Template:TNT weibull - Revision history

Richard House: Redirected page to Temperature-NonThermal Relationship#T-NT Weibull

2012-08-15T06:15:08Z

Redirected page to Temperature-NonThermal Relationship#T-NT Weibull

Athanasios Gerokostopoulos: /* Conditional Reliability Function */

2012-02-17T01:18:32Z

Conditional Reliability Function

← Older revision		Revision as of 01:18, 17 February 2012
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	<br>		<br>
	:or:		or:
	<br>		<br>
	::<math>R(T,t,U,V)={{e}^{-\left[ {{\left( \tfrac{\left( T+t \right){{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}-{{\left( \tfrac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }} \right]}}</math>		::<math>R(T,t,U,V)={{e}^{-\left[ {{\left( \tfrac{\left( T+t \right){{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}-{{\left( \tfrac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }} \right]}}</math>

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

2012-02-17T01:17:34Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 01:17, 17 February 2012
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	• <math>FI</math> is the number of interval data groups.		• <math>FI</math> is the number of interval data groups.
	<br>		<br>
	• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}		• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals.
	~~<br>~~
	• </math> group of data intervals.
	<br>		<br>
	• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.		• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

2012-02-17T01:16:33Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 01:16, 17 February 2012
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}{{\left( \frac{U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}{{T}_{i}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}{{T}_{i}} \right)}^{\beta }}}} \right] \\		& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}{{\left( \frac{U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}{{T}_{i}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}{{T}_{i}} \right)}^{\beta }}}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}T_{i}^{\prime } \right)}^{\beta }}+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
	~~& &~~ -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}T_{i}^{\prime } \right)}^{\beta }}+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
	\end{align}</math>		\end{align}</math>

	<br>		<br>
	:where:		where:
	<br>		<br>
	<br>		<br>
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	<br>		<br>
	:and:		and:
	<br>		<br>
	• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.		• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.
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	<br>		<br>
	• <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.		• <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.
	~~<br>~~
	•
	<br>		<br>
	• <math>FI</math> is the number of interval data groups.		• <math>FI</math> is the number of interval data groups.

Athanasios Gerokostopoulos: /* Median */

2012-02-17T01:13:47Z

Median

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	<br>		<br>
	::<math>\breve{T}=\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>		::<math>\breve{T}=\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>
	~~<br>~~
	<br>		<br>

Athanasios Gerokostopoulos: /* T-NT Weibull */

2012-02-17T01:12:59Z

T-NT Weibull

← Older revision		Revision as of 01:12, 17 February 2012
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	::<math>f(t,U,V)=\frac{\beta {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{\left( \frac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}</math>		::<math>f(t,U,V)=\frac{\beta {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{\left( \frac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}</math>

			<br>
	===T-NT Weibull Statistical Properties Summary===		===T-NT Weibull Statistical Properties Summary===
	<br>		<br>

Athanasios Gerokostopoulos: /* T-NT Weibull */

2012-02-17T01:12:35Z

T-NT Weibull

← Older revision		Revision as of 01:12, 17 February 2012
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	==T-NT Weibull==		==T-NT Weibull==
	<br>		<br>
	By setting <math>\eta =L(U,V)</math> ~~from Eqn. (Temp-Volt)~~, the T-NT Weibull model is given by:		By setting <math>\eta =L(U,V)</math>, the T-NT Weibull model is given by:

	<br>		<br>

Nicolette Young: Created page with '==T-NT Weibull==
By setting $\eta =L(U,V)$ from Eqn. (Temp-Volt), the T-NT Weibull model is given by:
:: $f(t,U,V)=\frac{\beta {{U}^{n}}{{e}^{-\tfrac{…'$

2012-01-16T18:27:22Z

Created page with '==T-NT Weibull== By setting <math>\eta =L(U,V)</math> from Eqn. (Temp-Volt), the T-NT Weibull model is given by: ::<math>f(t,U,V)=\frac{\beta {{U}^{n}}{{e}^{-\tfrac{…'

New page

==T-NT Weibull==
 
By setting <math>\eta =L(U,V)</math> from Eqn. (Temp-Volt), the T-NT Weibull model is given by:

 
::<math>f(t,U,V)=\frac{\beta {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{\left( \frac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}</math>

===T-NT Weibull Statistical Properties Summary===
 
====Mean or MTTF====
 
The mean, <math>\overline{T}</math> , for the T-NT Weibull model is given by:
 
::<math>\overline{T}=\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}\cdot \Gamma \left( \frac{1}{\beta }+1 \right)</math>

 
where <math>\Gamma \left( \tfrac{1}{\beta }+1 \right)</math> is the gamma function evaluated at the value of <math>\left( \tfrac{1}{\beta }+1 \right)</math> .

