Template:Example: 2P Weibull Distribution RRY - Revision history

Lisa Hacker: Redirected page to The Weibull Distribution

2012-08-08T08:35:41Z

Redirected page to The Weibull Distribution

← Older revision		Revision as of 08:35, 8 August 2012
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	~~'''2-Parameter Weibull Distribution RRY Example'''~~		#REDIRECT [[The Weibull Distribution]]

	~~Consider the data in~~ [[Weibull Example 1 Data\|Example 1]], where six units were tested to failure and the following failure times were recorded: 16, 34, 53, 75, 93 and 120 hours. Estimate the parameters and the correlation coefficient using rank regression on Y, assuming that the data follow the two-parameter Weibull distribution.

	~~'''Solution'''~~

	~~Construct a table as shown below.~~

	~~{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"~~
	\|-
	~~\|colspan="8" style="text-align:center"\| Table - Least Squares Analysis~~
	\|-
	~~!<math>N</math>~~
	~~!<math>T_{i}</math>~~
	~~!<math>ln(T_{i})</math>~~
	~~!<math>F(T_i)</math>~~
	~~!<math>y_{i}</math>~~
	~~!<math>(ln{T_i})^2</math>~~
	~~!<math>{y_i}^2</math>~~
	~~!<math>(ln{T_i})y_i</math>~~
	\|-
	~~\|1 \|\|16\|\|2.7726\|\|0.1091\|\|-2.1583\|\|7.6873\|\|4.6582\|\|-5.9840~~
	\|-
	~~\|2 \|\|34\|\|3.5264\|\|0.2645\|\|-1.1802\|\|12.4352\|\|1.393\|\|-4.1620~~
	\|-
	~~\|3 \|\|53\|\|3.9703\|\|0.4214\|\|-0.6030\|\|15.7632\|\|0.3637\|\|-2.3943~~
	\|-
	~~\|4 \|\|75\|\|4.3175\|\|0.5786\|\|-0.146\|\|18.6407\|\|0.0213\|\|-0.6303~~
	\|-
	~~\|5 \|\|93\|\|4.5326\|\|0.7355\|\|0.2851\|\|20.5445\|\|0.0813\|\|1.2923~~
	\|-
	~~\|6 \|\|120\|\|4.7875\|\|0.8909\|\|0.7955\|\|22.9201\|\|0.6328\|\|3.8083~~
	\|-
	~~\|<math>\sum</math>\|\| \|\|23.9068\|\| \|\|-3.007\|\|97.9909\|\|7.1502\|\|-8.0699~~
	\|}


	~~Utilizing the values from the table, calculate <math> \hat{a} </math> and <math> \hat{b} </math> using the following equations:~~
	::<math> \hat{b} =\frac{\sum\limits_{i=1}^{6}(\ln t_{i})y_{i}-(\sum\limits_{i=1}^{6}\ln t_{i})(\sum\limits_{i=1}^{6}y_{i})/6}{ \sum\limits_{i=1}^{6}(\ln t_{i})^{2}-(\sum\limits_{i=1}^{6}\ln t_{i})^{2}/6}
	~~</math>~~

	~~::<math> \hat{b}=\frac{-8.0699-(23.9068)(-3.0070)/6}{97.9909-(23.9068)^{2}/6} </math>~~

	or

	~~::<math> \hat{b}=1.4301 </math>~~

	~~and:~~

	~~::<math> \hat{a}=\overline{y}-\hat{b}\overline{T}=\frac{\sum \limits_{i=1}^{N}y_{i}}{N}-\hat{b}\frac{\sum\limits_{i=1}^{N}\ln t_{i}}{N } </math>~~

	~~or:~~

	~~::<math> \hat{a}=\frac{(-3.0070)}{6}-(1.4301)\frac{23.9068}{6}=-6.19935 </math>~~

	~~Therefore:~~

	~~::<math> \hat{\beta }=\hat{b}=1.4301 </math>~~

	~~and:~~

	~~::<math> \hat{\eta }=e^{-\frac{\hat{a}}{\hat{b}}}=e^{-\frac{(-6.19935)}{ 1.4301}} </math>~~

