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== Weibull-Bayesian Analysis ==
#REDIRECT [[Bayesian-Weibull_Analysis]]
In this section, the Bayesian methods are presented for the two-parameter Weibull distribution. Bayesian concepts were introduced in Chapter 3. This model considers prior knowledge on the shape (<span class="texhtml">β</span>) 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 <span class="texhtml">β</span> and <span class="texhtml">η</span> are independent, we obtain the following posterior :
 
::<math> f(\beta ,\eta |Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } </math> EQNREF WeibBayes
 
In this model, <span class="texhtml">η</span> is assumed to follow a noninformative prior distribution with the density function <math> \varphi (\eta )=\dfrac{1}{\eta } </math>. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on <span class="texhtml">η.</span> Specifically, since <span class="texhtml">η</span> is always positive, we can assume that ln(<span class="texhtml">η)</span> follows a uniform distribution, <span class="texhtml">''U''( − ∞, + ∞).</span> 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 <math> \varphi (\eta )=\dfrac{1}{\eta }. </math>
 
The prior distribution of <span class="texhtml">β</span>, denoted as <math> \varphi (\beta ) </math>, can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Weibull-Bayesian analysis is as follows: 
 
:*Collect the times-to-failure data.
:*Specify a prior distribution for <span class="texhtml">β</span> (the prior for <span class="texhtml">η</span> is assumed to be 1/<span class="texhtml">η).</span>
:*Obtain the posterior  from Eqn. (EQNREF WeibBayes ).
 
In other words, a distribution (the posterior ) 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  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.
 
[[Image:time-failedID.png|thumb|center|400px| ]]
 
The expected value of <span class="texhtml">β</span> is obtained by: 
::<math> E(\beta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\beta \cdot f(\beta ,\eta |Data)d\beta d\eta </math>
 
Similarly, the expected value of <span class="texhtml">η</span> is obtained by: 
::<math> E(\eta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\eta \cdot f(\beta ,\eta |Data)d\beta d\eta </math>
 
The median points are obtained by solving the following equations for <math> \breve{\beta} </math> and <math> \breve{\eta} </math> respectively:
 
::<math> \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\breve{\beta}}f(\beta ,\eta |Data)d\beta d\eta =0.5 </math>
 
:and
 
::<math> \int\nolimits_{0}^{\breve{\eta}}\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d\beta d\eta =0.5 </math>
 
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++.
 
=== Posterior Distributions for Functions of Parameters ===
As explained in Chapter 3, 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.
 
====<math>pdf</math> of the Times-to-Failure ====
The posterior distribution of the failure time  is given by:
 
::<math> f(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }f(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta </math> EQNREF WeibBayesPDF
 
:where:
 
::<math> f(T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1}e^{-\left( \dfrac{T}{\eta }\right) ^{\beta }} </math>
 
For the <math>pdf</math> 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  of the reliability. Since <span class="texhtml">''R''(''T'')</span> is a function of <span class="texhtml">β</span>, the density functions of <span class="texhtml">β</span> and <span class="texhtml">''R''(''T'')</span> have the following relationship:
 
::<math> \begin{align} f(R|Data,T)dR = & f(\beta |Data)d\beta)\\
          = & (\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d{\eta}) d{\beta} \\
=& \dfrac{\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }d\beta       
\end{align}
</math>  EQNREF Rpdf
 
The median value of the reliability is obtained by solving the following equation w.r.t. <math> \breve{R}: </math>
 
::<math> \int\nolimits_{0}^{\breve{R}}f(R|Data,T)dR=0.5 </math> EQNREF MedRel
 
The expected value of the reliability at time  is given by:
 
::<math> R(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }R(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta </math> where:
 
::<math> R(T,\beta ,\eta )=e^{-\left( \dfrac{T}{\eta }\right) ^{^{\beta }}} </math>
 
<br>
==== Failure Rate ====
The failure rate at time  is given by:
 
::<math> \lambda (T|Data)=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\lambda (T,\beta ,\eta )L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta } </math>
 
:where:
 
::<math> \lambda (T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1} </math>
 
<br>
=== 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 <math>pdf</math>s 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.
 
