Reliability Data - Modified Gompertz Model: Difference between revisions
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<noinclude>{{Banner RGA Examples}} | <noinclude>{{Banner RGA Examples}} | ||
''This example appears in the [[Gompertz_Models|Reliability Growth and Repairable System Analysis Reference | ''This example appears in the [[Gompertz_Models|Reliability Growth and Repairable System Analysis Reference]]''. | ||
</noinclude> | </noinclude> | ||
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\end{align}\,\!</math> | \end{align}\,\!</math> | ||
:<math>c(d)={{\left[ \frac{{{S}_{3}}(d)-{{S}_{2}}(d)}{{{S}_{2}}(d)-{{S}_{1}}(d)} \right]}^{\tfrac{1}{3}}}\,\!</math> | |||
:<math>a(d)={{e}^{\left[ \tfrac{1}{3}\left( {{S}_{1}}(d)+\tfrac{{{S}_{2}}(d)-{{S}_{1}}(d)}{1-{{[c(d)]}^{3}}} \right) \right]}}\,\!</math> | |||
:<math>b(d)={{e}^{\left[ \tfrac{({{S}_{2}}(d)-{{S}_{1}}(d))(c(d)-1)}{{{\left[ 1-{{[c(d)]}^{3}} \right]}^{2}}} \right]}}\,\!</math> | |||
and: | |||
:<math>{{R}_{0}}=d+a(d)\cdot b(d)\,\!</math> | |||
for <math>{{R}_{0}}=31%\,\!</math>, the equation above may be rewritten as: | for <math>{{R}_{0}}=31%\,\!</math>, the equation above may be rewritten as: | ||
:<math>d-31+a(d)\cdot b(d)=0\,\!</math> | |||
The equations for parameters <math>c\,\!</math>, <math>a\,\!</math> and <math>b\,\!</math> can now be solved simultaneously. One method for solving these equations numerically is to substitute different values of <math>d\,\!</math>, which must be less than <math>{{R}_{0}}\,\!</math>, into the last equation shown above, and plot the results along the y-axis with the value of <math>d\,\!</math> along the x-axis. The value of <math>d\,\!</math> can then be read from the x-intercept. This can be repeated for greater accuracy using smaller and smaller increments of <math>d\,\!</math>. Once the desired accuracy on <math>d\,\!</math> has been achieved, the value of <math>d\,\!</math> can then be used to solve for <math>a\,\!</math>, <math>b\,\!</math> and <math>c\,\!</math>. For this case, the initial estimates of the parameters are: | The equations for parameters <math>c\,\!</math>, <math>a\,\!</math> and <math>b\,\!</math> can now be solved simultaneously. One method for solving these equations numerically is to substitute different values of <math>d\,\!</math>, which must be less than <math>{{R}_{0}}\,\!</math>, into the last equation shown above, and plot the results along the y-axis with the value of <math>d\,\!</math> along the x-axis. The value of <math>d\,\!</math> can then be read from the x-intercept. This can be repeated for greater accuracy using smaller and smaller increments of <math>d\,\!</math>. Once the desired accuracy on <math>d\,\!</math> has been achieved, the value of <math>d\,\!</math> can then be used to solve for <math>a\,\!</math>, <math>b\,\!</math> and <math>c\,\!</math>. For this case, the initial estimates of the parameters are: | ||
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\widehat{d}= & 30.825 | \widehat{d}= & 30.825 | ||
\end{align}\,\!</math> | \end{align}\,\!</math> | ||
