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==Modified Gompertz Model==
#REDIRECT [[Gompertz_Models#Modified_Gompertz_Model]]
Sometimes reliability growth data with an S-shaped trend cannot be described accurately by the Gompertz or Logistic (Chapter 8) curves. Since these two models have fixed values of reliability at the inflection points, only a few reliability growth data sets following an S-shaped reliability growth curve can be fitted to them. A modification of the Gompertz curve, which overcomes this shortcoming, is given next [5].
<br>
If we apply a shift in the vertical coordinate, then the Gompertz model is defined by:
 
<br>
::<math>R=d+a{{b}^{{{c}^{T}}}}</math>
 
:where:
<br>
<br>
:::::<math>0<a+d\le 1</math>
::::<math>0<b<1,0<c<1,\text{and}T\ge 0</math>
<br>
<br>
::<math>R</math> = system's reliability at development time <math>T</math> or at launch number <math>T</math>, or stage number <math>T</math>.
::<math>d</math> = shift parameter.
:<math>d+a</math> = upper limit that the reliability approaches asymptotically as <math>T\to\infty</math>
:<math>d+ab</math> = initial reliability at <math>T=0</math>
::<math>c</math> = growth pattern indicator(small values of <math>c</math> indicate rapid early reliability growth and large values of <math>c</math> indicate slow reliability growth).
<br>
The Modified Gompertz model is more flexible than the original, especially when fitting growth data with S-shaped trends.
<br>
{{parameter estimation gumpz}}
 
===Confidence Bounds===
<br>
The approximate reliability confidence bounds under the Modified Gompertz model can be obtained using nonlinear regression. Additionally, the reliability is always between  <math>0</math>  and  <math>1</math> . In order to keep the endpoints of the confidence interval, the logit transformation can be used to obtain the confidence bounds on reliability.
 
::<math>CB=\frac{{{{\hat{R}}}_{i}}}{{{{\hat{R}}}_{i}}+(1-{{{\hat{R}}}_{i}}){{e}^{\pm {{z}_{\alpha }}{{{\hat{\sigma }}}_{R}}/\left[ {{{\hat{R}}}_{i}}(1-{{{\hat{R}}}_{i}}) \right]}}}</math>
<br>
::<math>{{\hat{\sigma }}^{2}}=\frac{SSE}{n-p}</math>
 
where  <math>p</math>  is the total number of groups (in this case 4) and  <math>n</math>  is the total number of items in each group.
 
'''Example 3'''
 
A reliability growth data set is given in Table 7.4, columns 1 and 2. Find the Modified Gompertz curve that represents the data and plot it comparatively with the raw data.
<br>
{|style= align="center" border="1"
|+'''Table 7.4 - The development time versus observed reliability data and predicted reliabilities'''
!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
|}
 
<br>
'''Solution'''
<br>
To determine the parameters of the Modified Gompertz curve, use:
 
::<math>\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>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:
<br>
::<math>{{R}_{0}}=d+a(d)\cdot b(d)</math>
<br>
where  <math>{{R}_{0}}=31%</math> . Then, Eqn. (eq27) may be rewritten as:
<br>
::<math>d-31+a(d)\cdot b(d)=0</math>
 
Eqns. (eq24), (eq25), (eq26) and (eq28) 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 Eqn. (eq28) and plot 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 substituted into Eqns. (eq24), (eq25) and (eq26). Now  <math>a</math> ,  <math>b</math>  and  <math>c</math>  can be calculated. For this case, the initial estimates of the parameters are:
 
::<math>\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>{{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  \\
  0.253873  \\
  -1.062940  \\
  0.565690  \\
  -0.845260  \\
  0.096737  \\
  0.076450  \\
  0.238155  \\
  -0.320890  \\
\end{matrix} \right]</math>
 
::<math>{{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>{{\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> 
<br>
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)}} \\
& = & \left[ \begin{matrix}
  -0.275569  \\
  -0.000549  \\
  -0.003202  \\
  0.209458  \\
\end{matrix} \right] 
\end{align}</math>
<br>
The revised estimated regression coefficients in matrix form are given by:
<br>
::<math>\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>
<br>
With the starting coefficients  <math>{{g}^{(0)}}</math> ,  <math>Q</math>  is:
<br>
::<math>\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>
<br>
With the coefficients at the end of the first iteration,  <math>{{g}^{(1)}}</math> ,  <math>Q</math>  is:
<br>
::<math>\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>
<br>
:Therefore:
<br>
::<math>{{Q}^{(1)}}<{{Q}^{(0)}}</math>
 
Hence, the Gauss-Newton method works in the right direction. The iterations are continued until the relationship of Eqn. (critir) has been satisfied. Using RGA, the estimators of the parameters are:
<br>
::<math>\begin{align}
  & \widehat{a}= & 0.6904 \\
& \widehat{b}= & 0.0020 \\
& \widehat{c}= & 0.4567 \\
& \widehat{d}= & 0.3104 
\end{align}</math>
<br>
Therefore, the Modified Gompertz model is:
<br>
::<math>R=0.3104+(0.6904){{(0.0020)}^{{{0.4567}^{T}}}}</math>
 
Using Eqn. (eq29), the predicted reliability is plotted in the following figure along with the raw data. It can be seen from the plot in Figure MGomp1 that the Modified Gompertz curve represents the data very well.
 
[[Image:rga7.5.png|thumb|center|400px|Modified Gompertz reliability growth curve.]]
<br>

Latest revision as of 02:14, 27 August 2012

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