use the prior knowledge gathered about β in the reliability analysis of the new product version.
Using the Bayesian-Weibull distribution and the β prior distribution, the new product test data set is entered into the Folio as follows.
The QCP can be used to estimate a warranty duration for the new product that would meet an 85% reliability at a 90% confidence level. The warranty duration is estimated to be 20.59 months.
The reliability plot is shown next.
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