Now, since the initial values have been determined, the Gauss-Newton method can be used. Therefore, substituting <math>{{Y}_{i}}={{R}_{i}},\,\!</math> <math>g_{1}^{(0)}=69.324,\,\!</math> <math>g_{2}^{(0)}=0.002524,\,\!</math> <math>g_{3}^{(0)}=0.46012,\,\!</math> and <math>g_{4}^{(0)}=30.825\,\!</math>, <math>{{Y}^{(0)}},{{D}^{(0)}},\,\!</math> <math>{{\nu }^{(0)}}\,\!</math> become: | Now, since the initial values have been determined, the Gauss-Newton method can be used. Therefore, substituting <math>{{Y}_{i}}={{R}_{i}},\,\!</math> <math>g_{1}^{(0)}=69.324,\,\!</math> <math>g_{2}^{(0)}=0.002524,\,\!</math> <math>g_{3}^{(0)}=0.46012,\,\!</math> and <math>g_{4}^{(0)}=30.825\,\!</math>, <math>{{Y}^{(0)}},{{D}^{(0)}},\,\!</math> <math>{{\nu }^{(0)}}\,\!</math> become: | ||
:<math>{{Y}^{(0)}}=\left[ \begin{matrix} | |||
0.000026 \\ | 0.000026 \\ | ||
0.253873 \\ | 0.253873 \\ | ||
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\end{matrix} \right]\,\!</math> | \end{matrix} \right]\,\!</math> | ||
:<math>{{D}^{(0)}}=\left[ \begin{matrix} | |||
0.002524 & 69.3240 & 0.0000 & 1 \\ | 0.002524 & 69.3240 & 0.0000 & 1 \\ | ||
0.063775 & 805.962 & -26.4468 & 1 \\ | 0.063775 & 805.962 & -26.4468 & 1 \\ | ||
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\end{matrix} \right]\,\!</math> | \end{matrix} \right]\,\!</math> | ||
:<math>{{\nu }^{(0)}}=\left[ \begin{matrix} | |||
g_{1}^{(0)} \\ | g_{1}^{(0)} \\ | ||
g_{2}^{(0)} \\ | g_{2}^{(0)} \\ | ||
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30.825 \\ | 30.825 \\ | ||
\end{matrix} \right]\,\!</math> | \end{matrix} \right]\,\!</math> | ||
The estimate of the parameters <math>{{\nu }^{(0)}}\,\!</math> is given by: | The estimate of the parameters <math>{{\nu }^{(0)}}\,\!</math> is given by: | ||
:<math>\begin{align} | |||
{{\widehat{\nu }}^{(0)}}= & {{\left( {{D}^{{{(0)}^{T}}}}{{D}^{(0)}} \right)}^{-1}}{{D}^{{{(0)}^{T}}}}{{Y}^{(0)}} \\ | {{\widehat{\nu }}^{(0)}}= & {{\left( {{D}^{{{(0)}^{T}}}}{{D}^{(0)}} \right)}^{-1}}{{D}^{{{(0)}^{T}}}}{{Y}^{(0)}} \\ | ||
= & \left[ \begin{matrix} | = & \left[ \begin{matrix} | ||
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\end{matrix} \right] | \end{matrix} \right] | ||
\end{align}\,\!</math> | \end{align}\,\!</math> | ||
The revised estimated regression coefficients in matrix form are given by: | The revised estimated regression coefficients in matrix form are given by: | ||
:<math>\begin{align} | |||
{{g}^{(1)}}= & {{g}^{(0)}}+{{\widehat{\nu }}^{(0)}}. \\ | {{g}^{(1)}}= & {{g}^{(0)}}+{{\widehat{\nu }}^{(0)}}. \\ | ||
= & \left[ \begin{matrix} | = & \left[ \begin{matrix} | ||
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\end{matrix} \right] | \end{matrix} \right] | ||
\end{align}\,\!</math> | \end{align}\,\!</math> | ||
With the starting coefficients <math>{{g}^{(0)}}\,\!</math>, <math>Q\,\!</math> is: | With the starting coefficients <math>{{g}^{(0)}}\,\!</math>, <math>Q\,\!</math> is: | ||
:<math>\begin{align} | |||
