Template:Grp confidence bounds - Revision history

Richard House: Redirected page to Recurrent Event Data Analysis#Confidence Bounds

2012-08-20T22:10:23Z

Redirected page to Recurrent Event Data Analysis#Confidence Bounds

Kate Racaza at 17:28, 8 March 2012

2012-03-08T17:28:20Z

← Older revision		Revision as of 17:28, 8 March 2012
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	::<math>{{\lambda }_{i}}(t)=\lambda \beta v_{t}^{\beta -1}</math>		::<math>{{\lambda }_{i}}(t)=\lambda \beta v_{t}^{\beta -1}</math>

	where <span class="texhtml">''v''<sub>''t''</sub></span> is the virtual age at time <span class="texhtml">''t''.</span> When <math>q\ne 1,</math> it is obtained from simulation. When <span class="texhtml">''q'' = 1</span> , <span class="texhtml">''v''<sub>''t''</sub> = ''t''</span> from model Type I and Type 2.		where <span class="texhtml">''v''<sub>''t''</sub></span> is the virtual age at time <span class="texhtml">''t''.</span> When <math>q\ne 1,</math> it is obtained from simulation. When <span class="texhtml">''q'' = 1</span> , <span class="texhtml">''v''<sub>''t''</sub> = ''t''</span> from model Type I and Type II.

	The variance of instantaneous failure intensity can be calculated by:		The variance of instantaneous failure intensity can be calculated by:

Kate Racaza at 17:27, 8 March 2012

2012-03-08T17:27:30Z

← Older revision		Revision as of 17:27, 8 March 2012
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	=== Confidence Bounds ===		=== Confidence Bounds ===

	In general, in order to obtain the virtual age, the exact occurrence time of each event (failure) should be available (see equations for Type I and Type 2 models). However, the times are unknown until the corresponding events occur. For this reason, there are no closed-form expressions for total failure number and failure intensity, which are functions of failure times and virtual age. Therefore, in Weibull++, a Monte Carlo simulation is used to predict values of virtual time, failure number, MTBF and failure rate. The approximate confidence bounds obtained from simulation are provided. The uncertainty of model parameters is also considered in the bounds.		In general, in order to obtain the virtual age, the exact occurrence time of each event (failure) should be available (see equations for Type I and Type II models). However, the times are unknown until the corresponding events occur. For this reason, there are no closed-form expressions for total failure number and failure intensity, which are functions of failure times and virtual age. Therefore, in Weibull++, a Monte Carlo simulation is used to predict values of virtual time, failure number, MTBF and failure rate. The approximate confidence bounds obtained from simulation are provided. The uncertainty of model parameters is also considered in the bounds.

	==== Bounds on Cumulative Failure (Event) Numbers ====		==== Bounds on Cumulative Failure (Event) Numbers ====

Kate Racaza at 16:15, 8 March 2012

2012-03-08T16:15:07Z

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Harry Guo: /* Bounds on Conditional Reliability */

2012-02-22T18:43:23Z

Bounds on Conditional Reliability

← Older revision		Revision as of 18:43, 22 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	& Var(R)= {{\left( \frac{\partial R}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial R}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\		& Var(R)= {{\left( \frac{\partial R}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial R}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
	& & +2\frac{\partial R}{\partial \beta }\frac{\partial R}{\partial \lambda }Cov(\hat{\beta },\hat{\lambda })+{{\left( \frac{\partial R}{\partial {{v}_{0}}} \right)}^{2}}Var({{{\hat{v}}}_{0}})		& +2\frac{\partial R}{\partial \beta }\frac{\partial R}{\partial \lambda }Cov(\hat{\beta },\hat{\lambda })+{{\left( \frac{\partial R}{\partial {{v}_{0}}} \right)}^{2}}Var({{{\hat{v}}}_{0}})
	\end{align}</math>		\end{align}</math>

Harry Guo: /* Bounds on Conditional Reliability */

2012-02-22T18:43:12Z

Bounds on Conditional Reliability

← Older revision		Revision as of 18:43, 22 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	& Var(R)= & {{\left( \frac{\partial R}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial R}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\		& Var(R)= {{\left( \frac{\partial R}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial R}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
	& & +2\frac{\partial R}{\partial \beta }\frac{\partial R}{\partial \lambda }Cov(\hat{\beta },\hat{\lambda })+{{\left( \frac{\partial R}{\partial {{v}_{0}}} \right)}^{2}}Var({{{\hat{v}}}_{0}})		& & +2\frac{\partial R}{\partial \beta }\frac{\partial R}{\partial \lambda }Cov(\hat{\beta },\hat{\lambda })+{{\left( \frac{\partial R}{\partial {{v}_{0}}} \right)}^{2}}Var({{{\hat{v}}}_{0}})
	\end{align}</math>		\end{align}</math>

