Logistic

The Logistic reliability growth model has an S-shaped curve and is given by Kececioglu [3]:


 * $$R = \frac{1}{1+be^{-kt}}, b > 0, k > 0, T \simeq 0$$

where $$b\,\!$$ and $$k\,\!$$ are parameters. Similar to the analysis given for the Gompertz curve, the following may be concluded:


 * 1)	The point of inflection is given by:


 * $${{T}_{i}}=\frac{\ln (b)}{k}\,\!$$


 * 2)	When $$b>1\,\!$$, then $${{T}_{i}}>0\,\!$$ and an S-shaped curve will be generated. However, when $$0<b\le 1\,\!$$, then $${{T}_{i}}\le 0\,\!$$ and the Logistic reliability growth model will not be described by an S-shaped curve.


 * 3)	The value of $$R\,\!$$ is equal to 0.5 at the inflection point.

=Parameter Estimation= In this section, we will demonstrate the parameter estimation method for the Logistic model using three examples for different types of data.

Example: Logistic for Grouped per Configuration Data
=Confidence Bounds= Least squares is used to estimate the parameters of the following Logistic model.


 * $$\ln (\frac{1}-1)=\ln (b)-k{{T}_{i}}\,\!$$

Thus, the confidence bounds on the parameter $$b\,\!$$ are given by:


 * $$b=\hat{b}{{e}^{{{t}_{n-2,\alpha /2}}SE(\ln \hat{b})}}\,\!$$

where:


 * $$\begin{align}

SE(\ln \hat{b})&=\sigma \cdot \sqrt{\frac{\underset{i=1}{\overset{n}{\mathop{\sum }}}\,{{({{T}_{i}})}^{2}}}{n\cdot {{S}_{xx}}}},\ \ \ {{S}_{xx}}=\left[ \underset{i=1}{\overset{n}{\mathop \sum }}\,{{({{T}_{i}})}^{2}} \right]-\frac{1}{n}{{\left( \underset{i=1}{\overset{n}{\mathop \sum }}\,{{T}_{i}} \right)}^{2}} \\ \\ \sigma &=\sqrt{SSE/(n-2)} \end{align}\,\!$$

and the confidence bounds on the parameter $$k\,\!$$ are:


 * $$k=\hat{k}\pm {{t}_{n-2,\alpha /2}}SE(\hat{k})\,\!$$

where:


 * $$SE(\hat{k})=\frac{\sigma }{\sqrt},\ \ {{S}_{xx}}=\left[ \underset{i=1}{\overset{n}{\mathop \sum }}\,{{({{T}_{i}})}^{2}} \right]-\frac{1}{n}{{\left( \underset{i=1}{\overset{n}{\mathop \sum }}\,{{T}_{i}} \right)}^{2}}\,\!$$

Since the reliability is always between 0 and 1, the logit transformation is used to obtain the confidence bounds on reliability, which is:


 * $$CB=\frac{{{{\hat{R}}}_{i}}+(1-{{{\hat{R}}}_{i}}){{e}^{\pm {{z}_{\alpha }}{{{\hat{\sigma }}}_{R}}/\left[ {{{\hat{R}}}_{i}}(1-{{{\hat{R}}}_{i}}) \right]}}}\,\!$$

Example: Logistic Confidence Bounds
=More Examples=