{{Q}^{(0)}}= & \underset{i=1}{\overset{N}{\mathop \sum }}\,{{\left( {{Y}_{i}}-f({{T}_{i}},{{g}^{(0)}}) \right)}^{2}} \\ | {{Q}^{(0)}}= & \underset{i=1}{\overset{N}{\mathop \sum }}\,{{\left( {{Y}_{i}}-f({{T}_{i}},{{g}^{(0)}}) \right)}^{2}} \\ | ||
= & 2.403672 | = & 2.403672 | ||
\end{align}\,\!</math> | \end{align}\,\!</math> | ||
With the coefficients at the end of the first iteration, <math>{{g}^{(1)}}\,\!</math>, <math>Q\,\!</math> is: | With the coefficients at the end of the first iteration, <math>{{g}^{(1)}}\,\!</math>, <math>Q\,\!</math> is: | ||
:<math>\begin{align} | |||
{{Q}^{(1)}}= & \underset{i=1}{\overset{N}{\mathop \sum }}\,{{\left[ {{Y}_{i}}-f\left( {{T}_{i}},{{g}^{(1)}} \right) \right]}^{2}} \\ | {{Q}^{(1)}}= & \underset{i=1}{\overset{N}{\mathop \sum }}\,{{\left[ {{Y}_{i}}-f\left( {{T}_{i}},{{g}^{(1)}} \right) \right]}^{2}} \\ | ||
= & 2.073964 | = & 2.073964 | ||
\end{align}\,\!</math> | \end{align}\,\!</math> | ||
Therefore: | Therefore: | ||
:<math>\begin{align} | |||
{{Q}^{(1)}}<{{Q}^{(0)}} | {{Q}^{(1)}}<{{Q}^{(0)}} | ||
\end{align}\,\!</math> | \end{align}\,\!</math> | ||
Hence, the Gauss-Newton method works in the right direction. The iterations are continued until the relationship of <math>{{Q}^{(s-1)}}-{{Q}^{(s)}}\simeq 0\,\!</math> has been satisfied. Using RGA, the estimators of the parameters are: | Hence, the Gauss-Newton method works in the right direction. The iterations are continued until the relationship of <math>{{Q}^{(s-1)}}-{{Q}^{(s)}}\simeq 0\,\!</math> has been satisfied. Using RGA, the estimators of the parameters are: | ||
:<math>\begin{align} | |||
\widehat{a}= & 0.6904 \\ | \widehat{a}= & 0.6904 \\ | ||
\widehat{b}= & 0.0020 \\ | \widehat{b}= & 0.0020 \\ | ||
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\widehat{d}= & 0.3104 | \widehat{d}= & 0.3104 | ||
\end{align}\,\!</math> | \end{align}\,\!</math> | ||
Therefore, the modified Gompertz model is: | Therefore, the modified Gompertz model is: | ||
:<math>R=0.3104+(0.6904){{(0.0020)}^{{{0.4567}^{T}}}}\,\!</math> | |||
Using this equation, the predicted reliability is plotted in the following figure along with the raw data. As you can see, the modified Gompertz curve represents the data very well. | Using this equation, the predicted reliability is plotted in the following figure along with the raw data. As you can see, the modified Gompertz curve represents the data very well. | ||
[[Image:rga7.5.png|center|400px|Modified Gompertz reliability growth curve.]] | [[Image:rga7.5.png|center|400px|Modified Gompertz reliability growth curve.]] | ||
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This example appears in the Reliability Growth and Repairable System Analysis Reference.
A reliability growth data set is given in columns 1 and 2 of the following table. Find the modified Gompertz curve that represents the data and plot it comparatively with the raw data.
| Time(months) | Raw Data Reliability(%) | Gompertz Reliability(%) | Logistic Reliability(%) | Modified Gompertz Reliability(%) |
|---|---|---|---|---|
| 0 | 31.00 | 25.17 | 22.70 | 31.18 |
| 1 | 35.50 | 38.33 | 38.10 | 35.08 |
| 2 | 49.30 | 51.35 | 56.40 | 49.92 |