	~~or:~~

	~~::<math> \hat{\eta }=76.318\text{ hr} </math>~~

	~~The correlation coefficient can be estimated as:~~

	~~::<math> \hat{\rho }=0.9956 </math>~~
	~~<br>~~
	~~This example can be repeated in the Weibull++ software, as shown next.~~ The ~~following picture shows a Weibull++ standard folio data sheet calculated with the 2P-Weibull distribution and rank regression on Y.~~

	~~[[Image:Weibull Distribution Example 3 RRY Result.png\|center\|550px\| ]]~~


	~~The following plot shows the Weibull probability plot for the data set (with 90% two-sided confidence bounds).~~

	~~[[Image:Weibull Distribution Example 3 RRY Confidence Plot.png\|center\|450px\| ]]~~

	~~If desired, the Weibull <math>pdf</math> representing the data set can be written as:~~

	~~::<math> f(t)={\frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t}{\eta }}\right) ^{\beta }} </math>~~

	~~or:~~

	~~::<math> f(t)={\frac{1.4302}{76.317}}\left( {\frac{t}{76.317}}\right) ^{0.4302}e^{-\left( {\frac{t}{76.317}}\right) ^{1.4302}} </math>~~

	~~You can also plot this result by selecting '''Pdf Plot''' on the Plot Type drop-down list on the control panel.~~

	~~[[Image:~~Weibull Distribution ~~Example 3 pdf Plot.png\|center\|450px~~]]

	~~From this point on, different results, reports and plots can be obtained.~~

Richard House at 02:10, 8 August 2012

2012-08-08T02:10:29Z

← Older revision		Revision as of 02:10, 8 August 2012
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	Consider the data in [[Weibull Example 1 Data\|Example 1]], where six units were tested to failure and the following failure times were recorded: 16, 34, 53, 75, 93 and 120 hours. Estimate the parameters and the correlation coefficient using rank regression on Y, assuming that the data follow the two-parameter Weibull distribution.		Consider the data in [[Weibull Example 1 Data\|Example 1]], where six units were tested to failure and the following failure times were recorded: 16, 34, 53, 75, 93 and 120 hours. Estimate the parameters and the correlation coefficient using rank regression on Y, assuming that the data follow the two-parameter Weibull distribution.


	'''Solution'''		'''Solution'''

Richard House at 02:10, 8 August 2012

2012-08-08T02:10:17Z

← Older revision		Revision as of 02:10, 8 August 2012
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	Construct a table as shown below.		Construct a table as shown below.

	{\|align="center" border=1 cellspacing=1		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|-		\|-
	\|colspan="8" style="text-align:center"\| Table - Least Squares Analysis		\|colspan="8" style="text-align:center"\| Table - Least Squares Analysis

Richard House at 05:37, 6 August 2012

2012-08-06T05:37:31Z

← Older revision		Revision as of 05:37, 6 August 2012
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	This example can be repeated in the Weibull++ software, as shown next. The following picture shows a Weibull++ standard folio data sheet calculated with the 2P-Weibull distribution and rank regression on Y.		This example can be repeated in the Weibull++ software, as shown next. The following picture shows a Weibull++ standard folio data sheet calculated with the 2P-Weibull distribution and rank regression on Y.

	[[Image:Weibull Distribution Example 3 RRY Result.png~~\|thumb~~\|center\|550px\| ]]		[[Image:Weibull Distribution Example 3 RRY Result.png\|center\|550px\| ]]


	The following plot shows the Weibull probability plot for the data set (with 90% two-sided confidence bounds).		The following plot shows the Weibull probability plot for the data set (with 90% two-sided confidence bounds).

	[[Image:Weibull Distribution Example 3 RRY Confidence Plot.png~~\|thumb~~\|center\|450px\| ]]		[[Image:Weibull Distribution Example 3 RRY Confidence Plot.png\|center\|450px\| ]]

	If desired, the Weibull <math>pdf</math> representing the data set can be written as:		If desired, the Weibull <math>pdf</math> representing the data set can be written as:
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	You can also plot this result by selecting '''Pdf Plot''' on the Plot Type drop-down list on the control panel.		You can also plot this result by selecting '''Pdf Plot''' on the Plot Type drop-down list on the control panel.

	[[Image:Weibull Distribution Example 3 pdf Plot.png~~\|thumb~~\|center\|450px]]		[[Image:Weibull Distribution Example 3 pdf Plot.png\|center\|450px]]

	From this point on, different results, reports and plots can be obtained.		From this point on, different results, reports and plots can be obtained.