Harry Guo: /* Confidence Bounds */

2012-02-22T18:42:44Z

Confidence Bounds

Harry Guo: /* Bounds of Conditional Reliability */

2012-02-22T17:50:54Z

Bounds of Conditional Reliability

← Older revision		Revision as of 17:50, 22 February 2012
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	::<math>{{\hat{m}}_{i}}{{(t)}_{L}}=\frac{1}{{{{\hat{\lambda }}}_{i}}{{(t)}_{U}}}</math>		::<math>{{\hat{m}}_{i}}{{(t)}_{L}}=\frac{1}{{{{\hat{\lambda }}}_{i}}{{(t)}_{U}}}</math>

	====Bounds of Conditional Reliability====		====Bounds on Conditional Reliability====
	Given mission start time <math>{{t}_{0}}</math> and mission time <math>T</math> , the conditional reliability can be calculated by:		Given mission start time <math>{{t}_{0}}</math> and mission time <math>T</math> , the conditional reliability can be calculated by:

Harry Guo: /* Bounds of Instantaneous Failure Intensity and MTBF */

2012-02-22T17:50:41Z

Bounds of Instantaneous Failure Intensity and MTBF

← Older revision		Revision as of 17:50, 22 February 2012
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	\end{align}</math>		\end{align}</math>

	====Bounds of Instantaneous Failure Intensity and MTBF====		====Bounds on Instantaneous Failure Intensity and MTBF====
	The instantaneous failure intensity is given by:		The instantaneous failure intensity is given by:

Harry Guo: /* Bounds of Cumulative Failure (Event) Numbers */

2012-02-22T17:50:28Z

Bounds of Cumulative Failure (Event) Numbers

← Older revision		Revision as of 17:50, 22 February 2012
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	In general, in order to obtain the virtual age, the exact occurrence time of each event (failure) should be available, see Eqns.(Type I) and (Type 2). However, the times are unknown until the corresponding events occur. For this reason, there are no closed-form expressions for total failure number and failure intensity, which are functions of failure times and virtual age. Therefore, in Weibull++, Monte Carlo simulation is used to predict values of virtual time, failure number, MTBF and failure rate. The approximate confidence bounds obtained from simulation are provided. The uncertainty of model parameters is also considered in the bounds.		In general, in order to obtain the virtual age, the exact occurrence time of each event (failure) should be available, see Eqns.(Type I) and (Type 2). However, the times are unknown until the corresponding events occur. For this reason, there are no closed-form expressions for total failure number and failure intensity, which are functions of failure times and virtual age. Therefore, in Weibull++, Monte Carlo simulation is used to predict values of virtual time, failure number, MTBF and failure rate. The approximate confidence bounds obtained from simulation are provided. The uncertainty of model parameters is also considered in the bounds.

	====Bounds of Cumulative Failure (Event) Numbers====		====Bounds on Cumulative Failure (Event) Numbers====
	The variance of the cumulative failure number <math>N(t)</math> is:		The variance of the cumulative failure number <math>N(t)</math> is:

← Older revision		Revision as of 18:42, 22 February 2012
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	===Confidence Bounds===		===Confidence Bounds===
	In general, in order to obtain the virtual age, the exact occurrence time of each event (failure) should be available, see ~~Eqns.(~~Type I) and (Type 2). However, the times are unknown until the corresponding events occur. For this reason, there are no closed-form expressions for total failure number and failure intensity, which are functions of failure times and virtual age. Therefore, in Weibull++, Monte Carlo simulation is used to predict values of virtual time, failure number, MTBF and failure rate. The approximate confidence bounds obtained from simulation are provided. The uncertainty of model parameters is also considered in the bounds.		In general, in order to obtain the virtual age, the exact occurrence time of each event (failure) should be available, see equations for Type I and Type 2 models. However, the times are unknown until the corresponding events occur. For this reason, there are no closed-form expressions for total failure number and failure intensity, which are functions of failure times and virtual age. Therefore, in Weibull++, Monte Carlo simulation is used to predict values of virtual time, failure number, MTBF and failure rate. The approximate confidence bounds obtained from simulation are provided. The uncertainty of model parameters is also considered in the bounds.

