Parametric Recurrent Event Data Analysis - Revision history

Lisa Hacker at 21:47, 18 September 2023

2023-09-18T21:47:52Z

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	<noinclude>{{Banner Weibull Articles}}		<noinclude>{{Banner Weibull Articles}}
	''This article appears in the [~~[Recurrent_Event_Data_Analysis#Parametric_Recurrent_Event_Data_Analysis\|~~Life ~~Data Analysis Reference book]~~].''		''This article appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference].''

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Sharon Honecker: /* The GRP Model */

2016-04-19T23:47:26Z

The GRP Model

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	::<math>\begin{align}		::<math>\begin{align}
	& \ln (L)= n(\ln \lambda +\ln \beta )-\lambda \left[ {{\left( T-{{t}_{n}}+{{v}_{n}} \right)}^{\beta }}-v_{n}^{\beta } \right] \\		& \ln (L)= n(\ln \lambda +\ln \beta )-\lambda \left[ {{\left( T-{{t}_{n}}+{{v}_{n}} \right)}^{\beta }}-v_{n}^{\beta } \right] \\
	& -\lambda \underset{i=1}{\overset{n}{\mathop \sum }}\,\left[ {{\left( {{x}_{i}}+{{v}_{i-1}} \right)}^{\beta }}-v_{i}^{\beta } \right]+(\beta -1)\underset{i=1}{\overset{n}{\mathop \sum }}\,\ln ({{x}_{i}}+{{v}_{i-1}})		& -\lambda \underset{i=1}{\overset{n}{\mathop \sum }}\,\left[ {{\left( {{x}_{i}}+{{v}_{i-1}} \right)}^{\beta }}-v_{i-1}^{\beta } \right]+(\beta -1)\underset{i=1}{\overset{n}{\mathop \sum }}\,\ln ({{x}_{i}}+{{v}_{i-1}})
	\end{align}\,\!</math>		\end{align}\,\!</math>

Miklos Szidarovszky: /* The GRP Model */

2015-04-02T17:19:59Z

The GRP Model

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	::<math>{{v}_{i}}=q({{v}_{i-1}}+{{x}_{i}})={{q}^{i}}{{x}_{1}}+{{q}^{i-1}}{{x}_{2}}+\cdots +{{q}{x}_{i}}\,\!</math>		::<math>{{v}_{i}}=q({{v}_{i-1}}+{{x}_{i}})={{q}^{i}}{{x}_{1}}+{{q}^{i-1}}{{x}_{2}}+\cdots +{{q}{x}_{i}}\,\!</math>

	where <math>{{v}_{i}}\,\!</math> is the virtual age of the system right after <math>i\,\!</math>th repair. The Type I model assumes that the <math>i\,\!</math>th repair cannot remove the damage incurred before the <math>{(}{i}-{1}{)}\,\!</math>th repair. It can only reduce the additional age <math>{{x}_{i}}\,\!</math> to <math>q{{x}_{i}}\,\!</math>. The Type II model assumes that at the <math>i\,\!</math>th repair, the virtual age has been accumulated to <math>v_{i-1} + {{x}_{i}}\,\!</math>. The <math>i\,\!</math>th repair will remove the cumulative damage from both current and previous failures by reducing the virtual age to <math>q(v_{i-1} + x_{i})\,\!</math>.		where <math>{{v}_{i}}\,\!</math> is the virtual age of the system right after <math>i\,\!</math>th repair. The Type I model assumes that the <math>i\,\!</math>th repair cannot remove the damage incurred before the <math>(i-1)\,\!</math> th repair. It can only reduce the additional age <math>{{x}_{i}}\,\!</math> to <math>{{qx}_{i}}\,\!</math>. The Type II model assumes that at the <math>i\,\!</math>th repair, the virtual age has been accumulated to <math>v_{i-1} + {{x}_{i}}\,\!</math>. The <math>i\,\!</math>th repair will remove the cumulative damage from both current and previous failures by reducing the virtual age to <math>q(v_{i-1} + x_{i})\,\!</math>.

