Template:Weibull fisher matrix confidence bounds - Revision history

Richard House: Redirected page to The Weibull Distribution#Fisher Matrix Confidence Bounds

2012-08-13T02:09:38Z

Redirected page to The Weibull Distribution#Fisher Matrix Confidence Bounds

Harry Guo: /* Bounds on Reliability */

2012-02-10T22:56:06Z

Bounds on Reliability

← Older revision		Revision as of 22:56, 10 February 2012
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	::<math> R(T)=e^{-e^{u}} </math>		::<math> R(T)=e^{-e^{u}} </math>

	The next step is to find the upper and lower bounds on <span class="texhtml">''u''.</span> Using the equations derived in [[Confidence Bounds ~~\|Chapter 6~~]], the bounds on are then estimated from [[Appendix: Weibull References\|[30]]]:		The next step is to find the upper and lower bounds on <span class="texhtml">''u''.</span> Using the equations derived in Chapter [[Confidence Bounds]], the bounds on are then estimated from [[Appendix: Weibull References\|[30]]]:

	::<math> u_{U} =\hat{u}+K_{\alpha }\sqrt{Var(\hat{u})}		::<math> u_{U} =\hat{u}+K_{\alpha }\sqrt{Var(\hat{u})}

Harry Guo at 22:53, 10 February 2012

2012-02-10T22:53:31Z

← Older revision		Revision as of 22:53, 10 February 2012
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	== Fisher Matrix Confidence Bounds ==		== Fisher Matrix Confidence Bounds ==

	One of the methods used by the application in estimating the different types of confidence bounds for Weibull data, the Fisher matrix method, is presented in this section. The complete derivations were presented in detail (for a general function) in chapter[[Confidence Bounds]].		One of the methods used by the application in estimating the different types of confidence bounds for Weibull data, the Fisher matrix method, is presented in this section. The complete derivations were presented in detail (for a general function) in chapter [[Confidence Bounds]].

	=== Bounds on the Parameters ===		=== Bounds on the Parameters ===

Harry Guo: /* Fisher Matrix Confidence Bounds */

2012-02-10T22:53:21Z

Fisher Matrix Confidence Bounds

← Older revision		Revision as of 22:53, 10 February 2012
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	== Fisher Matrix Confidence Bounds ==		== Fisher Matrix Confidence Bounds ==

	One of the methods used by the application in estimating the different types of confidence bounds for Weibull data, the Fisher matrix method, is presented in this section. The complete derivations were presented in detail (for a general function) in [[Confidence Bounds~~\|Chapter 6~~]].		One of the methods used by the application in estimating the different types of confidence bounds for Weibull data, the Fisher matrix method, is presented in this section. The complete derivations were presented in detail (for a general function) in chapter[[Confidence Bounds]].

	=== Bounds on the Parameters ===		=== Bounds on the Parameters ===

Harry Guo: /* Bounds on Time */

2012-02-09T22:59:05Z

Bounds on Time

← Older revision		Revision as of 22:59, 9 February 2012
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	::<math> u_{L} =\hat{u}-K_{\alpha }\sqrt{Var(\hat{u})} </math>		::<math> u_{L} =\hat{u}-K_{\alpha }\sqrt{Var(\hat{u})} </math>

	:where:		where:

	::<math> Var(\hat{u})=\left( \frac{\partial u}{\partial \beta }\right) ^{2}Var( \hat{\beta })+\left( \frac{\partial u}{\partial \eta }\right) ^{2}Var( \hat{\eta })+2\left( \frac{\partial u}{\partial \beta }\right) \left( \frac{\partial u}{\partial \eta }\right) Cov\left( \hat{\beta },\hat{ \eta }\right) </math>		::<math> Var(\hat{u})=\left( \frac{\partial u}{\partial \beta }\right) ^{2}Var( \hat{\beta })+\left( \frac{\partial u}{\partial \eta }\right) ^{2}Var( \hat{\eta })+2\left( \frac{\partial u}{\partial \beta }\right) \left( \frac{\partial u}{\partial \eta }\right) Cov\left( \hat{\beta },\hat{ \eta }\right) </math>

