Template:TNT Exponential - Revision history

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

2012-08-15T05:47:50Z

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

Chris Kahn: /* Maximum Likelihood Estimation Method */

2012-03-09T18:40:16Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 18:40, 9 March 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 i <math>^{th}</math> group of data intervals.		• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals.
	<br>		<br>
	• <math>T_{Li}^{\prime \prime }</math> is the beginning of the i <math>^{th}</math> interval.		• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.
	<br>		<br>
	• <math>T_{Ri}^{\prime \prime }</math> is the ending of the i <math>^{th}</math> interval.		• <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval.
	<br>		<br>
	<br>		<br>

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

2012-02-17T01:11:38Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 01:11, 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{U_{i}^{n}}{C}{{e}^{-\tfrac{B}{{{V}_{i}}}}}\cdot {{e}^{-\tfrac{U_{i}^{n}}{C}\left( {{e}^{-\tfrac{B}{{{V}_{i}}}}} \right){{T}_{i}}}} \right] \\		& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{U_{i}^{n}}{C}{{e}^{-\tfrac{B}{{{V}_{i}}}}}\cdot {{e}^{-\tfrac{U_{i}^{n}}{C}\left( {{e}^{-\tfrac{B}{{{V}_{i}}}}} \right){{T}_{i}}}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{U_{i}^{n}}{C}\left( {{e}^{-\tfrac{B}{{{V}_{i}}}}} \right)T_{i}^{\prime }+\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 }\frac{U_{i}^{n}}{C}\left( {{e}^{-\tfrac{B}{{{V}_{i}}}}} \right)T_{i}^{\prime }+\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>
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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.

Athanasios Gerokostopoulos: /* T-NT Exponential Reliability Function */

2012-02-17T01:09:57Z

T-NT Exponential Reliability Function

← Older revision		Revision as of 01:09, 17 February 2012
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	<br>		<br>
	::<math>R(T,U,V)=1-\~~mathop{}_~~{0}^{T}\frac{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{e}^{-\tfrac{T\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}dT={{e}^{-\tfrac{T\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}</math>		::<math>R(T,U,V)=1-\int_{0}^{T}\frac{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{e}^{-\tfrac{T\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}dT={{e}^{-\tfrac{T\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}</math>

	<br>		<br>

Athanasios Gerokostopoulos: /* T-NT Exponential Reliability Function */

2012-02-17T00:59:08Z

T-NT Exponential Reliability Function

← Older revision		Revision as of 00:59, 17 February 2012
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	<br>		<br>
	::<math>R(T,U,V)=1-Q(T,U,V)=1-\~~mathop{}_~~{0}^{T}f(T)dT</math>		::<math>R(T,U,V)=1-Q(T,U,V)=1-\int_{0}^{T}f(T)dT</math>

	<br>		<br>
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	<br>		<br>

	====Conditional Reliability====		====Conditional Reliability====
	<br>		<br>

Athanasios Gerokostopoulos: /* Mean or MTTF */

2012-02-17T00:46:21Z

Mean or MTTF

← Older revision		Revision as of 00:46, 17 February 2012
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \overline{T}= & \int\limits_{0}^{\infty }t\cdot f(t,U,V)dt \\		& \overline{T}= & \int\limits_{0}^{\infty }t\cdot f(t,U,V)dt = & \int\limits_{0}^{\infty }t\cdot \frac{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{e}^{-\tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}dt = & \frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}
	& = & \int\limits_{0}^{\infty }t\cdot \frac{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{e}^{-\tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}dt \\
	& = & \frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}
	\end{align}</math>		\end{align}</math>

Athanasios Gerokostopoulos: /* T-NT Exponential */

2012-02-17T00:45:58Z

T-NT Exponential

← Older revision		Revision as of 00:45, 17 February 2012
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	==T-NT Exponential==		==T-NT Exponential==
	<br>		<br>
	By setting <math>m=L(U,V)</math> ~~as given in Eqn. (Temp-Volt)~~, the exponential <math>pdf</math> becomes:		By setting <math>m=L(U,V)</math>, the exponential <math>pdf</math> becomes:

	<br>		<br>

Nicolette Young: Created page with '==T-NT Exponential==
By setting $m=L(U,V)$ as given in Eqn. (Temp-Volt), the exponential $pdf$ becomes:
:: $f(t,U,V)=\frac{{{U}^{n}}}{C}{…'$

