Bayesian-Weibull Lognormal Prior Example: Difference between revisions

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''This example also appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference]''.</noinclude>
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A manufacturer has tested prototypes of a modified product. The test was terminated at 2,000 hours, with only 2 failures observed from a sample size of 18. The following table shows the data.
A manufacturer has tested prototypes of a modified product. The test was terminated at 2,000 hours, with only 2 failures observed from a sample size of 18.


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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 2 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 2-parameter Weibull distribution have been used to model all prior tests results. The list of the estimated <span class="texhtml">β</span> parameter is as follows:  
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 the two failures is the same as the one that was observed in previous tests. In other words, it is expected that the shape of the distribution (beta) hasn't changed, but hopefully the scale (eta) has, indicating longer life. The 2-parameter Weibull distribution was used to model all prior tests results. The estimated beta (<math>\beta\,\!</math>) parameters of the prior test results are as follows:  


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!Betas Obtained for Similar Mode
!Betas Obtained for Similar Mode
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'''Solution'''
'''Solution'''


First, in order to fit the data to a Weibull-Bayesian model, a prior distribution for <span class="texhtml">β</span> needs to be determined. Based on the prior tests' <span class="texhtml">β</span> values, the prior distribution for <span class="texhtml">β</span> was found to be a lognormal distribution with <span class="texhtml">μ = 0.9064</span>, <span class="texhtml">σ = 0.3325</span> (obtained by entering the <span class="texhtml">β</span> values into a Weibull++ standard folio and analyzing it based on the RRX analysis method).
First, in order to fit the data to a Bayesian-Weibull model, a prior distribution for beta needs to be determined. Based on the beta values in the prior tests, the prior distribution for beta is found to be a lognormal distribution with <math>\mu = 0.9064\,\!</math>, <math>\sigma = 0.3325\,\!</math>. (The values of the parameters can be obtained by entering the beta values into a Weibull++ standard folio and analyzing it using the lognormal distribution and the RRX analysis method.)  


The test data is entered into a standard folio and calculated using the settings shown next.
Next, enter the data from the prototype testing into a standard folio. On the control panel, choose the '''Bayesian-Weibull > B-W Lognormal Prior''' distribution. Click '''Calculate''' and enter the parameters of the lognormal distribution, as shown next.
[[Image:Weibull Distribution Example 6 Data and Prior.png|thumb|center|500px| ]]
[[Image:Weibull Distribution Example 6 Data and Prior.png|center|600px| ]]


Suppose that the reliability at 3,000 hours is the metric of interest in this example. This metric can be obtained using the equation for median reliability, resulting in the median value of the posterior of the reliability at 3,000 hours. Using the QCP, this value is calculated to be 76.97%. (By default Weibull++ returns the median values of the posterior distribution.)
Click '''OK'''. The result is Beta (Median) = 2.361219 and Eta (Median) = 5321.631912 (by default Weibull++ returns the median values of the posterior distribution). Suppose that the reliability at 3,000 hours is the metric of interest in this example. Using the QCP, the reliability is calculated to be 76.97% at 3,000 hours. The following picture depicts the posterior ''pdf'' plot of the reliability at 3,000, with the corresponding median value as well as the 10th percentile value. The 10th percentile constitutes the 90% lower 1-sided bound on the reliability at 3,000 hours, which is calculated to be 50.77%.  
 
The posterior <math>pdf</math> of the reliability function at 3,000 hours can be obtained using the equation at the begining of this section. In Figure 6-10 the posterior <math>pdf</math> of the reliability at 3,000 hours 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 3,000 hours, which is calculated to be 50.77%.  


[[Image:WB.8 pdf median.png|center|400px| ]]  
[[Image:WB.8 pdf median.png|center|400px| ]]  


Notice that the <math>pdf</math> plotted in the above figure is of the reliability at 3,000 hours and not the <math>pdf</math> of the times-to-failure data. The <math>pdf</math> of the times-to-failure data can be obtained using the equation at the begining of this section and plotted using Weibull++, as shown next:  
The ''pdf'' of the times-to-failure data can be plotted in Weibull++, as shown next:  
 
[[Image:Weibull Distribution Example 6 pdf.png|thumb|center|400px| ]]
 
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[[Image:Weibull Distribution Example 6 pdf.png|center|500px| ]]

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This example also appears in the Life data analysis reference.

A manufacturer has tested prototypes of a modified product. The test was terminated at 2,000 hours, with only 2 failures observed from a sample size of 18. The following table shows the data.

Number of State State of 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 the two failures is the same as the one that was observed in previous tests. In other words, it is expected that the shape of the distribution (beta) hasn't changed, but hopefully the scale (eta) has, indicating longer life. The 2-parameter Weibull distribution was used to model all prior tests results. The estimated beta ([math]\displaystyle{ \beta\,\! }[/math]) parameters of the prior test results are as follows:

Betas Obtained for Similar Mode
1.7
2.1
2.4
3.1
3.5

Solution

First, in order to fit the data to a Bayesian-Weibull model, a prior distribution for beta needs to be determined. Based on the beta values in the prior tests, the prior distribution for beta is found to be a lognormal distribution with [math]\displaystyle{ \mu = 0.9064\,\! }[/math], [math]\displaystyle{ \sigma = 0.3325\,\! }[/math]. (The values of the parameters can be obtained by entering the beta values into a Weibull++ standard folio and analyzing it using the lognormal distribution and the RRX analysis method.)

Next, enter the data from the prototype testing into a standard folio. On the control panel, choose the Bayesian-Weibull > B-W Lognormal Prior distribution. Click Calculate and enter the parameters of the lognormal distribution, as shown next.

Weibull Distribution Example 6 Data and Prior.png

Click OK. The result is Beta (Median) = 2.361219 and Eta (Median) = 5321.631912 (by default Weibull++ returns the median values of the posterior distribution). Suppose that the reliability at 3,000 hours is the metric of interest in this example. Using the QCP, the reliability is calculated to be 76.97% at 3,000 hours. The following picture depicts the posterior pdf plot of the reliability at 3,000, with the corresponding median value as well as the 10th percentile value. The 10th percentile constitutes the 90% lower 1-sided bound on the reliability at 3,000 hours, which is calculated to be 50.77%.

WB.8 pdf median.png

The pdf of the times-to-failure data can be plotted in Weibull++, as shown next:

Weibull Distribution Example 6 pdf.png