Template:Weibull parameters rank regression on x - Revision history

Lisa Hacker: Redirected page to The Weibull Distribution

2012-08-08T08:41:57Z

Redirected page to The Weibull Distribution

← Older revision		Revision as of 08:41, 8 August 2012
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	~~=== Rank Regression on X ===~~		#REDIRECT [[The Weibull Distribution]]

	Performing a rank regression on X is similar to the process for rank regression on Y, with the difference being that the ''horizontal'' deviations from the points to the line are minimized rather than the vertical. Again, the first task is to bring the reliability function into a linear form. This step is exactly the same as in the regression on Y analysis and all the equations apply in this case too. The derivation from the previous analysis begins on the least squares fit part, where in this case we treat as the dependent variable and as the independent variable. The best-fitting straight line to the data, for regression on X (see [[~~Parameter Estimation\|Parameter Estimation]]), is the straight line:~~

	~~::<math> x= \hat{a}+\hat{b}y </math>~~

	The ~~corresponding equations for <math> \hat{a} </math> and <math> \hat{b} </math> are:~~

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

	~~and~~

	::<math> \hat{b}={\frac{\sum\limits_{i=1}^{N}x_{i}y_{i}-\frac{\sum \limits_{i=1}^{N}x_{i}\sum\limits_{i=1}^{N}y_{i}}{N}}{\sum \limits_{i=1}^{N}y_{i}^{2}-\frac{\left( \sum\limits_{i=1}^{N}y_{i}\right) ^{2}}{N}}} </math>

	~~where:~~

	~~::<math> y_{i}=\ln \left\{ -\ln [1-F(t_{i})]\right\} </math> and:~~

	~~<span class="texhtml">''x''<sub>''i''</sub> = ln(''t''<sub>''i''</sub>)</span> and the <span class="texhtml">''F''(''t''<sub>''i''</sub>)</span> values are again obtained from the median ranks.~~

	~~Once <math> \hat{a} </math> and <math> \hat{b} </math> are obtained, solve the linear equation for <span class="texhtml">''y'',</span> which corresponds to:~~

	~~::<math> y=-\frac{\hat{a}}{\hat{b}}+\frac{1}{\hat{b}}x </math> Solving for the parameters from above equations, we get:~~

	~~::<math> a=-\frac{\hat{a}}{\hat{b}}=-\beta \ln (\eta )</math>~~

	~~and~~

	~~::<math> b=\frac{1}{\hat{b}}=\beta</math>~~

	~~The correlation coefficient is evaluated as before.~~


	~~'''Example 4: '''~~
	~~{{Example: 2P~~ Weibull Distribution ~~RRX}}~~

Harry Guo: /* Rank Regression on X */

2012-02-10T15:58:56Z

Rank Regression on X

← Older revision		Revision as of 15:58, 10 February 2012
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	=== Rank Regression on X ===		=== Rank Regression on X ===

	Performing a rank regression on X is similar to the process for rank regression on Y, with the difference being that the ''horizontal'' deviations from the points to the line are minimized rather than the vertical. Again, the first task is to bring the reliability function into a linear form. This step is exactly the same as in the regression on Y analysis and all the equations apply in this case too. The derivation from the previous analysis begins on the least squares fit part, where in this case we treat as the dependent variable and as the independent variable. The best-fitting straight line to the data, for regression on X (see [[Parameter Estimation\|~~Chapter 4~~]]), is the straight line:		Performing a rank regression on X is similar to the process for rank regression on Y, with the difference being that the ''horizontal'' deviations from the points to the line are minimized rather than the vertical. Again, the first task is to bring the reliability function into a linear form. This step is exactly the same as in the regression on Y analysis and all the equations apply in this case too. The derivation from the previous analysis begins on the least squares fit part, where in this case we treat as the dependent variable and as the independent variable. The best-fitting straight line to the data, for regression on X (see [[Parameter Estimation\|Parameter Estimation]]), is the straight line:

	::<math> x= \hat{a}+\hat{b}y </math>		::<math> x= \hat{a}+\hat{b}y </math>

Harry Guo: /* Rank Regression on X */

2012-02-10T15:58:06Z

Rank Regression on X

← Older revision		Revision as of 15:58, 10 February 2012
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	=== Rank Regression on X ===		=== Rank Regression on X ===