	The power law function is used to model the rate of recurrence, which is:		The power law function is used to model the rate of recurrence, which is:

Miklos Szidarovszky: /* The GRP Model */

2015-04-02T17:05:14Z

The GRP Model

← Older revision		Revision as of 17:05, 2 April 2015
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	::<math>\begin{align}		::<math>\begin{align}
	v_{i}=v_{i}-1+qx_{i}=qt_{i}		v_{i}=v_{i-1}+qx_{i}=qt_{i}
	\end{align}\,\!</math>		\end{align}\,\!</math>

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	Type II:		Type II:

	::<math>{{v}_{i}}=q({{v}_{i-1}}+{{x}_{i}})={{q}^{i}}{{x}_{1}}+{{q}^{i-1}}{{x}_{2}}+\cdots +{{x}_{i}}\,\!</math>		::<math>{{v}_{i}}=q({{v}_{i-1}}+{{x}_{i}})={{q}^{i}}{{x}_{1}}+{{q}^{i-1}}{{x}_{2}}+\cdots +{{q}{x}_{i}}\,\!</math>

	where <math>{{v}_{i}}\,\!</math> is the virtual age of the system right after <math>i\,\!</math>th repair. The Type I model assumes that the <math>i\,\!</math>th repair cannot remove the damage incurred before the <math>i\,\!</math>th ~~failure~~. It can only reduce the additional age <math>{{x}_{i}}\,\!</math> to <math>q'{{x}_{i}}\,\!</math>. The Type II model assumes that at the <math>i\,\!</math>th repair, the virtual age has been accumulated to <math>v_{i-1} + {{x}_{i}}\,\!</math>. The <math>i\,\!</math>th repair will remove the cumulative damage from both current and previous failures by reducing the virtual age to <math>q(v_{i-1} + x_{i})\,\!</math>.		where <math>{{v}_{i}}\,\!</math> is the virtual age of the system right after <math>i\,\!</math>th repair. The Type I model assumes that the <math>i\,\!</math>th repair cannot remove the damage incurred before the <math>{(}{i}-{1}{)}\,\!</math>th repair. It can only reduce the additional age <math>{{x}_{i}}\,\!</math> to <math>q{{x}_{i}}\,\!</math>. The Type II model assumes that at the <math>i\,\!</math>th repair, the virtual age has been accumulated to <math>v_{i-1} + {{x}_{i}}\,\!</math>. The <math>i\,\!</math>th repair will remove the cumulative damage from both current and previous failures by reducing the virtual age to <math>q(v_{i-1} + x_{i})\,\!</math>.

	The power law function is used to model the rate of recurrence, which is:		The power law function is used to model the rate of recurrence, which is:

Richard House: /* Bounds on Conditional Reliability */

2012-09-26T22:39:52Z

Bounds on Conditional Reliability

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	=== Bounds on Conditional Reliability ===		=== Bounds on Conditional Reliability ===
	Given mission start time <~~span class="texhtml">''t''<sub~~>0</~~sub></span~~> and mission time <~~span class="texhtml"~~>''T''</~~span~~> , 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:

	::<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>

	<~~span class="texhtml">''v''<sub~~>0</~~sub></span~~> is the virtual age corresponding to time <~~span class="texhtml">''t''<sub~~>0</~~sub></span~~> . The expected value and the variance of <~~span class="texhtml">''v''<sub~~>0</~~sub></span~~> are obtained from Monte Carlo simulation. The variance of the conditional reliability <~~span class="texhtml"~~>''R''(''T'' \| ~~''t''<sub>~~0~~</sub>~~)</~~span~~> 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:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	Because of the simulation accuracy and the convergence problem in calculation of <math>Var(\hat{\beta }),Var(\hat{\lambda })\,\!</math> and <math>Cov(\hat{\beta },\hat{\lambda }),\,\!</math> <~~span class="texhtml"~~>~~''V''''a''''r''~~(''R'')</~~span~~> can be a negative value at some time points. When this case happens, the bounds are not provided.		Because of the simulation accuracy and the convergence problem in calculation of <math>Var(\hat{\beta }),Var(\hat{\lambda })\,\!</math> and <math>Cov(\hat{\beta },\hat{\lambda }),\,\!</math> <math>Var(R)\,\!</math> can be a negative value at some time points. When this case happens, the bounds are not provided.

	The bounds are based on:		The bounds are based on:

Richard House: /* Bounds on Instantaneous Failure Intensity and MTBF */

2012-09-26T22:36:52Z

Bounds on Instantaneous Failure Intensity and MTBF

Richard House: /* Bounds on Instantaneous Failure Intensity and MTBF */

2012-09-26T22:32:06Z

Bounds on Instantaneous Failure Intensity and MTBF

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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 II.		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 II.