	====Bounds on Cumulative Failure (Event) Numbers====		====Bounds on Cumulative Failure (Event) Numbers====
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	::<math>\begin{align}		::<math>\begin{align}
	& Var[\lambda v_{t}^{\beta }]= & {{\left( \frac{\partial (\lambda v_{t}^{\beta })}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial (\lambda v_{t}^{\beta })}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\		& Var[\lambda v_{t}^{\beta }]= & {{\left( \frac{\partial (\lambda v_{t}^{\beta })}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial (\lambda v_{t}^{\beta })}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
	& & +2\frac{\partial (\lambda v_{t}^{\beta })}{\partial \beta }\frac{\partial (\lambda v_{t}^{\beta })}{\partial \lambda }Cov(\hat{\beta },\hat{\lambda })		& +2\frac{\partial (\lambda v_{t}^{\beta })}{\partial \beta }\frac{\partial (\lambda v_{t}^{\beta })}{\partial \lambda }Cov(\hat{\beta },\hat{\lambda })
	\end{align}</math>		\end{align}</math>

	By conducting this approximation, the uncertainty of <math>\lambda </math> and <math>\beta </math> are considered. The value of <math>{{v}_{t}}</math> and the value of the second term in ~~Eqn.(VarN)~~ are obtained through the Monte Carlo simulation using parameters <math>\hat{\lambda },\hat{\beta },\hat{q},</math> which are the ML estimators. The same simulation is used to estimate the cumulative number of failures <math>\hat{N}(t)=E(N(t)\|\hat{\lambda },\hat{\beta },\hat{q})</math> .		By conducting this approximation, the uncertainty of <math>\lambda </math> and <math>\beta </math> are considered. The value of <math>{{v}_{t}}</math> and the value of the second term in the equation for the variance of number of failures are obtained through the Monte Carlo simulation using parameters <math>\hat{\lambda },\hat{\beta },\hat{q},</math> which are the ML estimators. The same simulation is used to estimate the cumulative number of failures <math>\hat{N}(t)=E(N(t)\|\hat{\lambda },\hat{\beta },\hat{q})</math> .

	Once the variance and the expected value of <math>N(t)</math> have been obtained, the bounds can be calculated by assuming <math>N(t)</math> is lognormally distributed as:		Once the variance and the expected value of <math>N(t)</math> have been obtained, the bounds can be calculated by assuming <math>N(t)</math> is lognormally distributed as:
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	::<math>N{{(t)}_{L}}=\hat{N}(t)-{{z}_{a}}\sqrt{Var(N(t))}</math>		::<math>N{{(t)}_{L}}=\hat{N}(t)-{{z}_{a}}\sqrt{Var(N(t))}</math>

	In Weibull++, the <math>N{{(t)}_{U}}</math> is the smaller of the upper bounds obtained from ~~Eqns. (LogNorm)~~ and ~~(Norm)~~. The <math>N{{(t)}_{L}}</math> is set to the largest of the lower bounds obtained from ~~Eqns. (LogNorm)~~ and ~~(Norm)~~. This combined method can prevent the out-of-range values of bounds for some small <math>t</math> values.		In Weibull++, the <math>N{{(t)}_{U}}</math> is the smaller of the upper bounds obtained from lognormal and normal distribution appoximation. The <math>N{{(t)}_{L}}</math> is set to the largest of the lower bounds obtained from lognormal and normal distribution appoximation. This combined method can prevent the out-of-range values of bounds for some small <math>t</math> values.
	Bounds of Cumulative Failure Intensity and MTBF		Bounds of Cumulative Failure Intensity and MTBF
	For a given time <math>t</math> , the expected value of cumulative MTBF <math>{{m}_{c}}(t)</math> and cumulative failure intensity <math>{{\lambda }_{c}}(t)</math> can be calculated using the following equations:		For a given time <math>t</math> , the expected value of cumulative MTBF <math>{{m}_{c}}(t)</math> and cumulative failure intensity <math>{{\lambda }_{c}}(t)</math> can be calculated using the following equations:
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	::<math>{{\lambda }_{i}}(t)=\lambda \beta v_{t}^{\beta -1}</math>		::<math>{{\lambda }_{i}}(t)=\lambda \beta v_{t}^{\beta -1}</math>

	where <math>{{v}_{t}}</math> is the virtual age at time <math>t.</math> When <math>q\ne 1,</math> it is obtained from simulation. When <math>q=1</math> , <math>{{v}_{t}}=t</math> from ~~Eqns. #(~~Type I) and (Type 2).		where <math>{{v}_{t}}</math> is the virtual age at time <math>t.</math> When <math>q\ne 1,</math> it is obtained from simulation. When <math>q=1</math> , <math>{{v}_{t}}=t</math> from model Type I and Type 2.