	Performing a rank regression on X is similar to the process for rank regression on Y, with the difference being that the ''horizontal'' deviations from the points to the line are minimized rather than the vertical. Again, the first task is to bring the reliability function into a linear form. This step is exactly the same as in the regression on Y analysis and all the equations apply in this case too. The derivation from the previous analysis begins on the least squares fit part, where in this case we treat as the dependent variable and as the independent variable. The best-fitting straight line to the data, for regression on X (see [[~~The~~ Parameter Estimation\|Chapter 4]]), is the straight line:		Performing a rank regression on X is similar to the process for rank regression on Y, with the difference being that the ''horizontal'' deviations from the points to the line are minimized rather than the vertical. Again, the first task is to bring the reliability function into a linear form. This step is exactly the same as in the regression on Y analysis and all the equations apply in this case too. The derivation from the previous analysis begins on the least squares fit part, where in this case we treat as the dependent variable and as the independent variable. The best-fitting straight line to the data, for regression on X (see [[Parameter Estimation\|Chapter 4]]), is the straight line:

	::<math> x= \hat{a}+\hat{b}y </math>		::<math> x= \hat{a}+\hat{b}y </math>

Harry Guo: /* Rank Regression on X */

2012-02-09T22:34:19Z

Rank Regression on X

← Older revision		Revision as of 22:34, 9 February 2012
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	=== Rank Regression on X ===		=== Rank Regression on X ===

	Performing a rank regression on X is similar to the process for rank regression on Y, with the difference being that the ''horizontal'' deviations from the points to the line are minimized rather than the vertical. Again, the first task is to bring ~~our~~ function~~, Eqn. (EQNREF Fw ),~~ into a linear form. This step is exactly the same as in the regression on Y analysis and ~~Eqns. (EQNREF logw ), (EQNREF yw ), (EQNREF aw ) and (EQNREF bw )~~ apply in this case too. The derivation from the previous analysis begins on the least squares fit part, where in this case we treat as the dependent variable and as the independent variable. The best-fitting straight line to the data, for regression on X (see Chapter 3), is the straight line:		Performing a rank regression on X is similar to the process for rank regression on Y, with the difference being that the ''horizontal'' deviations from the points to the line are minimized rather than the vertical. Again, the first task is to bring the reliability function into a linear form. This step is exactly the same as in the regression on Y analysis and all the equations apply in this case too. The derivation from the previous analysis begins on the least squares fit part, where in this case we treat as the dependent variable and as the independent variable. The best-fitting straight line to the data, for regression on X (see [[The Parameter Estimation\|Chapter 4]]), is the straight line:

	::<math> x= \hat{a}+\hat{b}y </math> ~~EQNREF xlinew~~		::<math> x= \hat{a}+\hat{b}y </math>

	The corresponding equations for <math> \hat{a} </math> and <math> \hat{b} </math> are:		The corresponding equations for <math> \hat{a} </math> and <math> \hat{b} </math> are:
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	::<math> \hat{a}=\overline{x}-\hat{b}\overline{y}=\frac{\sum\limits_{i=1}^{N}x_{i}}{N} -\hat{b}\frac{\sum\limits_{i=1}^{N}y_{i}}{N} </math>		::<math> \hat{a}=\overline{x}-\hat{b}\overline{y}=\frac{\sum\limits_{i=1}^{N}x_{i}}{N} -\hat{b}\frac{\sum\limits_{i=1}^{N}y_{i}}{N} </math>

	:and		and

	::<math> \hat{b}={\frac{\sum\limits_{i=1}^{N}x_{i}y_{i}-\frac{\sum \limits_{i=1}^{N}x_{i}\sum\limits_{i=1}^{N}y_{i}}{N}}{\sum \limits_{i=1}^{N}y_{i}^{2}-\frac{\left( \sum\limits_{i=1}^{N}y_{i}\right) ^{2}}{N}}} </math>		::<math> \hat{b}={\frac{\sum\limits_{i=1}^{N}x_{i}y_{i}-\frac{\sum \limits_{i=1}^{N}x_{i}\sum\limits_{i=1}^{N}y_{i}}{N}}{\sum \limits_{i=1}^{N}y_{i}^{2}-\frac{\left( \sum\limits_{i=1}^{N}y_{i}\right) ^{2}}{N}}} </math>