	The variance of instantaneous failure intensity can be calculated by:		The variance of instantaneous failure intensity can be calculated by:
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	The expected value and variance of <~~span class="texhtml"~~>~~''v''<sub>''~~t''</~~sub></span~~> are obtained from the Monte Carlo simulation with parameters <math>\hat{\lambda },\hat{\beta },\hat{q}.\,\!</math> Because of the simulation accuracy and the convergence problem in calculation of <math>Var(\hat{\beta }),Var(\hat{\lambda })\,\!</math> and <math>Cov(\hat{\beta },\hat{\lambda }),\,\!</math> <~~span class="texhtml"~~>~~''V''''a''''r''~~(~~λ<sub>''~~i~~''</sub>~~(''t''))</~~span~~> can be a negative value at some time points. When this case happens, the bounds of instantaneous failure intensity are not provided.		The expected value and variance of <math>v_{t}\,\!</math> are obtained from the Monte Carlo simulation with parameters <math>\hat{\lambda },\hat{\beta },\hat{q}.\,\!</math> Because of the simulation accuracy and the convergence problem in calculation of <math>Var(\hat{\beta }),Var(\hat{\lambda })\,\!</math> and <math>Cov(\hat{\beta },\hat{\lambda }),\,\!</math> <math>Var(\lambda_{i}(t))\,\!</math> can be a negative value at some time points. When this case happens, the bounds of instantaneous failure intensity are not provided.

	Once the variance and the expected value of <~~span class="texhtml"~~>~~λ<sub>''~~i~~''</sub>~~(''t'')</~~span~~> are obtained, the bounds can be calculated by assuming that  <~~span class="texhtml">λ<sub~~>''i~~''</sub>~~(''t'')</~~span~~> is lognormally distributed as:		Once the variance and the expected value of <math>\lambda_{i}(t)\,\!</math> are obtained, the bounds can be calculated by assuming that  <math>\lambda_{i}(t)\,\!</math> is lognormally distributed as:

	::<math>\frac{\ln {{\lambda }_{i}}(t)-\ln {{{\hat{\lambda }}}_{i}}(t)}{\sqrt{Var(\ln {{\lambda }_{i}}(t))}}\tilde{\ }N(0,1)\,\!</math>		::<math>\frac{\ln {{\lambda }_{i}}(t)-\ln {{{\hat{\lambda }}}_{i}}(t)}{\sqrt{Var(\ln {{\lambda }_{i}}(t))}}\tilde{\ }N(0,1)\,\!</math>

Richard House: /* Bounds of Cumulative Failure Intensity and MTBF */

2012-09-26T22:25:10Z

Bounds of Cumulative Failure Intensity and MTBF

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	=== Bounds of Cumulative Failure Intensity and MTBF ===		=== Bounds of Cumulative Failure Intensity and MTBF ===
	For a given time <~~span class="texhtml"~~>''t''</~~span~~> , the expected value of cumulative MTBF <~~span class="texhtml">''m''<sub~~>''c~~''</sub>~~(''t'')</~~span~~> and cumulative failure intensity <~~span class="texhtml"~~>~~λ<sub>''~~c~~''</sub>~~(''t'')</~~span~~> 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:

	::<math>{{\hat{\lambda }}_{c}}(t)=\frac{\hat{N}(t)}{t};{{\hat{m}}_{c}}(t)=\frac{t}{\hat{N}(t)}\,\!</math>		::<math>{{\hat{\lambda }}_{c}}(t)=\frac{\hat{N}(t)}{t};{{\hat{m}}_{c}}(t)=\frac{t}{\hat{N}(t)}\,\!</math>

	The bounds can be easily obtained from the corresponding bounds of <~~span class="texhtml"~~>''N''(''t'').</~~span~~>		The bounds can be easily obtained from the corresponding bounds of <math>N(t)\,\!</math>.

	::<math>\begin{align}		::<math>\begin{align}

Richard House: /* Bounds on Cumulative Failure (Event) Numbers */

2012-09-26T22:19:41Z

Bounds on Cumulative Failure (Event) Numbers

Richard House: /* Bounds on Cumulative Failure (Event) Numbers */

2012-09-26T22:14:40Z

Bounds on Cumulative Failure (Event) Numbers

← Older revision		Revision as of 22:14, 26 September 2012
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	=== Bounds on Cumulative Failure (Event) Numbers ===		=== Bounds on Cumulative Failure (Event) Numbers ===
	The variance of the cumulative failure number <~~span class="texhtml"~~>''N''(''t'')</~~span~~> is:		The variance of the cumulative failure number <math>N(t)\,\!</math> is:

	::<math>Var[N(t)]=Var\left[ E(N(t)\|\lambda ,\beta ,q) \right]+E\left[ Var(N(t)\|\lambda ,\beta ,q) \right]\,\!</math>		::<math>Var[N(t)]=Var\left[ E(N(t)\|\lambda ,\beta ,q) \right]+E\left[ Var(N(t)\|\lambda ,\beta ,q) \right]\,\!</math>