| 3 | 70.10 | 62.92 | 73.00 | 69.23 |
| 4 | 83.00 | 72.47 | 85.00 | 83.72 |
| 5 | 92.20 | 79.94 | 93.20 | 92.06 |
| 6 | 96.40 | 85.59 | 96.10 | 96.29 |
| 7 | 98.60 | 89.75 | 98.10 | 98.32 |
| 8 | 99.00 | 92.76 | 99.10 | 99.27 |
Solution
To determine the parameters of the modified Gompertz curve, use:
- [math]\displaystyle{ \begin{align} & {{S}_{1}}(d)= & \underset{i=0}{\overset{2}{\mathop \sum }}\,\ln ({{R}_{oi}}-d) \\ & {{S}_{2}}(d)= & \underset{i=3}{\overset{5}{\mathop \sum }}\,\ln ({{R}_{oi}}-d) \\ & {{S}_{3}}(d)= & \underset{i=6}{\overset{8}{\mathop \sum }}\,\ln ({{R}_{oi}}-d) \end{align}\,\! }[/math]
- [math]\displaystyle{ c(d)={{\left[ \frac{{{S}_{3}}(d)-{{S}_{2}}(d)}{{{S}_{2}}(d)-{{S}_{1}}(d)} \right]}^{\tfrac{1}{3}}}\,\! }[/math]
- [math]\displaystyle{ a(d)={{e}^{\left[ \tfrac{1}{3}\left( {{S}_{1}}(d)+\tfrac{{{S}_{2}}(d)-{{S}_{1}}(d)}{1-{{[c(d)]}^{3}}} \right) \right]}}\,\! }[/math]
- [math]\displaystyle{ b(d)={{e}^{\left[ \tfrac{({{S}_{2}}(d)-{{S}_{1}}(d))(c(d)-1)}{{{\left[ 1-{{[c(d)]}^{3}} \right]}^{2}}} \right]}}\,\! }[/math]
and:
- [math]\displaystyle{ {{R}_{0}}=d+a(d)\cdot b(d)\,\! }[/math]
for [math]\displaystyle{ {{R}_{0}}=31%\,\! }[/math], the equation above may be rewritten as:
- [math]\displaystyle{ d-31+a(d)\cdot b(d)=0\,\! }[/math]
The equations for parameters [math]\displaystyle{ c\,\! }[/math], [math]\displaystyle{ a\,\! }[/math] and [math]\displaystyle{ b\,\! }[/math] can now be solved simultaneously. One method for solving these equations numerically is to substitute different values of [math]\displaystyle{ d\,\! }[/math], which must be less than [math]\displaystyle{ {{R}_{0}}\,\! }[/math], into the last equation shown above, and plot the results along the y-axis with the value of [math]\displaystyle{ d\,\! }[/math] along the x-axis. The value of [math]\displaystyle{ d\,\! }[/math] can then be read from the x-intercept. This can be repeated for greater accuracy using smaller and smaller increments of [math]\displaystyle{ d\,\! }[/math]. Once the desired accuracy on [math]\displaystyle{ d\,\! }[/math] has been achieved, the value of [math]\displaystyle{ d\,\! }[/math] can then be used to solve for [math]\displaystyle{ a\,\! }[/math], [math]\displaystyle{ b\,\! }[/math] and [math]\displaystyle{ c\,\! }[/math]. For this case, the initial estimates of the parameters are:
- [math]\displaystyle{ \begin{align} \widehat{a}= & 69.324 \\ \widehat{b}= & 0.002524 \\ \widehat{c}= & 0.46012 \\ \widehat{d}= & 30.825 \end{align}\,\! }[/math]
Now, since the initial values have been determined, the Gauss-Newton method can be used. Therefore, substituting [math]\displaystyle{ {{Y}_{i}}={{R}_{i}},\,\! }[/math] [math]\displaystyle{ g_{1}^{(0)}=69.324,\,\! }[/math] [math]\displaystyle{ g_{2}^{(0)}=0.002524,\,\! }[/math] [math]\displaystyle{ g_{3}^{(0)}=0.46012,\,\! }[/math] and [math]\displaystyle{ g_{4}^{(0)}=30.825\,\! }[/math], [math]\displaystyle{ {{Y}^{(0)}},{{D}^{(0)}},\,\! }[/math] [math]\displaystyle{ {{\nu }^{(0)}}\,\! }[/math] become:
- [math]\displaystyle{ {{Y}^{(0)}}=\left[ \begin{matrix} 0.000026 \\ 0.253873 \\ -1.062940 \\ 0.565690 \\ -0.845260 \\ 0.096737 \\ 0.076450 \\ 0.238155 \\ -0.320890 \\ \end{matrix} \right]\,\! }[/math]