	:where:		where:

	::<math> y_{i}=\ln \left\{ -\ln [1-F(T_{i})]\right\} </math> and:		::<math> y_{i}=\ln \left\{ -\ln [1-F(t_{i})]\right\} </math> and:

	<span class="texhtml">''x''<sub>''i''</sub> = ln(''T''<sub>''i''</sub>)</span> and the <span class="texhtml">''F''(''T''<sub>''i''</sub>)</span> values are again obtained from the median ranks.		<span class="texhtml">''x''<sub>''i''</sub> = ln(''t''<sub>''i''</sub>)</span> and the <span class="texhtml">''F''(''t''<sub>''i''</sub>)</span> values are again obtained from the median ranks.

	Once <math> \hat{a} </math> and <math> \hat{b} </math> are obtained, solve ~~Eqn. (EQNREF xlinew )~~ for <span class="texhtml">''y'',</span> which corresponds to:		Once <math> \hat{a} </math> and <math> \hat{b} </math> are obtained, solve the linear equation for <span class="texhtml">''y'',</span> which corresponds to:

	::<math> y=-\frac{\hat{a}}{\hat{b}}+\frac{1}{\hat{b}}x </math> Solving for the parameters from ~~Eqns. (EQNREF aw ) and (EQNREF bw )~~ we get		::<math> y=-\frac{\hat{a}}{\hat{b}}+\frac{1}{\hat{b}}x </math> Solving for the parameters from above equations, we get:

	::<math> a=-\frac{\hat{a}}{\hat{b}}=-\beta \ln (\eta )</math> ~~EQNREF awx~~		::<math> a=-\frac{\hat{a}}{\hat{b}}=-\beta \ln (\eta )</math>

	:and		and

	::<math> b=\frac{1}{\hat{b}}=\beta ~~EQNREF bwx~~ </math> The correlation coefficient is evaluated as before ~~using Eqn. (EQNREF RHOw )~~.		::<math> b=\frac{1}{\hat{b}}=\beta</math>

			The correlation coefficient is evaluated as before.


	'''Example 4: '''		'''Example 4: '''
	{{Example: 2P Weibull Distribution RRX}}		{{Example: 2P Weibull Distribution RRX}}

Harry Guo at 19:11, 9 February 2012

2012-02-09T19:11:47Z

← Older revision		Revision as of 19:11, 9 February 2012
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			'''Example 4: '''
	{{Example: 2P Weibull Distribution RRX}}		{{Example: 2P Weibull Distribution RRX}}

Harry Guo: moved Template:Example: 2P Weibull RRX to Template:Weibull parameters rank regression on x over redirect

2012-02-09T17:43:39Z

moved Template:Example: 2P Weibull RRX to Template:Weibull parameters rank regression on x over redirect

← Older revision	Revision as of 17:43, 9 February 2012
(No difference)

Harry Guo at 17:42, 9 February 2012

2012-02-09T17:42:33Z

← Older revision		Revision as of 17:42, 9 February 2012
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	::<math> b=\frac{1}{\hat{b}}=\beta EQNREF bwx </math> The correlation coefficient is evaluated as before using Eqn. (EQNREF RHOw ).		::<math> b=\frac{1}{\hat{b}}=\beta EQNREF bwx </math> The correlation coefficient is evaluated as before using Eqn. (EQNREF RHOw ).

	~~==== Example 4: 2P Weibull Distribution RRX Example ====~~
	~~Repeat Example 1 using rank regression on X.~~

	~~'''Solution'''~~		{{Example: 2P Weibull Distribution RRX}}

	~~Solution to Example 4 Table 8.1, constructed in Example 3, can also be applied to this example.~~

	~~Using the values from this table we get:~~

	~~::<math> \hat~~{~~b} =~~{\frac{\sum\limits_{i=1}^{6}(\ln T_{i})y_{i}-\frac{ \sum\limits_{i=1}^{6}\ln T_{i}\sum\limits_{i=1}^{6}y_{i}}{6}}{ \sum\limits_{i=1}^{6}y_{i}^{2}-\frac{\left( \sum\limits_{i=1}^{6}y_{i}\right) ^{2}}{6}}}
	~~</math>~~