=== Confidence Bounds on <span class="texhtml">''R''(''T'')</span> ===
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 <span class="texhtml">β</span> is considered instead of an non-informative prior. The Bayesian one-sided upper bound estimate for <span class="texhtml">''R''(''T'')</span> is given by:
 
::<math> \int\nolimits_{0}^{R_{U}(T)}f(R|Data,T)dR=CL </math>
 
Using Eqns. (EQNREF WeibBayes ) and (EQNREF Rpdf ) the following is obtained:
 
::<math> \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{T\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL </math> EQNREF 1CLRU
 
Eqn. (EQNREF 1CLRU ) can be solved for <span class="texhtml">''R''<sub>''U''</sub>(''T'')</span>. The Bayesian one-sided lower bound estimate for <math> \ R(T) </math> is given by:
 
::<math> \int\nolimits_{0}^{R_{L}(T)}f(R|Data,T)dR=1-CL </math>
 
Using Eqns. (EQNREF WeibBayes ) and (EQNREF Rpdf ) the following is obtained:
 
::<math> \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{L})}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=1-CL </math> EQNREF 1CLRL
 
Eqn. (EQNREF 1CLRL ) can be solved for <span class="texhtml">''R''<sub>''L''</sub>(''T'')</span>. The Bayesian two-sided bounds estimate for <span class="texhtml">''R''(''T'')</span> is given by:
 
::<math> \int\nolimits_{R_{L}(T)}^{R_{U}(T)}f(R|Data,T)dR=CL </math> which is equivalent to:
 
::<math> \int\nolimits_{0}^{R_{U}(T)}f(R|Data,T)dR=(1+CL)/2 </math>
 
:and
 
::<math> \int\nolimits_{0}^{R_{L}(T)}f(R|Data,T)dR=(1-CL)/2 </math>
 
Using the same method for one-sided bounds, <span class="texhtml">''R''<sub>''U''</sub>(''T'')</span>and <span class="texhtml">''R''<sub>''L''</sub>(''T'')</span> can be computed.
 
=== Confidence Bounds on Time <math>T</math> ===
Following the same procedure described for bounds on Reliability, the bounds of time  can be calculated, given . The Bayesian one-sided upper bound estimate for <span class="texhtml">''T''(''R'')</span> is given by:
 
::<math> \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=CL </math>
 
Using Eqns. (EQNREF WeibBayes ) and. (EQNREF WeibBayesPDF ), we obtain:
 
::<math> \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{\ln (-\ln R)}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL </math> EQNREF 1CLTU
 
Eqn. (EQNREF 1CLTU ) can be solved for <span class="texhtml">''T''<sub>''U''</sub>(''R'')</span>. The Bayesian one-sided lower bound estimate for <span class="texhtml">''T''(''R'')</span> is given by:
 
::<math> \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=1-CL </math>
 
:or:
 
::<math> \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{T_{L}\exp (\dfrac{-\ln (-\ln R)}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL </math> EQNREF 1CLTL
 
Eqn. (EQNREF 1CLTL ) can be solved for <span class="texhtml">''T''<sub>''L''</sub>(''R'')</span>. The Bayesian two-sided lower bounds estimate for <span class="texhtml">''T''(''R'')</span> is:
 
::<math> \int\nolimits_{T_{L}(R)}^{T_{U}(R)}f(T|Data,R)dT=CL </math>
 
which is equivalent to:
 
::<math> \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=(1+CL)/2 </math>
 
:and:
 
::<math> \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=(1-CL)/2 </math>
 
<br>
====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.
 
{|  border=1 cellspacing=1 align="center"
|-
!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 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 <span class="texhtml">β</span> parameter is as follows:
 
{| border=1 cellspacing=1 align="center"
|-
!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 <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.)
 
the test data is entered into a ''Standard Folio'', the Weibull-Bayesian is selected under '' Distribution'' and the <span class="texhtml">β</span> prior distribution is entered after clicking the ''Calculate'' button.
 
[[Image:chp6folio2data1.gif|thumb|center|300px| ]]
 
Suppose that the reliability at 3000hr is the metric of interest in this example. This reliability can be obtained using Eqn. (EQNREF MedRel ), resulting in the median value of the posterior  of the reliability at 3000hr. Using the ''QCP'', this value is calculated to be 76.97. ( By default Weibull++ returns the median values of the posterior distribution. )
 
The posterior <math>pdf</math> of the reliability function at 3000hrs can be obtained using Eqn. (EQNREF Rpdf ). In Figure 6-10 the posterior <math>pdf</math> 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.
 
[[Image:lda6.10.gif|thumb|center|300px| ]]
 
Notice that the <math>pdf</math> plotted in Fig. 6-10 is of the reliability at 3000hrs, 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 Eqn. (EQNREF WeibBayesPDF ) and plotted using Weibull++, as shown next:
 
[[Image:chp6plotofdata1.gif|thumb|center|400px| ]]
 
<br>
{{RS Copyright}} 
 
[[Category:Life_Data_Analysis_Reference]]

Latest revision as of 03:18, 10 August 2012

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