Template:Example: 2P Weibull Distribution RRY: Difference between revisions

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Utilizing the values from above table, calculate <math> \hat{a} </math> and <math> \hat{b} </math> using the following equations:  
Utilizing the values from the table, calculate <math> \hat{a} </math> and <math> \hat{b} </math> using the following equations:  
::<math> \hat{b} =\frac{\sum\limits_{i=1}^{6}(\ln t_{i})y_{i}-(\sum\limits_{i=1}^{6}\ln t_{i})(\sum\limits_{i=1}^{6}y_{i})/6}{ \sum\limits_{i=1}^{6}(\ln t_{i})^{2}-(\sum\limits_{i=1}^{6}\ln t_{i})^{2}/6}
::<math> \hat{b} =\frac{\sum\limits_{i=1}^{6}(\ln t_{i})y_{i}-(\sum\limits_{i=1}^{6}\ln t_{i})(\sum\limits_{i=1}^{6}y_{i})/6}{ \sum\limits_{i=1}^{6}(\ln t_{i})^{2}-(\sum\limits_{i=1}^{6}\ln t_{i})^{2}/6}
</math>  
</math>  
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::<math> \hat{\rho }=0.9956 </math>  
::<math> \hat{\rho }=0.9956 </math>  
The above example can be repeated using Weibull++. Start Weibull++ and create a new ''Data Folio''.
[[Image:projectwizardweibull.png|thumb|center|400px|]]
Select the '' Times-to-failure data'' option.
[[Image:times--to-failure.png|thumb|center|400px| The effect of the Weibull shape parameter on the <math>pdf</math>. ]]
Enter the times-to-failure in the datasheet (ignore the Subset ID column), as shown next. The times-to-failure need not be sorted, Weibull++ will automatically sort the data.
<br>
<br>
[[Image:Weibull Distribution Example 3 RRY Data.png|thumb|center|400px| ]]
This example can be repeated in the Weibull++ software, as shown next. The following picture shows a Weibull++ standard folio data sheet calculated with the 2P-Weibull distribution and rank regression on Y.  
 
Select the desired method of analysis. Note that we are assuming that the underlying distribution is the Weibull, so make sure that the 2P-Weibull distribution is selected.
<br>
[[Image:parameterweibull.png|thumb|center|400px|]]
 
Also, so that you get the same results as this example, switch to the ''Analysis'' page and make sure you are using the ''Rank Regression on Y (RRY)'' calculation method with this example, as shown next.
 
[[Image:Weibull Distribution Example 3 Select RRY.png|thumb|center|400px| ]]
 
Note that this can also be done from the ''Main'' page by clicking the left bottom box under the Results area. Each time you click that box you will see the method switch between MLE, RRX, and RRY. Click the ''Calculate'' icon,
 
[[Image:calculateicon.png|thumb|center|400px|]]
or select ''Calculate'' from the ''Data'' menu. The results will appear in the Data Folio's ''Results area''. The next figure shows the results for this example.  


[[Image:Weibull Distribution Example 3 RRY Result.png|thumb|center|400px| ]]
[[Image:Weibull Distribution Example 3 RRY Result.png|thumb|center|400px| ]]


<br>
You can now plot the results by clicking the ''Plot'' icon,
[[Image:ploticon.png|thumb|center|400px| ]]
or by selecting ''Plot Probability'' from the ''Data'' menu.
The Weibull probability plot for these data is shown next.
[[Image: Weibull Distribution Example 3 RRY Plot.png|thumb|center|400px|]]
The confidence bounds, as determined from the Fisher matrix, can also be plotted. Select ''Confidence Bounds'' from the ''Plot'' menu, choose ''Two-Sided'' under ''Sides,'' ''Reliability (Type II)'' under ''Type'' and enter ''90'' for the ''Confidence level.


[[Image:weibullconfidencebounds.png|thumb|center|400px| ]]
The following plot shows the Weibull probability plot for the data set (with 90% two-sided confidence bounds).
The plot will appear as follows,


[[Image:Weibull Distribution Example 3 RRY Confidence Plot.png|thumb|center|400px| ]]
[[Image:Weibull Distribution Example 3 RRY Confidence Plot.png|thumb|center|400px| ]]


If desired, the Weibull  <math>pdf</math> representing these data can be written as:  
If desired, the Weibull  <math>pdf</math> representing the data set can be written as:  


::<math> f(t)={\frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t}{\eta }}\right) ^{\beta }} </math>  
::<math> f(t)={\frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t}{\eta }}\right) ^{\beta }} </math>  
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::<math> f(t)={\frac{1.4302}{76.317}}\left( {\frac{t}{76.317}}\right) ^{0.4302}e^{-\left( {\frac{t}{76.317}}\right) ^{1.4302}} </math>  
::<math> f(t)={\frac{1.4302}{76.317}}\left( {\frac{t}{76.317}}\right) ^{0.4302}e^{-\left( {\frac{t}{76.317}}\right) ^{1.4302}} </math>  


You can also plot the Weibull  by selecting ''Pdf Plot'' from the ''Plot Type'' drop-down menu on the control panel to the right of the plot area.  
You can also plot this result by selecting '''Pdf Plot''' on the Plot Type drop-down list on the control panel.  