Richard House at 05:05, 6 August 2012

2012-08-06T05:05:53Z

← Older revision		Revision as of 05:05, 6 August 2012
Line 73:		Line 73:
	This example can be repeated in the Weibull++ software, as shown next. The following picture shows a Weibull++ standard folio data sheet calculated with the 2P-Weibull distribution and rank regression on Y.		This example can be repeated in the Weibull++ software, as shown next. The following picture shows a Weibull++ standard folio data sheet calculated with the 2P-Weibull distribution and rank regression on Y.

	[[Image:Weibull Distribution Example 3 RRY Result.png\|thumb\|center\|~~250px~~\| ]]		[[Image:Weibull Distribution Example 3 RRY Result.png\|thumb\|center\|550px\| ]]


	The following plot shows the Weibull probability plot for the data set (with 90% two-sided confidence bounds).		The following plot shows the Weibull probability plot for the data set (with 90% two-sided confidence bounds).

	[[Image:Weibull Distribution Example 3 RRY Confidence Plot.png\|thumb\|center\|~~250px~~\| ]]		[[Image:Weibull Distribution Example 3 RRY Confidence Plot.png\|thumb\|center\|450px\| ]]

	If desired, the Weibull <math>pdf</math> representing the data set can be written as:		If desired, the Weibull <math>pdf</math> representing the data set can be written as:
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	You can also plot this result by selecting '''Pdf Plot''' on the Plot Type drop-down list on the control panel.		You can also plot this result by selecting '''Pdf Plot''' on the Plot Type drop-down list on the control panel.

	[[Image:Weibull Distribution Example 3 pdf Plot.png\|thumb\|center\|~~250px~~]]		[[Image:Weibull Distribution Example 3 pdf Plot.png\|thumb\|center\|450px]]

	From this point on, different results, reports and plots can be obtained.		From this point on, different results, reports and plots can be obtained.

Chris Kahn at 05:34, 23 July 2012

2012-07-23T05:34:45Z

← Older revision		Revision as of 05:34, 23 July 2012
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	'''2P-Weibull Distribution RRY Example'''		'''2-Parameter Weibull Distribution RRY Example'''

	Consider the data in [[Weibull Example 1 Data\|Example 1]], where six units were tested to failure and the following failure times were recorded: 16, 34, 53, 75, 93 and 120 hours. Estimate the parameters and the correlation coefficient using rank regression on Y, assuming that the data follow the two-parameter Weibull distribution.		Consider the data in [[Weibull Example 1 Data\|Example 1]], where six units were tested to failure and the following failure times were recorded: 16, 34, 53, 75, 93 and 120 hours. Estimate the parameters and the correlation coefficient using rank regression on Y, assuming that the data follow the two-parameter Weibull distribution.

Chris Kahn at 04:41, 23 July 2012

2012-07-23T04:41:08Z

← Older revision		Revision as of 04:41, 23 July 2012
Line 1:		Line 1:
	'''2P Weibull Distribution RRY Example'''		'''2P-Weibull Distribution RRY Example'''

	Consider the data in [[Weibull Example 1 Data\|Example 1]], where six units were tested to failure and the following failure times were recorded: 16, 34, 53, 75, 93 and 120 hours. Estimate the parameters and the correlation coefficient using rank regression on Y, assuming that the data follow the two-parameter Weibull distribution.		Consider the data in [[Weibull Example 1 Data\|Example 1]], where six units were tested to failure and the following failure times were recorded: 16, 34, 53, 75, 93 and 120 hours. Estimate the parameters and the correlation coefficient using rank regression on Y, assuming that the data follow the two-parameter Weibull distribution.

Nicolette Young at 18:13, 25 April 2012

2012-04-25T18:13:23Z

← Older revision		Revision as of 18:13, 25 April 2012
Line 73:		Line 73:
	This example can be repeated in the Weibull++ software, as shown next. The following picture shows a Weibull++ standard folio data sheet calculated with the 2P-Weibull distribution and rank regression on Y.		This example can be repeated in the Weibull++ software, as shown next. The following picture shows a Weibull++ standard folio data sheet calculated with the 2P-Weibull distribution and rank regression on Y.