	:or:		or:
	::<math> Var(\hat{u}) =\frac{1}{\hat{\beta }^{4}}\left[ \ln (-\ln R)\right] ^{2}Var(\hat{\beta })+\frac{1}{\hat{\eta }^{2}}Var(\hat{\eta })+2\left( -\frac{1}{\hat{\beta }^{2}}\right) \left( \frac{\ln (-\ln R)}{ \hat{\eta }}\right) Cov\left( \hat{\beta },\hat{\eta }\right) </math>		::<math> Var(\hat{u}) =\frac{1}{\hat{\beta }^{4}}\left[ \ln (-\ln R)\right] ^{2}Var(\hat{\beta })+\frac{1}{\hat{\eta }^{2}}Var(\hat{\eta })+2\left( -\frac{1}{\hat{\beta }^{2}}\right) \left( \frac{\ln (-\ln R)}{ \hat{\eta }}\right) Cov\left( \hat{\beta },\hat{\eta }\right) </math>

Harry Guo: /* Bounds on Time */

2012-02-09T22:58:16Z

Bounds on Time

← Older revision		Revision as of 22:58, 9 February 2012
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	=== Bounds on Time ===		=== Bounds on Time ===

	The bounds around the time estimate or reliable life estimate, for a given Weibull percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows [24, 30]:		The bounds around the time estimate or reliable life estimate, for a given Weibull percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows [[Appendix: Weibull References \|[24, 30]]]:

	::<math> \ln R =-\left( \frac{T}{\eta }\right) ^{\beta }		::<math> \ln R =-\left( \frac{t}{\eta }\right) ^{\beta }
	</math>		</math>

	::<math> \ln (-\ln R) =\beta \ln \left( \frac{T}{\eta }\right) </math>		::<math> \ln (-\ln R) =\beta \ln \left( \frac{t}{\eta }\right) </math>

	::<math> \ln (-\ln R) =\beta (\ln T-\ln \eta )</math>		::<math> \ln (-\ln R) =\beta (\ln t-\ln \eta )</math>

	:or:		or:

	::<math> u=\frac{1}{\beta }\ln (-\ln R)+\ln \eta </math>		::<math> u=\frac{1}{\beta }\ln (-\ln R)+\ln \eta </math>

	where <span class="texhtml">''u'' = ln''T''.</span>		where <span class="texhtml">''u'' = ln''t''.</span>

	The upper and lower bounds on are estimated from:		The upper and lower bounds on are estimated from:

Harry Guo: /* Bounds on Reliability */

2012-02-09T22:55:40Z

Bounds on Reliability

← Older revision		Revision as of 22:55, 9 February 2012
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	with:		with:

	::<math> R(T)=e^{-e^{\beta \left( \ln T-\ln \eta \right) }}</math>		::<math> R(T)=e^{-e^{\beta \left( \ln t-\ln \eta \right) }}</math>

	set <math> u=\beta \left( \ln T-\ln \eta \right). </math> The reliability function now becomes:		set:

			:: <math> u=\beta \left( \ln t-\ln \eta \right). </math>

			The reliability function now becomes:

	::<math> R(T)=e^{-e^{u}} </math>		::<math> R(T)=e^{-e^{u}} </math>

Harry Guo: /* Bounds on Reliability */

2012-02-09T22:54:58Z

Bounds on Reliability

← Older revision		Revision as of 22:54, 9 February 2012
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	::<math> R(t)=e^{-e^{\left( \frac{t-p_{1}}{p_{2}}\right) }} </math>		::<math> R(t)=e^{-e^{\left( \frac{t-p_{1}}{p_{2}}\right) }} </math>

	By transforming <span class="texhtml">''t'' = ln''T''</span> and converting <math> p=\ln{\eta}</math>, <math> p_{2}=\frac{1}{ \beta } </math>, the above equation becomes the Weibull reliability function:		By transforming <span class="texhtml">''t'' = ln''t''</span> and converting <math> p=\ln({\eta})</math>, <math> p_{2}=\frac{1}{ \beta } </math>, the above equation becomes the Weibull reliability function:

	::<math> R(T)=e^{-e^{\beta \left( \ln T-\ln \eta \right) }}=e^{-e^{\ln \left( \frac{T }{\eta }\right) ^{\beta }}}=e^{-\left( \frac{T}{\eta }\right) ^{\beta }} </math> ~~With:~~		::<math> R(t)=e^{-e^{\beta \left( \ln t-\ln \eta \right) }}=e^{-e^{\ln \left( \frac{t }{\eta }\right) ^{\beta }}}=e^{-\left( \frac{t}{\eta }\right) ^{\beta }} </math>

