Lloyd-Lipow - Revision history

David J. Groebel: /* Lloyd-Lipow Confidence Bounds Example */

2014-06-03T22:55:44Z

Lloyd-Lipow Confidence Bounds Example

← Older revision		Revision as of 22:55, 3 June 2014
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	[[Image:rga6.2.png\|center\|450px\|Predicted reliability with 90% confidence bounds.]]		[[Image:rga6.2.png\|center\|450px\|Predicted reliability with 90% confidence bounds.]]

	===Lloyd-Lipow Confidence Bounds ~~Example~~===		===Example - Lloyd-Lipow Confidence Bounds===
	{{:Lloyd-Lipow_Confidence_Bounds_Example}}		{{:Lloyd-Lipow_Confidence_Bounds_Example}}

David J. Groebel at 22:54, 3 June 2014

2014-06-03T22:54:23Z

← Older revision		Revision as of 22:54, 3 June 2014
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	The confidence bounds can be obtained by solving for the three equations shown above and the equation for <math>Var(\hat{R}_{k})\,\!</math>, and then substituting the values into the Fisher Matrix.		The confidence bounds can be obtained by solving for the three equations shown above and the equation for <math>Var(\hat{R}_{k})\,\!</math>, and then substituting the values into the Fisher Matrix.

	As an example, you can calculate and plot the confidence bounds for the data set given above in the [[Lloyd-Lipow#~~Least_Squares_Example~~\|Least Squares example]] as:		As an example, you can calculate and plot the confidence bounds for the data set given above in the [[Lloyd-Lipow#Example_-_Least_Squares\|Least Squares example]] as:

	:<math>\begin{align}		:<math>\begin{align}
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	The confidence bounds on reliability can now be calculated. The associated confidence bounds at the 90% confidence level are plotted in the figure below with the predicted reliability, <math>{{R}_{k}}\,\!</math>.		The confidence bounds on reliability can now be calculated. The associated confidence bounds at the 90% confidence level are plotted in the figure below with the predicted reliability, <math>{{R}_{k}}\,\!</math>.

	[[Image:rga6.2.png\|center\|~~400px~~\|Predicted reliability with 90% confidence bounds.]]		[[Image:rga6.2.png\|center\|450px\|Predicted reliability with 90% confidence bounds.]]

	===Lloyd-Lipow Confidence Bounds Example===		===Lloyd-Lipow Confidence Bounds Example===

David J. Groebel: /* Least Squares Example */

2014-06-03T22:52:25Z

Least Squares Example

← Older revision		Revision as of 22:52, 3 June 2014
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	:<math>\hat{\alpha }=\frac{\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k}\underset{k=1}{\overset{N}{\mathop{\sum }}}\,{{R}_{k}}-N\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{{{R}_{k}}}{k}}{N\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{{{k}^{2}}}-{{\left( \underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k} \right)}^{2}}}\,\!</math>		:<math>\hat{\alpha }=\frac{\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k}\underset{k=1}{\overset{N}{\mathop{\sum }}}\,{{R}_{k}}-N\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{{{R}_{k}}}{k}}{N\underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{{{k}^{2}}}-{{\left( \underset{k=1}{\overset{N}{\mathop{\sum }}}\,\tfrac{1}{k} \right)}^{2}}}\,\!</math>

	====Least Squares ~~Example~~====<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->		====Example - Least Squares====<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
	{{:Lloyd-Lipow_Least_Squares_Example}}		{{:Lloyd-Lipow_Least_Squares_Example}}

David J. Groebel at 19:40, 29 May 2014

2014-05-29T19:40:29Z

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

2014-01-21T19:35:26Z

Sequential Data with Failure Modes Example

← Older revision		Revision as of 19:35, 21 January 2014
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	===Sequential Data with Failure Modes Example===		===Sequential Data with Failure Modes Example===
	Use least squares to find the Lloyd-Lipow model that represents the data in the following table. This data set includes information about the failure mode that was responsible for each failure, so that the probability of each failure mode recurring is taken into account in the analysis.		{{:Sequential_Data_with_Modes_-_Lloyd-Lipow_Model}}