====Median====
 
The median, <math>\breve{T},</math>
for the T-NT Weibull model is given by:

 
::<math>\breve{T}=\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>
 
 

====Mode====
 
The mode, <math>\tilde{T},</math> for the T-NT Weibull model is given by:

 
::<math>\tilde{T}=\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}</math>

 

====Standard Deviation====
 
The standard deviation, <math>{{\sigma }_{T}},</math> for the T-NT Weibull model is given by:

 
::<math>{{\sigma }_{T}}=\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}}}</math>
 
 
====T-NT Weibull Reliability Function====
 
The T-NT Weibull reliability function is given by:

 
::<math>R(T,U,V)={{e}^{-{{\left( \tfrac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}</math>

 
====Conditional Reliability Function====
 
The T-NT Weibull conditional reliability function at a specified stress level is given by:

 
::<math>R(T,t,U,V)=\frac{R(T+t,U,V)}{R(T,U,V)}=\frac{{{e}^{-{{\left( \tfrac{\left( T+t \right){{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}}</math>

 
:or:
 
::<math>R(T,t,U,V)={{e}^{-\left[ {{\left( \tfrac{\left( T+t \right){{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}-{{\left( \tfrac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }} \right]}}</math>

 

====Reliable Life====
 
For the T-NT Weibull model, the reliable life, <math>{{T}_{R}}</math> , of a unit for a specified reliability and starting the mission at age zero is given by:

 
::<math>{{T}_{R}}=\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{{\left\{ -\ln \left[ R\left( {{T}_{R}},U,V \right) \right] \right\}}^{\tfrac{1}{\beta }}}</math>

 
====T-NT Weibull Failure Rate Function====
 
The T-NT Weibull failure rate function, <math>\lambda (T)</math> , is given by:

 
::<math>\lambda \left( T,U,V \right)=\frac{f\left( T,U,V \right)}{R\left( T,U,V \right)}=\frac{\beta {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{\left( \frac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta -1}}</math>

===Parameter Estimation===
 
====Maximum Likelihood Estimation Method====
 
Substituting the T-NT relationship into the Weibull log-likelihood function yields:

 
::<math>\begin{align}
& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}{{\left( \frac{U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}{{T}_{i}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}{{T}_{i}} \right)}^{\beta }}}} \right] \\
& & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{U_{i}^{n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}{C}T_{i}^{\prime } \right)}^{\beta }}+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
\end{align}</math>

 
:where:
 
 
::<math>R_{Li}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{C}U_{i}^{\prime \prime n}{{e}^{-\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}}}</math>
 
 
::<math>R_{Ri}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C}U_{i}^{\prime \prime n}{{e}^{-\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}}}</math>

 
:and:
 
• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.
 
• <math>{{N}_{i}}</math> is the number of times-to-failure data points in the <math>{{i}^{th}}</math> time-to-failure data group.
 
• <math>\beta </math> is the Weibull shape parameter (unknown, the first of four parameters to be estimated).
 
• <math>B</math> is the first T-NT parameter (unknown, the second of four parameters to be estimated).
 
• <math>C</math> is the second T-NT parameter (unknown, the third of four parameters to be estimated).
 
• <math>n</math> is the third T-NT parameter (unknown, the fourth of four parameters to be estimated).
 
• <math>{{V}_{i}}</math> is the temperature level of the <math>{{i}^{th}}</math> group.
 
• <math>{{U}_{i}}</math> is the non-thermal stress level of the <math>{{i}^{th}}</math> group.
 
• <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.
 
• <math>S</math> is the number of groups of suspension data points.
 
• <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points.
 
• <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.
 
•
 
• <math>FI</math> is the number of interval data groups.
 
• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}
 
• </math> group of data intervals.
 
• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.
 
• <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval.
 
 
The solution (parameter estimates) will be found by solving for the parameters <math>B,</math> <math>C,</math> <math>n</math> and <math>\beta </math> so that <math>\tfrac{\partial \Lambda }{\partial B}=0,</math> <math>\tfrac{\partial \Lambda }{\partial C}=0,</math> <math>\tfrac{\partial \Lambda }{\partial n}=0</math> and <math>\tfrac{\partial \Lambda }{\partial \beta }=0</math> .

Template:TNT weibull - Revision history

Richard House: Redirected page to Temperature-NonThermal Relationship#T-NT Weibull

Athanasios Gerokostopoulos: /* Conditional Reliability Function */

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

Athanasios Gerokostopoulos: /* Median */

Athanasios Gerokostopoulos: /* T-NT Weibull */

Athanasios Gerokostopoulos: /* T-NT Weibull */

Nicolette Young: Created page with '==T-NT Weibull== By setting \eta =L(U,V) from Eqn. (Temp-Volt), the T-NT Weibull model is given by: ::f(t,U,V)=\frac{\beta {{U}^{n}}{{e}^{-\tfrac{…'

Nicolette Young: Created page with '==T-NT Weibull==
By setting $\eta =L(U,V)$ from Eqn. (Temp-Volt), the T-NT Weibull model is given by:
:: $f(t,U,V)=\frac{\beta {{U}^{n}}{{e}^{-\tfrac{…'$