	::<math> R(T)=e^{-e^{\beta \left( \ln T-\ln \eta \right) }} ~~EQNREF eq27~~ </math>		with:

			::<math> R(T)=e^{-e^{\beta \left( \ln T-\ln \eta \right) }}</math>

	set <math> u=\beta \left( \ln T-\ln \eta \right). </math> The reliability function now becomes:		set <math> u=\beta \left( \ln T-\ln \eta \right). </math> The reliability function now becomes:
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	::<math> R(T)=e^{-e^{u}} </math>		::<math> R(T)=e^{-e^{u}} </math>

	The next step is to find the upper and lower bounds on <span class="texhtml">''u''.</span> Using the equations derived in Chapter 5, the bounds on are then estimated from [30]:		The next step is to find the upper and lower bounds on <span class="texhtml">''u''.</span> Using the equations derived in [[Confidence Bounds \|Chapter 6]], the bounds on are then estimated from [[Appendix: Weibull References\|[30]]]:

	::<math> u_{U} =\hat{u}+K_{\alpha }\sqrt{Var(\hat{u})}		::<math> u_{U} =\hat{u}+K_{\alpha }\sqrt{Var(\hat{u})}
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	</math>		</math>

	:where:		where:

	::<math> Var(\hat{u}) =\left( \frac{\partial u}{\partial \beta }\right) ^{2}Var( \hat{\beta })+\left( \frac{\partial u}{\partial \eta }\right) ^{2}Var( \hat{\eta }) +2\left( \frac{\partial u}{\partial \beta }\right) \left( \frac{\partial u }{\partial \eta }\right) Cov\left( \hat{\beta },\hat{\eta }\right) </math>		::<math> Var(\hat{u}) =\left( \frac{\partial u}{\partial \beta }\right) ^{2}Var( \hat{\beta })+\left( \frac{\partial u}{\partial \eta }\right) ^{2}Var( \hat{\eta }) +2\left( \frac{\partial u}{\partial \beta }\right) \left( \frac{\partial u }{\partial \eta }\right) Cov\left( \hat{\beta },\hat{\eta }\right) </math>

	:or:		or:

	::<math> Var(\hat{u}) =\frac{\hat{u}^{2}}{\hat{\beta }^{2}}Var(\hat{ \beta })+\frac{\hat{\beta }^{2}}{\hat{\eta }^{2}}Var(\hat{\eta }) -\left( \frac{2u}{\hat{\eta }}\right) Cov\left( \hat{\beta }, \hat{\eta }\right). ~~EQNREF eq32~~</math>		::<math> Var(\hat{u}) =\frac{\hat{u}^{2}}{\hat{\beta }^{2}}Var(\hat{ \beta })+\frac{\hat{\beta }^{2}}{\hat{\eta }^{2}}Var(\hat{\eta }) -\left( \frac{2u}{\hat{\eta }}\right) Cov\left( \hat{\beta }, \hat{\eta }\right). </math>

	The upper and lower bounds on reliability are:		The upper and lower bounds on reliability are:
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	::<math> R_{L} =e^{-e^{u_{U}}}\text{ (lower bound)}</math>		::<math> R_{L} =e^{-e^{u_{U}}}\text{ (lower bound)}</math>


	'''Other Weibull Forms'''		'''Other Weibull Forms'''

	Weibull++ makes the following assumptions/substitutions ~~in Eqn. (EQNREF eq27 ) to Eqn. (EQNREF eq32 )~~ when using the three-parameter or one-parameter forms:		Weibull++ makes the following assumptions/substitutions when using the three-parameter or one-parameter forms:


	:*For the three-parameter case, substitute <math> t=\ln (T-\hat{\gamma }) </math> (and by definition <span class="texhtml">γ < ''T''</span>)<span class="texhtml">,</span> instead of <span class="texhtml">ln''T''.</span> (Note that this is an approximation since it eliminates the third parameter and assumes that <math> Var( \hat{\gamma })=0. </math>)		:*For the three-parameter case, substitute <math> t=\ln (t-\hat{\gamma }) </math> (and by definition <span class="texhtml">γ < ''t''</span>)<span class="texhtml">,</span> instead of <span class="texhtml">ln''t''.</span> (Note that this is an approximation since it eliminates the third parameter and assumes that <math> Var( \hat{\gamma })=0. </math>)
	:*For the one-parameter, <math> Var(\hat{\beta })=0, </math> thus:		:*For the one-parameter, <math> Var(\hat{\beta })=0, </math> thus:

	::<math> Var(\hat{u})=\left( \frac{\partial u}{\partial \eta }\right) ^{2}Var( \hat{\eta })=\left( \frac{\hat{\beta }}{\hat{\eta }}\right) ^{2}Var(\hat{\eta }) </math>		::<math> Var(\hat{u})=\left( \frac{\partial u}{\partial \eta }\right) ^{2}Var( \hat{\eta })=\left( \frac{\hat{\beta }}{\hat{\eta }}\right) ^{2}Var(\hat{\eta }) </math>

	Also note that the time axis (x-axis) in the three-parameter Weibull plot in Weibull++ 7 is not but <span class="texhtml">''T'' − γ.</span> This means that one must be cautious when obtaining confidence bounds from the plot. If one desires to estimate the confidence bounds on reliability for a given time <span class="texhtml">''T''<sub>0</sub></span> from the adjusted plotted line, then these bounds should be obtained for a <span class="texhtml">''T''<sub>0</sub> − γ</span> entry on the time axis.		Also note that the time axis (x-axis) in the three-parameter Weibull plot in Weibull++ is not but <span class="texhtml">''t'' − γ.</span> This means that one must be cautious when obtaining confidence bounds from the plot. If one desires to estimate the confidence bounds on reliability for a given time <span class="texhtml">''t''<sub>0</sub></span> from the adjusted plotted line, then these bounds should be obtained for a <span class="texhtml">''t''<sub>0</sub> − γ</span> entry on the time axis.

	=== Bounds on Time ===		=== Bounds on Time ===

Harry Guo: /* Bounds on the Parameters */

2012-02-09T22:49:20Z

Bounds on the Parameters

← Older revision		Revision as of 22:49, 9 February 2012
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	-\frac{\partial ^{2}\Lambda }{\partial \beta \partial \eta } & -\frac{ \partial ^{2}\Lambda }{\partial \eta ^{2}} \end{array} \right) _{\beta =\hat{\beta },\text{ }\eta =\hat{\eta }}^{-1} </math>		-\frac{\partial ^{2}\Lambda }{\partial \beta \partial \eta } & -\frac{ \partial ^{2}\Lambda }{\partial \eta ^{2}} \end{array} \right) _{\beta =\hat{\beta },\text{ }\eta =\hat{\eta }}^{-1} </math>


	'''Fisher Matrix Confidence Bounds and Regression Analysis'''		'''Fisher Matrix Confidence Bounds and Regression Analysis'''

Harry Guo: /* Bounds on the Parameters */

2012-02-09T22:48:26Z

Bounds on the Parameters

← Older revision		Revision as of 22:48, 9 February 2012
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	</math>		</math>

	:and:		and:

	::<math> \eta _{U} =\hat{\eta }\cdot e^{\frac{K_{\alpha }\sqrt{Var(\hat{ \eta })}}{\hat{\eta }}}\text{ (upper bound)}		::<math> \eta _{U} =\hat{\eta }\cdot e^{\frac{K_{\alpha }\sqrt{Var(\hat{ \eta })}}{\hat{\eta }}}\text{ (upper bound)}
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	::<math> \eta _{L} =\frac{\hat{\eta }}{e^{\frac{K_{\alpha }\sqrt{Var(\hat{ \eta })}}{\hat{\eta }}}}\text{ (lower bound)} </math>		::<math> \eta _{L} =\frac{\hat{\eta }}{e^{\frac{K_{\alpha }\sqrt{Var(\hat{ \eta })}}{\hat{\eta }}}}\text{ (lower bound)} </math>

	:where <math> K_{\alpha}</math> is defined by:		where <math> K_{\alpha}</math> is defined by:

	::<math> \alpha =\frac{1}{\sqrt{2\pi }}\int_{K_{\alpha }}^{\infty }e^{-\frac{t^{2}}{2} }dt=1-\Phi (K_{\alpha }) </math>		::<math> \alpha =\frac{1}{\sqrt{2\pi }}\int_{K_{\alpha }}^{\infty }e^{-\frac{t^{2}}{2} }dt=1-\Phi (K_{\alpha }) </math>