	The first term accounts for the uncertainty of the parameter estimation. The second term considers the uncertainty caused by the renewal process even when model parameters are fixed. However, unless <~~span class="texhtml"~~>''q'' = 1</~~span~~> , <math>Var\left[ E(N(t)\|\lambda ,\beta ,q) \right]\,\!</math> cannot be calculated because <~~span class="texhtml"~~>''E''(''N''(''t''))</~~span~~> cannot be expressed as a closed-form function of <~~span class="texhtml"~~>λ,β,</~~span~~> and <~~span class="texhtml"~~>''q''</~~span~~> . In order to consider the uncertainty of the parameter estimation, <math>Var\left[ E(N(t)\|\lambda ,\beta ,q) \right]\,\!</math> is approximated by:		The first term accounts for the uncertainty of the parameter estimation. The second term considers the uncertainty caused by the renewal process even when model parameters are fixed. However, unless <math>q = 1\,\!</math> , <math>Var\left[ E(N(t)\|\lambda ,\beta ,q) \right]\,\!</math> cannot be calculated because <math>E(N(t))\,\!</math> cannot be expressed as a closed-form function of <math>\lambda,\beta\,\,</math>, and <math>q\,\!</math>. In order to consider the uncertainty of the parameter estimation, <math>Var\left[ E(N(t)\|\lambda ,\beta ,q) \right]\,\!</math> is approximated by:

	::<math>Var\left[ E(N(t)\|\lambda ,\beta ,q) \right]=Var[E(N({{v}_{t}})\|\lambda ,\beta )]=Var[\lambda v_{t}^{\beta }]\,\!</math>		::<math>Var\left[ E(N(t)\|\lambda ,\beta ,q) \right]=Var[E(N({{v}_{t}})\|\lambda ,\beta )]=Var[\lambda v_{t}^{\beta }]\,\!</math>

	where <~~span class="texhtml"~~>~~''v''<sub>''~~t~~''</sub>~~</~~span~~> is the expected virtual age at time <~~span class="texhtml"~~>''t''</~~span~~> and <math>Var[\lambda v_{t}^{\beta }]\,\!</math> is:		where <math>v_{t}\,\!</math> is the expected virtual age at time <math>t\,\!</math> and <math>Var[\lambda v_{t}^{\beta }]\,\!</math> is:

	::<math>\begin{align}		::<math>\begin{align}

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	::<math>\frac{\ln {{\lambda }_{i}}(t)-\ln {{{\hat{\lambda }}}_{i}}(t)}{\sqrt{Var(\ln {{\lambda }_{i}}(t))}}\tilde{\ }N(0,1)\,\!</math>		::<math>\frac{\ln {{\lambda }_{i}}(t)-\ln {{{\hat{\lambda }}}_{i}}(t)}{\sqrt{Var(\ln {{\lambda }_{i}}(t))}}\tilde{\ }N(0,1)\,\!</math>

	The upper and lower bounds for a given confidence level <~~span class="texhtml"~~>α</~~span~~> can be calculated by:		The upper and lower bounds for a given confidence level <math>\alpha\,\!</math> can be calculated by:

	::<math>{{\lambda }_{i}}(t)={{\hat{\lambda }}_{i}}(t){{e}^{\pm {{z}_{a}}\sqrt{Var({{\lambda }_{i}}(t))}/{{{\hat{\lambda }}}_{i}}(t)}}\,\!</math>		::<math>{{\lambda }_{i}}(t)={{\hat{\lambda }}_{i}}(t){{e}^{\pm {{z}_{a}}\sqrt{Var({{\lambda }_{i}}(t))}/{{{\hat{\lambda }}}_{i}}(t)}}\,\!</math>

	where <~~span class="texhtml"~~>~~''z''<sub>''~~a~~''</sub>~~</~~span~~> is the standard normal distribution.		where <math>z_{a}\,\!</math> is the standard normal distribution.

	If <~~span class="texhtml"~~>~~λ<sub>''~~i~~''</sub>~~(''t'')</~~span~~> is assumed to be normally distributed, the bounds can be calculated by:		If <math>\lambda_{i}(t)\,\!</math> is assumed to be normally distributed, the bounds can be calculated by:

	::<math>{{\lambda }_{i}}{{(t)}_{U}}={{\hat{\lambda }}_{i}}(t)+{{z}_{a}}\sqrt{Var(N(t))}\,\!</math>		::<math>{{\lambda }_{i}}{{(t)}_{U}}={{\hat{\lambda }}_{i}}(t)+{{z}_{a}}\sqrt{Var(N(t))}\,\!</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++, <~~span class="texhtml"~~>~~λ<sub>''~~i~~''</sub>~~(''t'')~~<sub>''~~U~~''</sub>~~</~~span~~> is set to the smaller of the two upper bounds obtained from the above lognormal and normal distribution appoximation. <~~span class="texhtml">λ<sub~~>''i~~''</sub>~~(''t'')~~<sub>''~~L~~''</sub>~~</~~span~~> 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 <~~span class="texhtml"~~>''t''</~~span~~> 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 <~~span class="texhtml"~~>''t''</~~span~~> , the expected value of cumulative MTBF <~~span class="texhtml">''m''<sub~~>''i~~''</sub>~~(''t'')</~~span~~> is:		For a given time <math>t\,\!</math>, the expected value of cumulative MTBF <math>m_{i}(t)\,\!</math> is:

	::<math>{{\hat{m}}_{i}}(t)=\frac{1}{{{{\hat{\lambda }}}_{i}}(t)}\text{ }\,\!</math>		::<math>{{\hat{m}}_{i}}(t)=\frac{1}{{{{\hat{\lambda }}}_{i}}(t)}\text{ }\,\!</math>

	The upper and lower bounds can be easily obtained from the corresponding bounds of <~~span class="texhtml"~~>~~λ<sub>''~~i~~''</sub>~~(''t''):</~~span~~>		The upper and lower bounds can be easily obtained from the corresponding bounds of <math>\lambda_{i}(t)\,\!</math>:

	::<math>{{\hat{m}}_{i}}{{(t)}_{U}}=\frac{1}{{{{\hat{\lambda }}}_{i}}{{(t)}_{L}}}\,\!</math>		::<math>{{\hat{m}}_{i}}{{(t)}_{U}}=\frac{1}{{{{\hat{\lambda }}}_{i}}{{(t)}_{L}}}\,\!</math>

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	\end{align}\,\!</math>		\end{align}\,\!</math>

	By conducting this approximation, the uncertainty of <~~span class="texhtml"~~>λ</~~span~~> and <~~span class="texhtml"~~>β</~~span~~> are considered. The value of <~~span class="texhtml"~~>~~''v''<sub>''~~t~~''</sub>~~</~~span~~> 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>.		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 <~~span class="texhtml"~~>''N''(''t'')</~~span~~> have been obtained, the bounds can be calculated by assuming that <~~span class="texhtml"~~>''N''(''t'')</~~span~~> 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 that <math>N(t)\,\!</math> is lognormally distributed as:

	::<math>\frac{\ln N(t)-\ln \hat{N}(t)}{\sqrt{Var(\ln N(t))}}\tilde{\ }N(0,1)\,\!</math>		::<math>\frac{\ln N(t)-\ln \hat{N}(t)}{\sqrt{Var(\ln N(t))}}\tilde{\ }N(0,1)\,\!</math>

	The upper and lower bounds for a given confidence level <~~span class="texhtml"~~>α</~~span~~> can be calculated by:		The upper and lower bounds for a given confidence level <math>\alpha\,\!</math> can be calculated by:

	::<math>N{{(t)}_{U,L}}=\hat{N}(t){{e}^{\pm {{z}_{a}}\sqrt{Var(N(t))}/\hat{N}(t)}}\,\!</math>		::<math>N{{(t)}_{U,L}}=\hat{N}(t){{e}^{\pm {{z}_{a}}\sqrt{Var(N(t))}/\hat{N}(t)}}\,\!</math>

	where <~~span class="texhtml"~~>~~''z''<sub>''~~a~~''</sub>~~</~~span~~> is the standard normal distribution.		where <math>z_{a}\,\!</math> is the standard normal distribution.

	If <~~span class="texhtml"~~>''N''(''t'')</~~span~~> is assumed to be normally distributed, the bounds can be calculated by:		If <math>N(t)\,\!</math> is assumed to be normally distributed, the bounds can be calculated by:

	::<math>N{{(t)}_{U}}=\hat{N}(t)+{{z}_{a}}\sqrt{Var(N(t))}\,\!</math>		::<math>N{{(t)}_{U}}=\hat{N}(t)+{{z}_{a}}\sqrt{Var(N(t))}\,\!</math>
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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 <~~span class="texhtml"~~>''N''(''t'')~~<sub>''~~U~~''</sub>~~</~~span~~> is the smaller of the upper bounds obtained from lognormal and normal distribution appoximation. The <~~span class="texhtml"~~>''N''(''t'')~~<sub>''~~L~~''</sub>~~</~~span~~> 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 <~~span class="texhtml"~~>''t''</~~span~~> 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 ===