	[[Image:Weibull Distribution Example 3 RRY Result.png\|thumb\|center\|~~400px~~\| ]]		[[Image:Weibull Distribution Example 3 RRY Result.png\|thumb\|center\|250px\| ]]


	The following plot shows the Weibull probability plot for the data set (with 90% two-sided confidence bounds).		The following plot shows the Weibull probability plot for the data set (with 90% two-sided confidence bounds).

	[[Image:Weibull Distribution Example 3 RRY Confidence Plot.png\|thumb\|center\|~~400px~~\| ]]		[[Image:Weibull Distribution Example 3 RRY Confidence Plot.png\|thumb\|center\|250px\| ]]

	If desired, the Weibull <math>pdf</math> representing the data set can be written as:		If desired, the Weibull <math>pdf</math> representing the data set can be written as:
Line 90:		Line 90:
	You can also plot this result by selecting '''Pdf Plot''' on the Plot Type drop-down list on the control panel.		You can also plot this result by selecting '''Pdf Plot''' on the Plot Type drop-down list on the control panel.

	[[Image:Weibull Distribution Example 3 pdf Plot.png\|thumb\|center\|~~400px~~]]		[[Image:Weibull Distribution Example 3 pdf Plot.png\|thumb\|center\|250px]]

	From this point on, different results, reports and plots can be obtained.		From this point on, different results, reports and plots can be obtained.

Kate Racaza at 23:24, 6 April 2012

2012-04-06T23:24:55Z

← Older revision		Revision as of 23:24, 6 April 2012
Line 37:		Line 37:


	Utilizing the values from ~~above~~ table, calculate <math> \hat{a} </math> and <math> \hat{b} </math> using the following equations:		Utilizing the values from the table, calculate <math> \hat{a} </math> and <math> \hat{b} </math> using the following equations:
	::<math> \hat{b} =\frac{\sum\limits_{i=1}^{6}(\ln t_{i})y_{i}-(\sum\limits_{i=1}^{6}\ln t_{i})(\sum\limits_{i=1}^{6}y_{i})/6}{ \sum\limits_{i=1}^{6}(\ln t_{i})^{2}-(\sum\limits_{i=1}^{6}\ln t_{i})^{2}/6}		::<math> \hat{b} =\frac{\sum\limits_{i=1}^{6}(\ln t_{i})y_{i}-(\sum\limits_{i=1}^{6}\ln t_{i})(\sum\limits_{i=1}^{6}y_{i})/6}{ \sum\limits_{i=1}^{6}(\ln t_{i})^{2}-(\sum\limits_{i=1}^{6}\ln t_{i})^{2}/6}
	</math>		</math>
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	::<math> \hat{\rho }=0.9956 </math>		::<math> \hat{\rho }=0.9956 </math>

	~~The above example can be repeated using Weibull++. Start Weibull++ and create a new ''Data Folio''.~~

	~~[[Image:projectwizardweibull.png\|thumb\|center\|400px\|]]~~

	~~Select the '' Times-to-failure data'' option.~~

	~~[[Image:times--to-failure.png\|thumb\|center\|400px\| The effect of the Weibull shape parameter on the <math>pdf</math>. ]]~~

	~~Enter the times-to-failure in the datasheet (ignore the Subset ID column), as shown next. The times-to-failure need not be sorted, Weibull++ will automatically sort the data.~~
	<br>		<br>
	~~[[Image:~~Weibull ~~Distribution Example 3 RRY Data~~.~~png\|thumb\|center\|400px\| ]]~~		This example can be repeated in the Weibull++ software, as shown next. The following picture shows a Weibull++ standard folio data sheet calculated with the 2P-Weibull distribution and rank regression on Y.

	~~Select the desired method of analysis. Note that we are assuming that the underlying distribution is the~~ Weibull~~, so make sure that~~ the 2P-Weibull distribution ~~is selected.~~
	~~<br>~~
	~~[[Image:parameterweibull.png\|thumb\|center\|400px\|]]~~

	~~Also, so that you get the same results as this example, switch to the ''Analysis'' page~~ and ~~make sure you are using the ''Rank Regression~~ on Y ~~(RRY)'' calculation method with this example, as shown next.~~

	~~[[Image:Weibull Distribution Example 3 Select RRY.png\|thumb\|center\|400px\| ]]~~

	Note that this can also be done from the ''Main'' page by clicking the left bottom box under the Results area. Each time you click that box you will see the method switch between MLE, RRX, and RRY. Click the ''Calculate'' icon,

	~~[[Image:calculateicon.png\|thumb\|center\|400px\|]]~~

	~~or select ''Calculate'' from the ''Data'' menu. The results will appear in the Data Folio's ''Results area''. The next figure shows the results for this example~~.

