Logistic - Revision history

David J. Groebel at 18:18, 29 May 2014

2014-05-29T18:18:43Z

← Older revision		Revision as of 18:18, 29 May 2014
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	:<math>b=\hat{b}{{e}^{{{t}_{n-2,\alpha /2}}SE(\ln \hat{b})}}\,\!</math>		:<math>b=\hat{b}{{e}^{{{t}_{n-2,\alpha /2}}SE(\ln \hat{b})}}\,\!</math>


	where:		where:

	::<math>\begin{align}		:<math>\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}} \\ \\		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)}		\sigma &=\sqrt{SSE/(n-2)}
	\end{align}\,\!</math>		\end{align}\,\!</math>


	and the confidence bounds on the parameter <math>k\,\!</math> are:		and the confidence bounds on the parameter <math>k\,\!</math> are:

	::<math>k=\hat{k}\pm {{t}_{n-2,\alpha /2}}SE(\hat{k})\,\!</math>		:<math>k=\hat{k}\pm {{t}_{n-2,\alpha /2}}SE(\hat{k})\,\!</math>


	where:		where:

	::<math>SE(\hat{k})=\frac{\sigma }{\sqrt{{{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}}\,\!</math>		:<math>SE(\hat{k})=\frac{\sigma }{\sqrt{{{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}}\,\!</math>


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

	::<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>		:<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>


	===Example: Logistic Confidence Bounds===		===Example: Logistic Confidence Bounds===

David J. Groebel at 18:18, 29 May 2014

2014-05-29T18:18:11Z

David J. Groebel at 18:17, 29 May 2014

2014-05-29T18:17:01Z

← Older revision		Revision as of 18:17, 29 May 2014
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	The Logistic reliability growth model has an S-shaped curve and is given by Kececioglu [[RGA_References\|[3]]]:		The Logistic reliability growth model has an S-shaped curve and is given by Kececioglu [[RGA_References\|[3]]]:

	::<math>R = \frac{1}{1+be^{-kt}}, b > 0, k > 0, T \simeq 0</math>		:<math>R = \frac{1}{1+be^{-kt}}, b > 0, k > 0, T \simeq 0</math>


	where <math>b\,\!</math> and <math>k\,\!</math> are parameters. Similar to the analysis given for the [[Gompertz Models\|Gompertz curve]], the following may be concluded:		where <math>b\,\!</math> and <math>k\,\!</math> are parameters. Similar to the analysis given for the [[Gompertz Models\|Gompertz curve]], the following may be concluded:

	~~:1)~~ The point of inflection is given by:		<ol>
			<li>The point of inflection is given by:

	::<math>{{T}_{i}}=\frac{\ln (b)}{k}\,\!</math>		:<math>{{T}_{i}}=\frac{\ln (b)}{k}\,\!</math>
			</li>
	~~:2)~~ When <math>b>1\,\!</math>, then <math>{{T}_{i}}>0\,\!</math> and an S-shaped curve will be generated. However, when <math>0<b\le 1\,\!</math> , then <math>{{T}_{i}}\le 0\,\!</math> and the Logistic reliability growth model will not be described by an S-shaped curve.		<li>When <math>b>1\,\!</math>, then <math>{{T}_{i}}>0\,\!</math> and an S-shaped curve will be generated. However, when <math>0<b\le 1\,\!</math> , then <math>{{T}_{i}}\le 0\,\!</math> and the Logistic reliability growth model will not be described by an S-shaped curve.
			</li>
	~~:3)~~ The value of <math>R\,\!</math> is equal to 0.5 at the inflection point.		<li>The value of <math>R\,\!</math> is equal to 0.5 at the inflection point.
			</li>
			</ol>

	=Parameter Estimation=		=Parameter Estimation=
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	Least squares is used to estimate the parameters of the following Logistic model.		Least squares is used to estimate the parameters of the following Logistic model.