	The variance of instantaneous failure intensity can be calculated by:		The variance of instantaneous failure intensity can be calculated by:

	::<math>\begin{align}		::<math>\begin{align}
	& Var({{\lambda }_{i}}(t))= & {{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\		& Var({{\lambda }_{i}}(t))= {{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
	& & +2\frac{\partial {{\lambda }_{i}}(t)}{\partial \beta }\frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda }Cov(\hat{\beta },\hat{\lambda })+{{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial v(t)} \right)}^{2}}Var({{{\hat{v}}}_{t}})		& +2\frac{\partial {{\lambda }_{i}}(t)}{\partial \beta }\frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda }Cov(\hat{\beta },\hat{\lambda })+{{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial v(t)} \right)}^{2}}Var({{{\hat{v}}}_{t}})
	\end{align}</math>		\end{align}</math>

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	::<math>{{\lambda }_{i}}{{(t)}_{L}}={{\hat{\lambda }}_{i}}(t)-{{z}_{a}}\sqrt{Var(N(t))}</math>		::<math>{{\lambda }_{i}}{{(t)}_{L}}={{\hat{\lambda }}_{i}}(t)-{{z}_{a}}\sqrt{Var(N(t))}</math>

	In Weibull++, <math>{{\lambda }_{i}}{{(t)}_{U}}</math> is set to the smaller of the two upper bounds obtained from ~~Eqns.#(LognorIF)~~and ~~(normIF)~~. <math>{{\lambda }_{i}}{{(t)}_{L}}</math> is set to the largest of the two lower bounds obtained from ~~Eqns.#(LognorIF)~~ and ~~(normIF)~~. This combination method can prevent the out of range values of bounds when <math>t</math> values are small.		In Weibull++, <math>{{\lambda }_{i}}{{(t)}_{U}}</math> is set to the smaller of the two upper bounds obtained from the above lognormal and normal distribution appoximation. <math>{{\lambda }_{i}}{{(t)}_{L}}</math> is set to the largest of the two lower bounds obtained from the above lognormal and normal distribution appoximation. This combination method can prevent the out of range values of bounds when <math>t</math> values are small.

	For a given time <math>t</math> , the expected value of cumulative MTBF <math>{{m}_{i}}(t)</math> is:		For a given time <math>t</math> , the expected value of cumulative MTBF <math>{{m}_{i}}(t)</math> is:
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	::<math>R(T\|{{t}_{0}})=\frac{R(T+{{v}_{0}})}{R({{v}_{0}})}={{e}^{-\lambda [{{({{v}_{0}}+T)}^{\beta }}-{{v}_{0}}]}}</math>		::<math>R(T\|{{t}_{0}})=\frac{R(T+{{v}_{0}})}{R({{v}_{0}})}={{e}^{-\lambda [{{({{v}_{0}}+T)}^{\beta }}-{{v}_{0}}]}}</math>


	<math>{{v}_{0}}</math> is the virtual age corresponding to time <math>{{t}_{0}}</math> . The expected value and the variance of <math>{{v}_{0}}</math> are obtained from Monte Carlo simulation. The variance of the conditional reliability <math>R(T\|{{t}_{0}})</math> is:		<math>{{v}_{0}}</math> is the virtual age corresponding to time <math>{{t}_{0}}</math> . The expected value and the variance of <math>{{v}_{0}}</math> are obtained from Monte Carlo simulation. The variance of the conditional reliability <math>R(T\|{{t}_{0}})</math> is:
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	::<math>R=\hat{R}{{e}^{\pm {{z}_{a}}\sqrt{Var(R)}/\hat{R}}}</math>		::<math>R=\hat{R}{{e}^{\pm {{z}_{a}}\sqrt{Var(R)}/\hat{R}}}</math>


	The smaller of the two upper bounds will be the final upper bound and the larger of the two lower bounds will be the final lower bound.		The smaller of the two upper bounds will be the final upper bound and the larger of the two lower bounds will be the final lower bound.