	:~~:<math>\hat{b} =\frac{-8.0699-(23.9068)(-3.0070)/6}{7.1502-(-3.0070)^{2}/6} </math>~~

	~~:or:~~

	~~::<math> \hat{b}=0.6931 </math>~~

	~~:and:~~

	~~::<math> \hat{a}=\overline{x}-\hat{b}\overline{y}=\frac{\sum\limits_{i=1}^{6}\ln T_{i} }{6}-\hat{b}\frac{\sum\limits_{i=1}^{6}y_{i}}{6} </math>~~

	~~:or:~~

	~~::<math> \hat{a}=\frac{23.9068}{6}-(0.6931)\frac{(-3.0070)}{6}=4.3318 </math>~~

	~~Therefore, from Eqn. (EQNREF bwx ):~~

	~~::<math> \hat{\beta~~ }~~=\frac{1~~}~~{\hat{b}}=\frac{1}{0.6931}=1.4428 </math>~~

	~~:and from Eqn. (EQNREF awx )~~

	~~::<math> \hat{\eta }=e^{\frac{\hat{a}}{\hat{b}}\cdot \frac{1}{\hat{ \beta }}}=e^{\frac{4.3318}{0.6931}\cdot \frac{1}{1.4428}}=76.0811\text{ hr} </math>~~

	~~The correlation coefficient is found using Eqn. (EQNREF RHOw ):~~

	~~::<math> \hat{\rho }=0.9956 </math>~~

	The results and the associated graph using Weibull++ are given next. Note that the slight variation in the results is due to the number of significant figures used in the estimation of the median ranks. Weibull++ by default uses double precision accuracy when computing the median ranks.

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

	~~<br>~~

Harry Guo: moved Template:Weibull parameters rank regression on x to Template:Example: 2P Weibull RRX

2012-02-09T17:37:20Z

moved Template:Weibull parameters rank regression on x to Template:Example: 2P Weibull RRX

← Older revision	Revision as of 17:37, 9 February 2012
(No difference)

Harry Guo: /* Weibull Rank Regression on X Example */

2012-02-09T17:36:09Z

Weibull Rank Regression on X Example

← Older revision		Revision as of 17:36, 9 February 2012
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	::<math> b=\frac{1}{\hat{b}}=\beta EQNREF bwx </math> The correlation coefficient is evaluated as before using Eqn. (EQNREF RHOw ).		::<math> b=\frac{1}{\hat{b}}=\beta EQNREF bwx </math> The correlation coefficient is evaluated as before using Eqn. (EQNREF RHOw ).

	==== Weibull ~~Rank Regression on X~~ Example ====		==== Example 4: 2P Weibull Distribution RRX Example ====
	Repeat Example 1 using rank regression on X.		Repeat Example 1 using rank regression on X.

	~~=====~~ Solution ~~=====~~		'''Solution'''
	Solution to Example 4 Table 6.1, constructed in Example 3, can also be applied to this example.
			Solution to Example 4 Table 8.1, constructed in Example 3, can also be applied to this example.

	Using the values from this table we get:		Using the values from this table we get:

Nicolette Young: /* Example 4 */

2012-02-08T19:53:06Z

Example 4

← Older revision		Revision as of 19:53, 8 February 2012
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	::<math> b=\frac{1}{\hat{b}}=\beta EQNREF bwx </math> The correlation coefficient is evaluated as before using Eqn. (EQNREF RHOw ).		::<math> b=\frac{1}{\hat{b}}=\beta EQNREF bwx </math> The correlation coefficient is evaluated as before using Eqn. (EQNREF RHOw ).

	==== Example 4 ====		==== Weibull Rank Regression on X Example ====
	Repeat Example 1 using rank regression on X.		Repeat Example 1 using rank regression on X.

	===== Solution ~~to Example 4~~ =====		===== Solution =====
	Solution to Example 4 Table 6.1, constructed in Example 3, can also be applied to this example.		Solution to Example 4 Table 6.1, constructed in Example 3, can also be applied to this example.