	[[Image:Weibull Distribution Example 3 RRY Result.png\|thumb\|center\|400px\| ]]		[[Image:Weibull Distribution Example 3 RRY Result.png\|thumb\|center\|400px\| ]]

	~~<br>~~
	~~You can now plot the results by clicking the ''Plot'' icon,~~

	~~[[Image:ploticon.png\|thumb\|center\|400px\| ]]~~

	~~or by selecting ''Plot Probability'' from the ''Data'' menu.~~

	~~The Weibull probability plot for these data is shown next.~~

	~~[[Image: Weibull Distribution Example 3 RRY Plot.png\|thumb\|center\|400px\|]]~~


	The confidence bounds, as determined from the Fisher matrix, can also be plotted. Select ''Confidence Bounds'' from the ''Plot'' menu, choose ''Two-Sided'' under ''Sides,'' ''Reliability (Type II)'' under ''Type'' and enter ''90'' for the ''Confidence level.

	~~[[Image:weibullconfidencebounds.png\|thumb\|center\|400px\| ]]~~		The following plot shows the Weibull probability plot for the data set (with 90% two-sided confidence bounds).

	The plot ~~will appear as follows,~~

	[[Image:Weibull Distribution Example 3 RRY Confidence Plot.png\|thumb\|center\|400px\| ]]		[[Image:Weibull Distribution Example 3 RRY Confidence Plot.png\|thumb\|center\|400px\| ]]

	If desired, the Weibull <math>pdf</math> representing ~~these~~ data can be written as:		If desired, the Weibull <math>pdf</math> representing the data set can be written as:

	::<math> f(t)={\frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t}{\eta }}\right) ^{\beta }} </math>		::<math> f(t)={\frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t}{\eta }}\right) ^{\beta }} </math>
Line 127:		Line 88:
	::<math> f(t)={\frac{1.4302}{76.317}}\left( {\frac{t}{76.317}}\right) ^{0.4302}e^{-\left( {\frac{t}{76.317}}\right) ^{1.4302}} </math>		::<math> f(t)={\frac{1.4302}{76.317}}\left( {\frac{t}{76.317}}\right) ^{0.4302}e^{-\left( {\frac{t}{76.317}}\right) ^{1.4302}} </math>

	You can also plot ~~the Weibull~~ by selecting ''Pdf Plot'' ~~from~~ the ''Plot Type'' drop-down ~~menu~~ on the control panel ~~to the right of the plot area~~.		You can also plot this result by selecting '''Pdf Plot''' on the Plot Type drop-down list on the control panel.

	[[Image:Weibull Distribution Example 3 pdf Plot.png\|thumb\|center\|400px]]		[[Image:Weibull Distribution Example 3 pdf Plot.png\|thumb\|center\|400px]]

	From this point on, different results, reports and plots can be obtained.		From this point on, different results, reports and plots can be obtained.

Harry Guo at 22:44, 1 March 2012

2012-03-01T22:44:53Z

← Older revision		Revision as of 22:44, 1 March 2012
Line 81:		Line 81:
	Enter the times-to-failure in the datasheet (ignore the Subset ID column), as shown next. The times-to-failure need not be sorted, Weibull++ will automatically sort the data.		Enter the times-to-failure in the datasheet (ignore the Subset ID column), as shown next. The times-to-failure need not be sorted, Weibull++ will automatically sort the data.
	<br>		<br>
	[[Image:~~weibullfolio16~~.png\|thumb\|center\|400px\| ]]		[[Image:Weibull Distribution Example 3 RRY Data.png\|thumb\|center\|400px\| ]]

	Select the desired method of analysis. Note that we are assuming that the underlying distribution is the Weibull, so make sure that the 2P-Weibull distribution is selected.		Select the desired method of analysis. Note that we are assuming that the underlying distribution is the Weibull, so make sure that the 2P-Weibull distribution is selected.