[[Image:Weibull Distribution Example 3 pdf Plot.png|thumb|center|400px]]
[[Image:Weibull Distribution Example 3 pdf Plot.png|thumb|center|400px]]


From this point on, different results, reports and plots can be obtained.
From this point on, different results, reports and plots can be obtained.

Revision as of 23:24, 6 April 2012

2P Weibull Distribution RRY Example

Consider the data in Example 1, where six units were tested to failure and the following failure times were recorded: 16, 34, 53, 75, 93 and 120 hours. Estimate the parameters and the correlation coefficient using rank regression on Y, assuming that the data follow the two-parameter Weibull distribution.


Solution

Construct a table as shown below.

Table - Least Squares Analysis
[math]\displaystyle{ N }[/math] [math]\displaystyle{ T_{i} }[/math] [math]\displaystyle{ ln(T_{i}) }[/math] [math]\displaystyle{ F(T_i) }[/math] [math]\displaystyle{ y_{i} }[/math] [math]\displaystyle{ (ln{T_i})^2 }[/math] [math]\displaystyle{ {y_i}^2 }[/math] [math]\displaystyle{ (ln{T_i})y_i }[/math]
1 16 2.7726 0.1091 -2.1583 7.6873 4.6582 -5.9840
2 34 3.5264 0.2645 -1.1802 12.4352 1.393 -4.1620
3 53 3.9703 0.4214 -0.6030 15.7632 0.3637 -2.3943
4 75 4.3175 0.5786 -0.146 18.6407 0.0213 -0.6303
5 93 4.5326 0.7355 0.2851 20.5445 0.0813 1.2923
6 120 4.7875 0.8909 0.7955 22.9201 0.6328 3.8083
[math]\displaystyle{ \sum }[/math] 23.9068 -3.007 97.9909 7.1502 -8.0699


Utilizing the values from the table, calculate [math]\displaystyle{ \hat{a} }[/math] and [math]\displaystyle{ \hat{b} }[/math] using the following equations:

[math]\displaystyle{ \hat{b} =\frac{\sum\limits_{i=1}^{6}(\ln t_{i})y_{i}-(\sum\limits_{i=1}^{6}\ln t_{i})(\sum\limits_{i=1}^{6}y_{i})/6}{ \sum\limits_{i=1}^{6}(\ln t_{i})^{2}-(\sum\limits_{i=1}^{6}\ln t_{i})^{2}/6} }[/math]
[math]\displaystyle{ \hat{b}=\frac{-8.0699-(23.9068)(-3.0070)/6}{97.9909-(23.9068)^{2}/6} }[/math]

or

[math]\displaystyle{ \hat{b}=1.4301 }[/math]

and:

[math]\displaystyle{ \hat{a}=\overline{y}-\hat{b}\overline{T}=\frac{\sum \limits_{i=1}^{N}y_{i}}{N}-\hat{b}\frac{\sum\limits_{i=1}^{N}\ln t_{i}}{N } }[/math]

or:

[math]\displaystyle{ \hat{a}=\frac{(-3.0070)}{6}-(1.4301)\frac{23.9068}{6}=-6.19935 }[/math]

Therefore:

[math]\displaystyle{ \hat{\beta }=\hat{b}=1.4301 }[/math]

and:

[math]\displaystyle{ \hat{\eta }=e^{-\frac{\hat{a}}{\hat{b}}}=e^{-\frac{(-6.19935)}{ 1.4301}} }[/math]

or:

[math]\displaystyle{ \hat{\eta }=76.318\text{ hr} }[/math]

The correlation coefficient can be estimated as:

[math]\displaystyle{ \hat{\rho }=0.9956 }[/math]


This example can be repeated in the Weibull++ software, as shown next. The following picture shows a Weibull++ standard folio data sheet calculated with the 2P-Weibull distribution and rank regression on Y.

Weibull Distribution Example 3 RRY Result.png


The following plot shows the Weibull probability plot for the data set (with 90% two-sided confidence bounds).

Weibull Distribution Example 3 RRY Confidence Plot.png

If desired, the Weibull [math]\displaystyle{ pdf }[/math] representing the data set can be written as:

[math]\displaystyle{ f(t)={\frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t}{\eta }}\right) ^{\beta }} }[/math]

or:

[math]\displaystyle{ f(t)={\frac{1.4302}{76.317}}\left( {\frac{t}{76.317}}\right) ^{0.4302}e^{-\left( {\frac{t}{76.317}}\right) ^{1.4302}} }[/math]

You can also plot this result by selecting Pdf Plot on the Plot Type drop-down list on the control panel.

Weibull Distribution Example 3 pdf Plot.png

From this point on, different results, reports and plots can be obtained.

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