	::<math>\ln (\frac{1}{{{{\hat{R}}}_{i}}}-1)=\ln (b)-k{{T}_{i}}\,\!</math>		:<math>\ln (\frac{1}{{{{\hat{R}}}_{i}}}-1)=\ln (b)-k{{T}_{i}}\,\!</math>


	Thus, the confidence bounds on the parameter <math>b\,\!</math> are given by:		Thus, the confidence bounds on the parameter <math>b\,\!</math> are given by:

	::<math>b=\hat{b}{{e}^{{{t}_{n-2,\alpha /2}}SE(\ln \hat{b})}}\,\!</math>		:<math>b=\hat{b}{{e}^{{{t}_{n-2,\alpha /2}}SE(\ln \hat{b})}}\,\!</math>

Kate Racaza: /* Sequential Data with Failure Modes */

2014-01-21T21:29:41Z

Sequential Data with Failure Modes

← Older revision		Revision as of 21:29, 21 January 2014
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	==Sequential Data with Failure Modes==		==Sequential Data with Failure Modes==
	Consider the data given for the missile launch test in the [[Logistic#Sequential_Data_from_Missile_Launch_Test\|previous example]]. Now suppose that the engineers assigned failure modes to each failure and that the appropriate corrective actions were taken.		{{:Sequential_Data_with_Failure_Modes_-_Logistic_Model}}

	~~The table below presents the data.~~
	~~:1) Find the Logistic reliability growth curve that best represents the data.~~
	~~:2) Plot it comparatively with the raw data.~~
	~~:3) If design changes continue to be incorporated and the testing continues, when will the reliability goal of 99.50% be achieved?~~
	~~:4) If design changes continue to be incorporated and the testing continues, what will be the attainable reliability at the end of the 35th launch?~~
	~~<br>~~

	{~~\|border="1" align="center" style="border-collapse~~: ~~collapse;" cellpadding="5" cellspacing="5"~~
	~~\|+'''Sequential success/failure data with modes'''~~
	~~!Launch Number~~
	~~!Result~~
	~~!Mode~~
	\|-
	~~\|1\|\| F\|\| 2~~
	\|-
	~~\|2\|\| F\|\| 1~~
	\|-
	~~\|3\|\| S\|\|~~
	\|-
	~~\|4\|\| F\|\| 3~~
	\|-
	~~\|5\|\| F\|\| 3~~
	\|-
	~~\|6\|\| S\|\|~~
	\|-
	~~\|7\|\| S\|\|~~
	\|-
	~~\|8\|\| S\|\|~~
	\|-
	~~\|9\|\| F\|\| 2~~
	\|-
	~~\|10\|\| S\|\|~~
	\|-
	~~\|11\|\| F\|\| 1~~
	\|-
	~~\|12\|\| S\|\|~~
	\|-
	~~\|13\|\| S\|\|~~
	\|-
	~~\|14\|\| S\|\|~~
	\|-
	~~\|15\|\| S\|\|~~
	\|-
	~~\|16\|\| S\|\|~~
	\|-
	~~\|17\|\| S\|\|~~
	\|-
	~~\|18\|\| S\|\|~~
	\|-
	~~\|19\|\| S\|\|~~
	\|-
	~~\|20\|\| S\|\|~~
	\|}

	~~'''Solution'''~~
	~~<br>~~
	~~:1) The next figure shows the estimated parameters.~~
	~~<br>~~
	~~[[Image:rga8.15.png\|thumb\|center\|400px\|Entered data and the estimated Logistic parameters.]]~~
	~~<br>~~
	~~:2) The next figure displays the Reliability vs. Time plot.~~
	~~<br>~~
	~~[[Image:rga8.16.png\|thumb\|center\|400px\|Reliability vs. Time plot.]]~~
	~~<br>~~
	~~:3) The next figure displays when the reliability goal of 99.5% will be achieved.~~
	~~<br>~~
	~~[[Image:rga8.17.png\|thumb\|center\|400px\|Calculate when the reliability goal of 99.5% will be achieved.]]~~
	~~<br>~~
	~~:4) The last figure displays the reliability after the 35th launch.~~
	~~<br>~~
	~~[[Image:rga8.18.png\|thumb\|center\|400px\|Calculate the reliability at the end of the 35th launch.]]~~

Kate Racaza: /* Sequential Data from Missile Launch Test */

2014-01-21T21:24:04Z

Sequential Data from Missile Launch Test

@@ Line 68: / Line 68: @@
 ==Sequential Data from Missile Launch Test==
+{{:Logistic_Model_-_Missile_Launch_Test_Example}}
 ==Sequential Data with Failure Modes==