	~~<center>'''Sequential with Mode Data'''</center>~~
	{~~\|border="1" align="center" style="border-collapse~~: ~~collapse;" cellpadding="5" cellspacing="5"~~
	~~!Run Number~~
	~~!Result~~
	~~!Mode~~
	\|-
	~~\|1\|\| S\|\|~~
	\|-
	~~\|2\|\| F\|\| 1~~
	\|-
	~~\|3\|\| F\|\| 2~~
	\|-
	~~\|4\|\| F\|\| 3~~
	\|-
	~~\|5\|\| S\|\|~~
	\|-
	~~\|6\|\| S\|\|~~
	\|-
	~~\|7\|\| S\|\|~~
	\|-
	~~\|8\|\| F\|\| 3~~
	\|-
	~~\|9\|\| F\|\| 2~~
	\|-
	~~\|10\|\| S\|\|~~
	\|-
	~~\|11\|\| F\|\| 2~~
	\|-
	~~\|12\|\| S\|\|~~
	\|-
	~~\|13\|\| S\|\|~~
	\|-
	~~\|14\|\| S\|\|~~
	\|-
	~~\|15\|\| S\|\|~~
	\|}

	~~'''Solution'''~~

	~~The following figure shows the analysis.~~

	~~[[Image:rga6.12.png\|thumb\|center\|400px\|Estimated Lloyd-Lipow parameters using least squares.]]~~

Kate Racaza: /* Sequential Data Example */

2014-01-21T19:33:56Z

Sequential Data Example

@@ Line 185: / Line 185: @@
 ===Sequential Data Example===
-Use MLE to find the Lloyd-Lipow model that represents the data in the following table, and plot it along with the 95% 2-sided confidence bounds. Does the model follow the data?
+{{:Sequential_Data_-_Lloyd-Lipow_Model}}
 ===Sequential Data with Failure Modes Example===

Kate Racaza: /* Reliability Data Example */

2014-01-21T19:32:33Z

Reliability Data Example

@@ Line 182: / Line 182: @@
 ===Reliability Data Example===
-Given the reliability data in the table below, do the following:
+{{:Reliability_Data_-_Lloyd-Lipow_Model}}
 ===Sequential Data Example===

Kate Racaza: /* Grouped per Configuration Example */

2014-01-21T19:00:57Z

Grouped per Configuration Example

@@ Line 179: / Line 179: @@
 ==More Examples==
 ===Grouped per Configuration Example===
-A 15-stage reliability development test program was performed. The grouped per configuration data set is shown in the following table. Do the following:
+{{:Grouped_per_Configuration_-_Lloyd-Lipow_Model}}
 ===Reliability Data Example===

Kate Racaza: /* Lloyd-Lipow Confidence Bounds Example */

2014-01-21T18:56:04Z

Lloyd-Lipow Confidence Bounds Example

@@ Line 175: / Line 175: @@
 ===Lloyd-Lipow Confidence Bounds Example===
-Consider the success/failure data given in the following table. Solve for the Lloyd-Lipow parameters using least squares analysis, and plot the Lloyd-Lipow reliability with 2-sided confidence bounds at the 90% confidence level.
+{{:Lloyd-Lipow_Confidence_Bounds_Example}}
 ==More Examples==

Kate Racaza: /* Least Squares Example */

2014-01-21T18:54:46Z

Least Squares Example

@@ Line 83: / Line 83: @@
 ====Least Squares Example====<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
-After a 20-stage reliability development test program, 20 groups of success/failure data were obtained and are given in the table below. Do the following:
+{{:Lloyd-Lipow_Least_Squares_Example}}
 ==Confidence Bounds==