Weibull-Bayesian Analysis

In this section, the Bayesian methods are presented for the two-parameter Weibull distribution. Bayesian concepts were introduced in the Statistical Background chapter. This model considers prior knowledge on the shape (β) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the one-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Weibull-Bayesian model in Weibull++ (which is actually a true "WeiBayes" model, unlike the one-parameter Weibull that is commonly referred to as such) offers an alternative to the one-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter.

Applying Bayes's rule on the two-parameter Weibull distribution and assuming the prior distributions of β and η are independent, we obtain the following posterior pdf:

In this model, η is assumed to follow a noninformative prior distribution with the density function

. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on η. Specifically, since η is always positive, we can assume that ln(η) follows a uniform distribution, U(-

, +

). Applying Jeffrey's rule [ 9] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information", yields

The prior distribution of β, denoted as f(β), can be selected from the following distributions: normal, lognormal, exponential and uniform.

In other words, a distribution (the posterior pdf) is obtained, rather than a point estimate as in classical statistics (i.e. as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior pdf needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the Analysis page, under the Results As area, as shown next.

The median points are obtained by solving the following equations for $\breve{\beta}$ and $\breve{\eta}$ respectively:

Of course, other points of the posterior distribution can be calculated as well. For example, one may want to calculate the 10th percentile of the joint posterior distribution (w.r.t. one of the parameters). The procedure for obtaining other points of the posterior distribution is similar to the one for obtaining the median values, where instead of 0.5 the percentage of interest is given. This procedure actually provides the confidence bounds on the parameters, which in the Bayesian framework are called Credible Bounds. However, since the engineering interpretation is the same, and to avoid confusion, we refer to them as confidence bounds in this reference and in Weibull++.

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Posterior Distributions for Functions of Parameters in Weibull-Bayesian Analysis

As explained in the Statistical Background chapter, in Bayesian analysis, all the functions of the parameters are distributed. In other words, a posterior distribution is obtained for functions such as reliability and failure rate, instead of point estimate as in classical statistics. Therefore, in order to obtain a point estimate for these functions, a point on the posterior distributions needs to be calculated. Again, the expected value (mean) or median value are used.

pdf of the Times-to-Failure

For the pdf of the times-to-failure, only the expected value is calculated and reported in Weibull++.

Reliability

In order to calculate the median value of the reliability function, we first need to obtain posterior pdf of the reliability. Since R(T) is a function of β, the density functions of β and R(T) have the following relationship:

The median value of the reliability is obtained by solving the following equation w.r.t. $\breve{R}:$

Failure Rate

Note on Calculated Results

As mentioned above, in order to obtain point estimates for the parameters of functions of the parameters in Bayesian analysis, the Median or Mean values of the different posterior pdfs are calculated. It is important to note that the Median value is preferable and is the default in Weibull++. This is because the Median value always corresponds to the 50th percentile of the distribution. On the other hand, the Mean is not a fixed point on the distribution, which could cause issues, especially when comparing results across different data sets.

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Confidence Bounds on R(T) for Weibull-Bayesian Analysis

The confidence bounds calculation under the Weibull-Bayesian analysis is very similar to the Bayesian Confidence Bounds method described in the previous section, with the exception that in the case of the Weibull-Bayesian Analysis the specified prior of β is considered instead of an non-informative prior.

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Confidence Bounds on Time T for Weibull-Bayesian Analysis

Following the same procedure described for bounds on Reliability, the bounds of time T can be calculated, given R.

Weibull Distribution Example 6

A manufacturer has tested prototypes of a modified product. The test was terminated at 2000 hours, with only two failures observed from a sample size of eighteen.

Number in State	State F or S	State End Time
1	F	1180
1	F	1842
16	S	2000

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. This decision was made because failure analysis indicated that the failure mode of these two failures is the same as the one observed in previous tests. In other words, it is expected that the shape of the distribution hasn't changed, but hopefully the scale has, indicating longer life. The two-parameter Weibull distribution have been used to model all prior tests results. The list of the estimated β parameter is as follows:

Betas Obtained for Similar Mode

1.7

2.1

2.4

3.1

3.5

First, in order to fit the data to a Weibull-Bayesian model, a prior distribution for β needs to be determined. Based on the prior tests' β values, the prior distribution for β was found to be a lognormal distribution with μ = 0.9064, σ = 0.3325 (obtained by entering the β values into a Weibull++ Standard Folio and analyzing it based on the RRX analysis method.)

The test data is entered into a Standard Folio, the Weibull-Bayesian is selected under Distribution and the β prior distribution is entered after clicking the Calculate button.

Suppose that the reliability at 3000hr is the metric of interest in this example. This reliability can be obtained using Eqn. ( 48), resulting in the median value of the posterior pdf of the reliability at 3000hr. Using the QCP, this value is calculated to be 76.97%. (Note: By default Weibull++ returns the median values of the posterior distribution.)

The posterior pdf of the reliability function at 3000hrs can be obtained using Eqn. ( 47). In Figure 6-10 the posterior pdf of the reliability at 3000hrs is plotted, with the corresponding median value as well as the 10th percentile value shown. The 10th percentile constitutes the 90% Lower 1-Sided bound on the reliability at 3000hrs, which is calculated to be 50.77%.

Notice that the pdf plotted in Fig. 6-10 is of the reliability at 3000hrs, and not the pdf of the times-to-failure data. The pdf of the times-to-failure data can be obtained using Eqn. ( 46) and plotted using Weibull++, as shown next: