Duane Confidence Bounds Example: Difference between revisions

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<noinclude>{{Banner RGA Examples}}
<noinclude>{{Banner RGA Examples}}
''This example appears in the [[Duane Model|Reliability Growth and Repairable System Analysis Reference book]]''.
''This example appears in the [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference]''.


</noinclude>
</noinclude>


Using the values of <math>\widehat{b}\,\!</math> and <math>\widehat{\alpha }\,\!</math> estimated from the least squares analysis in <noinclude>[[Duane Linear Regression Examples|
Using the values of <math>\hat{b}\,\!</math> and <math>\hat{\alpha }\,\!</math> estimated from the least squares analysis in <noinclude>[[Duane Linear Regression Examples|
Least Squares Example 2]]</noinclude><includeonly>Least Squares Example 2</includeonly>:
Least Squares Example 2]]</noinclude><includeonly>Least Squares Example 2</includeonly>:
:<math>\widehat{b}=1.9453\,\!</math>
:<math>\hat{b}=1.9453\,\!</math>
:<math>\widehat{\alpha}=0.6133\,\!</math>
:<math>\hat{\alpha}=0.6133\,\!</math>


calculate the 90% confidence bounds for:
Calculate the 90% confidence bounds for the following:


#The parameters <math>\alpha\,\!</math> and <math>b\,\!</math>.
#The parameters <math>\alpha\,\!</math> and <math>b\,\!</math>.
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'''Solution'''
'''Solution'''


1. Use the values of <math>\widehat{b}\,\!</math> and <math>\widehat{\alpha }\,\!</math> estimated from the least squares analysis. Then:  
<ol>
<li>Use the values of <math>\hat{b}\,\!</math> and <math>\hat{\alpha }\,\!</math> estimated from the least squares analysis. Then:  


:<math>\begin{align}
:<math>\begin{align}
Line 53: Line 54:
   {{b}_{U}}= & 2.1231   
   {{b}_{U}}= & 2.1231   
\end{align}\,\!</math>
\end{align}\,\!</math>
 
</li>
2. The cumulative failure intensity is:  
<li>The cumulative failure intensity is:  


:<math>\begin{align}
:<math>\begin{align}
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:<math>\begin{align}
:<math>\begin{align}
   {{[{{\lambda }_{c}}(t)]}_{L}}= & 0.00100254 \\  
   {{[{{\lambda }_{c}}(t)]}_{L}}= & 0.00106780 \\  
   {{[{{\lambda }_{c}}(t)]}_{U}}= & 0.00124429  
   {{[{{\lambda }_{c}}(t)]}_{U}}= & 0.00116825  
\end{align}\,\!</math>
\end{align}\,\!</math>


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:<math>\begin{align}
:<math>\begin{align}
   {{[{{\lambda }_{i}}(t)]}_{L}}= & 0.00038775 \\  
   {{[{{\lambda }_{i}}(t)]}_{L}}= & 0.00041299 \\  
   {{[{{\lambda }_{c}}(t)]}_{U}}= & 0.00048125  
   {{[{{\lambda }_{c}}(t)]}_{U}}= & 0.00045184  
\end{align}\,\!</math>
\end{align}\,\!</math>


The following figures show the graphs of the cumulative and instantaneous failure intensity. Both are plotted with confidence bounds.
The following figures show the graphs of the cumulative and instantaneous failure intensity. Both are plotted with confidence bounds.


[[Image:rga4.7.png|center|400px|Cumulative Failure Intensity plot with 2-sided 90% confidence bounds.]]
[[Image:rga4.7.png|center|450px]]
 
[[Image:rga4.8.png|center|400px|Instantaneous Failure Intensity plot with 2-sided 90% confidence bounds.]]
 


3. The cumulative MTBF is:  
[[Image:rga4.8.png|center|450px]]
</li>
<li>The cumulative MTBF is:  


:<math>\begin{align}
:<math>\begin{align}
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:<math>\begin{align}
:<math>\begin{align}
   {{m}_{c}}{{(t)}_{l}}= & 803.6695 \\  
   {{m}_{c}}{{(t)}_{l}}= & 855.9815 \\  
   {{m}_{c}}{{(t)}_{u}}= & 997.4658  
   {{m}_{c}}{{(t)}_{u}}= & 936.5071  
\end{align}\,\!</math>
\end{align}\,\!</math>


Line 113: Line 113:


:<math>\begin{align}
:<math>\begin{align}
   {{m}_{i}}{{(t)}_{l}}= & 2077.9204 \\  
   {{m}_{i}}{{(t)}_{l}}= & 2213.1753 \\  
   {{m}_{i}}{{(t)}_{u}}= & 2578.9886  
   {{m}_{i}}{{(t)}_{u}}= & 2421.3776  
\end{align}\,\!</math>
\end{align}\,\!</math>


The figure below displays the cumulative MTBF.
The figure below displays the cumulative MTBF.


[[Image:rga4.9.png|center|400px|Cumulative MTBF plot with 2-sided 90% condfidence bounds.]]
[[Image:rga4.9.png|center|450px]]


The next figure displays the instantaneous MTBF. Both are plotted with confidence bounds.
The next figure displays the instantaneous MTBF. Both are plotted with confidence bounds.


[[Image:rga4.10.png|center|400px|Instantaneous MTBF plot with 2-sided 90% confidence bounds.]]
[[Image:rga4.10.png|center|450px]]
</li>
</ol>

Latest revision as of 21:23, 18 September 2023

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This example appears in the Reliability growth reference.


Using the values of [math]\displaystyle{ \hat{b}\,\! }[/math] and [math]\displaystyle{ \hat{\alpha }\,\! }[/math] estimated from the least squares analysis in Least Squares Example 2:

[math]\displaystyle{ \hat{b}=1.9453\,\! }[/math]
[math]\displaystyle{ \hat{\alpha}=0.6133\,\! }[/math]

Calculate the 90% confidence bounds for the following:

  1. The parameters [math]\displaystyle{ \alpha\,\! }[/math] and [math]\displaystyle{ b\,\! }[/math].
  2. The cumulative and instantaneous failure intensity.
  3. The cumulative and instantaneous MTBF.

Solution

  1. Use the values of [math]\displaystyle{ \hat{b}\,\! }[/math] and [math]\displaystyle{ \hat{\alpha }\,\! }[/math] estimated from the least squares analysis. Then:
    [math]\displaystyle{ \begin{align} {{S}_{xx}}&=\left[ \underset{i=1}{\overset{n}{\mathop \sum }}\,{{(\ln {{t}_{i}})}^{2}} \right]-\frac{1}{n}{{\left( \underset{i=1}{\overset{n}{\mathop \sum }}\,\ln ({{t}_{i}}) \right)}^{2}} \\ & = 1400.9084-1301.4545 \\ & = 99.4539 \end{align}\,\! }[/math]
    [math]\displaystyle{ \begin{align} SE(\hat{\alpha })= & \frac{\sigma }{\sqrt{{{S}_{xx}}}} \\ = & \frac{0.08428}{9.9727} \\ = & 0.008452 \end{align}\,\! }[/math]
    [math]\displaystyle{ \begin{align} SE(\ln \hat{b})= & \sigma \cdot \sqrt{\frac{\underset{i=1}{\overset{n}{\mathop{\sum }}}\,{{(\ln {{T}_{i}})}^{2}}}{n\cdot {{S}_{xx}}}} \\ = & 0.065960 \end{align}\,\! }[/math]
    Thus, the 90% confidence bounds on parameter [math]\displaystyle{ \alpha \,\! }[/math] are:
    [math]\displaystyle{ C{{B}_{\alpha }}=\hat{\alpha }\pm {{t}_{n-2,\alpha /2}}SE(\hat{\alpha })\,\! }[/math]
    [math]\displaystyle{ \begin{align} {{\alpha }_{L}}= & 0.602050 \\ {{\alpha }_{U}}= & 0.624417 \end{align}\,\! }[/math]
    And 90% confidence bounds on parameter [math]\displaystyle{ b\,\! }[/math] are:
    [math]\displaystyle{ C{{B}_{b}}=\hat{b}{{e}^{\pm {{t}_{n-2,\alpha /2}}SE\left[ \ln (\hat{b}) \right]}}\,\! }[/math]
    [math]\displaystyle{ \begin{align} {{b}_{L}}= & 1.7831 \\ {{b}_{U}}= & 2.1231 \end{align}\,\! }[/math]
  2. The cumulative failure intensity is:
    [math]\displaystyle{ \begin{align} {{\lambda }_{c}}= & \frac{1}{1.9453}\cdot {{22000}^{-0.6133}} \\ = & 0.00111689 \end{align}\,\! }[/math]
    And the instantaneous failure intensity is equal to:
    [math]\displaystyle{ \begin{align} {{\lambda }_{i}}= & \frac{1}{1.9453}\cdot (1-0.6133)\cdot {{22000}^{-0.6133}} \\ = & 0.00043198 \end{align}\,\! }[/math]
    So, at the 90% confidence level and for [math]\displaystyle{ T=22,000\,\! }[/math] hours, the confidence bounds on cumulative failure intensity are:
    [math]\displaystyle{ \begin{align} {{[{{\lambda }_{c}}(t)]}_{L}}= & 0.00106780 \\ {{[{{\lambda }_{c}}(t)]}_{U}}= & 0.00116825 \end{align}\,\! }[/math]
    For the instantaneous failure intensity:
    [math]\displaystyle{ \begin{align} {{[{{\lambda }_{i}}(t)]}_{L}}= & 0.00041299 \\ {{[{{\lambda }_{c}}(t)]}_{U}}= & 0.00045184 \end{align}\,\! }[/math]
    The following figures show the graphs of the cumulative and instantaneous failure intensity. Both are plotted with confidence bounds.
    Rga4.7.png
    Rga4.8.png
  3. The cumulative MTBF is:
    [math]\displaystyle{ \begin{align} {{m}_{c}}(T)= & 1.9453\cdot {{22000}^{0.6133}} \\ = & 895.3395 \end{align}\,\! }[/math]
    And the instantaneous MTBF is:
    [math]\displaystyle{ \begin{align} {{m}_{i}}(T)= & \frac{1.9453}{1-0.6133}\cdot {{22000}^{0.6133}} \\ = & 2314.9369 \end{align}\,\! }[/math]
    So, at 90% confidence level and for [math]\displaystyle{ T=22,000\,\! }[/math] hours, the confidence bounds on the cumulative MTBF are:
    [math]\displaystyle{ \begin{align} {{m}_{c}}{{(t)}_{l}}= & 855.9815 \\ {{m}_{c}}{{(t)}_{u}}= & 936.5071 \end{align}\,\! }[/math]
    The confidence bounds for the instantaneous MTBF are:
    [math]\displaystyle{ \begin{align} {{m}_{i}}{{(t)}_{l}}= & 2213.1753 \\ {{m}_{i}}{{(t)}_{u}}= & 2421.3776 \end{align}\,\! }[/math]
    The figure below displays the cumulative MTBF.
    Rga4.9.png

    The next figure displays the instantaneous MTBF. Both are plotted with confidence bounds.

    Rga4.10.png