- [math]\displaystyle{ {{D}^{(0)}}=\left[ \begin{matrix} 0.002524 & 69.3240 & 0.0000 & 1 \\ 0.063775 & 805.962 & -26.4468 & 1 \\ 0.281835 & 1638.82 & -107.552 & 1 \\ 0.558383 & 1493.96 & -147.068 & 1 \\ 0.764818 & 941.536 & -123.582 & 1 \\ 0.883940 & 500.694 & -82.1487 & 1 \\ 0.944818 & 246.246 & -48.4818 & 1 \\ 0.974220 & 116.829 & -26.8352 & 1 \\ 0.988055 & 54.5185 & -14.3117 & 1 \\ \end{matrix} \right]\,\! }[/math]
- [math]\displaystyle{ {{\nu }^{(0)}}=\left[ \begin{matrix} g_{1}^{(0)} \\ g_{2}^{(0)} \\ g_{3}^{(0)} \\ g_{4}^{(0)} \\ \end{matrix} \right]=\left[ \begin{matrix} 69.324 \\ 0.002524 \\ 0.46012 \\ 30.825 \\ \end{matrix} \right]\,\! }[/math]
The estimate of the parameters [math]\displaystyle{ {{\nu }^{(0)}}\,\! }[/math] is given by:
- [math]\displaystyle{ \begin{align} {{\widehat{\nu }}^{(0)}}= & {{\left( {{D}^{{{(0)}^{T}}}}{{D}^{(0)}} \right)}^{-1}}{{D}^{{{(0)}^{T}}}}{{Y}^{(0)}} \\ = & \left[ \begin{matrix} -0.275569 \\ -0.000549 \\ -0.003202 \\ 0.209458 \\ \end{matrix} \right] \end{align}\,\! }[/math]
The revised estimated regression coefficients in matrix form are given by:
- [math]\displaystyle{ \begin{align} {{g}^{(1)}}= & {{g}^{(0)}}+{{\widehat{\nu }}^{(0)}}. \\ = & \left[ \begin{matrix} 69.324 \\ 0.002524 \\ 0.46012 \\ 30.825 \\ \end{matrix} \right]+\left[ \begin{matrix} -0.275569 \\ -0.000549 \\ -0.003202 \\ 0.209458 \\ \end{matrix} \right] \\ = & \left[ \begin{matrix} 69.0484 \\ 0.00198 \\ 0.45692 \\ 31.0345 \\ \end{matrix} \right] \end{align}\,\! }[/math]
With the starting coefficients [math]\displaystyle{ {{g}^{(0)}}\,\! }[/math], [math]\displaystyle{ Q\,\! }[/math] is:
- [math]\displaystyle{ \begin{align} {{Q}^{(0)}}= & \underset{i=1}{\overset{N}{\mathop \sum }}\,{{\left( {{Y}_{i}}-f({{T}_{i}},{{g}^{(0)}}) \right)}^{2}} \\ = & 2.403672 \end{align}\,\! }[/math]
With the coefficients at the end of the first iteration, [math]\displaystyle{ {{g}^{(1)}}\,\! }[/math], [math]\displaystyle{ Q\,\! }[/math] is:
- [math]\displaystyle{ \begin{align} {{Q}^{(1)}}= & \underset{i=1}{\overset{N}{\mathop \sum }}\,{{\left[ {{Y}_{i}}-f\left( {{T}_{i}},{{g}^{(1)}} \right) \right]}^{2}} \\ = & 2.073964 \end{align}\,\! }[/math]
Therefore:
- [math]\displaystyle{ \begin{align} {{Q}^{(1)}}\lt {{Q}^{(0)}} \end{align}\,\! }[/math]
Hence, the Gauss-Newton method works in the right direction. The iterations are continued until the relationship of [math]\displaystyle{ {{Q}^{(s-1)}}-{{Q}^{(s)}}\simeq 0\,\! }[/math] has been satisfied. Using RGA, the estimators of the parameters are:
- [math]\displaystyle{ \begin{align} \widehat{a}= & 0.6904 \\ \widehat{b}= & 0.0020 \\ \widehat{c}= & 0.4567 \\ \widehat{d}= & 0.3104 \end{align}\,\! }[/math]
Therefore, the modified Gompertz model is:
- [math]\displaystyle{ R=0.3104+(0.6904){{(0.0020)}^{{{0.4567}^{T}}}}\,\! }[/math]
Using this equation, the predicted reliability is plotted in the following figure along with the raw data. As you can see, the modified Gompertz curve represents the data very well.