Kate Racaza: /* Auto Transmission Reliability Data */

2014-01-21T21:20:50Z

Auto Transmission Reliability Data

@@ Line 65: / Line 65: @@
 =More Examples=
 ==Auto Transmission Reliability Data==
-The following table presents the reliabilities observed monthly for an automobile transmission that was tested for one year.
+{{:Logistic_Model_-_Auto_Transmission_Example}}
 ==Sequential Data from Missile Launch Test==

Kate Racaza: /* Example: Logistic Confidence Bounds */

2014-01-21T21:18:54Z

Example: Logistic Confidence Bounds

@@ Line 61: / Line 61: @@
 ==Example: Logistic Confidence Bounds==
-For the data given above for the [[Logistic#Example:_Logistic_for_Reliability_Data|reliability data example]], calculate the 2-sided 90% confidence bounds under the Logistic model for the following:
+{{:Logistic_Confidence_Bounds_Example}}
 =More Examples=

Kate Racaza: /* Example: Logistic for Grouped per Configuration Data */

2014-01-21T20:52:52Z

Example: Logistic for Grouped per Configuration Data

@@ Line 24: / Line 24: @@
 ==Example: Logistic for Grouped per Configuration Data==
-Some equipment underwent testing in different stages. The testing may have been performed in subsequent days, weeks or months with an unequal number of units tested every day. Each group was tested and several failures occurred. The data set is given in columns 1 and 2 of the following table. Find the Logistic model that best fits the data, and plot it along with the reliability observed from the raw data.
+{{:Grouped_per_Configuration_Data_-_Logistic_Model}}
 =Confidence Bounds=

Kate Racaza: /* Example: Logistic for Sequential Success/Failure Data */

2014-01-21T20:50:50Z

Example: Logistic for Sequential Success/Failure Data

@@ Line 21: / Line 21: @@
 ==Example: Logistic for Sequential Success/Failure Data==
-A prototype was tested under a success/failure pattern. The test consisted of 15 runs. The following table presents the data from the test. Find the Logistic model that best fits the data set, and plot it along with the reliability observed from the raw data.
+{{:Sequential_Data_-_Logistic_Model}}
 ==Example: Logistic for Grouped per Configuration Data==

Kate Racaza: /* Example: Logistic for Reliability Data */

2014-01-21T20:47:55Z

Example: Logistic for Reliability Data

@@ Line 18: / Line 18: @@
 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 Reliability Data==
-Using the reliability growth data given in the table below, do the following:
+{{:Reliability_Data_-_Logistic_Model}}
 ==Example: Logistic for Sequential Success/Failure Data==

← Older revision		Revision as of 18:18, 29 May 2014
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	</ol>		</ol>

	=Parameter Estimation=		==Parameter Estimation==
	In this section, we will demonstrate the parameter estimation method for the Logistic model using three examples for different types of data.		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 Reliability Data==		==Example: Logistic for Reliability Data==
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	{{:Sequential_Data_-_Logistic_Model}}		{{:Sequential_Data_-_Logistic_Model}}

	==Example: Logistic for Grouped per Configuration Data==		===Example: Logistic for Grouped per Configuration Data===
	{{:Grouped_per_Configuration_Data_-_Logistic_Model}}		{{:Grouped_per_Configuration_Data_-_Logistic_Model}}

	=Confidence Bounds=		==Confidence Bounds==
	Least squares is used to estimate the parameters of the following Logistic model.		Least squares is used to estimate the parameters of the following Logistic model.

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	==Example: Logistic Confidence Bounds==		===Example: Logistic Confidence Bounds===
	{{:Logistic_Confidence_Bounds_Example}}		{{:Logistic_Confidence_Bounds_Example}}

	=More Examples=		==More Examples==
	==Auto Transmission Reliability Data==		===Auto Transmission Reliability Data===
	{{:Logistic_Model_-_Auto_Transmission_Example}}		{{:Logistic_Model_-_Auto_Transmission_Example}}

	==Sequential Data from Missile Launch Test==		===Sequential Data from Missile Launch Test===
	{{:Logistic_Model_-_Missile_Launch_Test_Example}}		{{:Logistic_Model_-_Missile_Launch_Test_Example}}

	==Sequential Data with Failure Modes==		===Sequential Data with Failure Modes===
	{{:Sequential_Data_with_Failure_Modes_-_Logistic_Model}}		{{:Sequential_Data_with_Failure_Modes_-_Logistic_Model}}