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Two methods are presented below that yield the same posterior distribution.

 

 

Method 1:  Image Removed simulation model Turbine Blade Simulation

We tested 18 * 30 = 540 blades, of which 1 failed. That leads to an estimate of:

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P(TurbineFail) = 1-(1- P(BladeFail))30

 

 

The object of this analysis was to determine P(BladeFail) and from that determine P(TurbineFail). A Bayesian estimate of P(BladeFail) required the parameter F, which was uncertain. The distribution for F is a hyperparameter. By using simulation to arrive at P(BladeFail) we have a natural way of integrating the extra uncertainty that F introduces into the calculation.

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Turbine Blade Simulation
Turbine Blade Simulation

The links to the Turbine Blade Simulation software specific models are provided here:

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titleCrystal Ball

 Image Added Turbine_blade_simulation

 

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Here is teh result of the above model:

 

Image Added

 

The graph shows that we are 90% confident that the probability a turbine fails within period T is less than 28%.

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title@Risk

 Image Added Turbine_blade_simulation

 

Here is the result of the above model:

Image Added

 

The graph shows that we are 90% confident that the probability a turbine fails within period T is less than 29.4%.

 

Method 2: Turbine Blade Construction

 

We can construct a confidence distribution for P(BladeFail) = q as follows:

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This leads to the following posterior distribution:

 

From that analysis, Crystal Ball's Custom distribution can then be used to sample values from the uncertainty distribution, as shown in the Image RemovedTurbine blade construction model. The analysis has not yet taken into account the uncertainty around P(detect), but we can do this with numerical integration. We can create a spreadsheet cell that simulates simulation for P(detect), as before:

 

P(detect) = Beta(4+1,5-4+1,1) = Beta(5,2,1)

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and then link it to the BINOMDIST likelihood calculation. Crystal Ball's CB.GetForeStatFN(..,2) function can then be used to calculate Then we calculate the posterior after this hyperparameter has been taken into account, by simply taking the mean of the simulated posterior density:

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If you did not need a posterior distribution for P(BladeFail), the extra step of using the CB.GetForeStatFN(..,2) function is unnecessary: it is sufficient to simply run a simulation that samples from the Beta distribution for P(detect), constructs the General distribution for P(BladeFail), samples from that General distribution and calculates a value for P(TurbineFail), your output. All options are demonstrated in the accompanying model.

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Turbine Blade Construction
Turbine Blade Construction

The links to the Turbine Blade Construction software specific models are provided here:

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titleCrystal Ball

 Image Added Turbine_blade_construction

 

In the model, Crystal Ball's Custom distribution was used to sample values from the uncertainty distribution.

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title@Risk

 Image Added Turbine_blade_construction

 

In the model, @Risk's General distribution was used to sample values from the uncertainty distribution.