2012-01-16T18:35:40Z

Created page with '==T-NT Exponential== By setting <math>m=L(U,V)</math> as given in Eqn. (Temp-Volt), the exponential <math>pdf</math> becomes: ::<math>f(t,U,V)=\frac{{{U}^{n}}}{C}{…'

New page

==T-NT Exponential==
 
By setting <math>m=L(U,V)</math> as given in Eqn. (Temp-Volt), the exponential <math>pdf</math> becomes:

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

 
===T-NT Exponential Statistical Properties Summary===
 
====Mean or MTTF====
 
The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) for the T-NT exponential model is given by:

 
::<math>\begin{align}
& \overline{T}= & \int\limits_{0}^{\infty }t\cdot f(t,U,V)dt \\
& = & \int\limits_{0}^{\infty }t\cdot \frac{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{e}^{-\tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}dt \\
& = & \frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}
\end{align}</math>

 

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

 
::<math>\breve{T}=\frac{1}{\lambda }0.693=0.693\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}</math>

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

 
::<math>\tilde{T}=0</math>

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

 
::<math>{{\sigma }_{T}}=\frac{1}{\lambda }=m=\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}</math>

 
====T-NT Exponential Reliability Function====
 
The T-NT exponential reliability function is given by:

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

 
This function is the complement of the T-NT exponential cumulative distribution function or:

 
::<math>R(T,U,V)=1-Q(T,U,V)=1-\mathop{}_{0}^{T}f(T)dT</math>

 
and,

 
::<math>R(T,U,V)=1-\mathop{}_{0}^{T}\frac{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{e}^{-\tfrac{T\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}dT={{e}^{-\tfrac{T\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}</math>

 
====Conditional Reliability====
 
The conditional reliability function for the T-NT exponential model is given by,

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

 
====Reliable Life====
 
For the T-NT exponential model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}}</math> , is given by:

 
::<math>R({{t}_{R}},U,V)={{e}^{-\tfrac{{{t}_{R}}\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}</math>

 
::<math>\ln [R({{t}_{R}},U,V)]{{=}^{-\tfrac{{{t}_{R}}\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}</math>

 
:or:

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

 

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

 
::<math>\begin{align}
& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{U_{i}^{n}}{C}{{e}^{-\tfrac{B}{{{V}_{i}}}}}\cdot {{e}^{-\tfrac{U_{i}^{n}}{C}\left( {{e}^{-\tfrac{B}{{{V}_{i}}}}} \right){{T}_{i}}}} \right] \\
& & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{U_{i}^{n}}{C}\left( {{e}^{-\tfrac{B}{{{V}_{i}}}}} \right)T_{i}^{\prime }+\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}^{-\tfrac{T_{Li}^{\prime \prime }}{C}U_{i}^{\prime \prime n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}}</math>
 
 
::<math>R_{Ri}^{\prime \prime }={{e}^{-\tfrac{T_{Ri}^{\prime \prime }}{C}U_{i}^{\prime \prime n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}}</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>B</math> is the T-NT parameter (unknown, the first of three parameters to be estimated).
 
• <math>C</math> is the second T-NT parameter (unknown, the second of three parameters to be estimated).
 
• <math>n</math> is the third T-NT parameter (unknown, the third of three 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 i <math>^{th}</math> group of data intervals.
 
• <math>T_{Li}^{\prime \prime }</math> is the beginning of the i <math>^{th}</math> interval.
 
• <math>T_{Ri}^{\prime \prime }</math> is the ending of the i <math>^{th}</math> interval.
 
 
The solution (parameter estimates) will be found by solving for the parameters <math>B,</math> <math>C</math> and <math>n</math> so that <math>\tfrac{\partial \Lambda }{\partial B}=0,</math> <math>\tfrac{\partial \Lambda }{\partial C}=0</math> and <math>\tfrac{\partial \Lambda }{\partial n}=0</math> .
 

{{TNT weibull}}

{{TNT Lognormal}}

Template:TNT Exponential - Revision history

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

Chris Kahn: /* Maximum Likelihood Estimation Method */

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

Athanasios Gerokostopoulos: /* T-NT Exponential Reliability Function */

Athanasios Gerokostopoulos: /* T-NT Exponential Reliability Function */

Athanasios Gerokostopoulos: /* Mean or MTTF */

Athanasios Gerokostopoulos: /* T-NT Exponential */

Nicolette Young: Created page with '==T-NT Exponential== By setting m=L(U,V) as given in Eqn. (Temp-Volt), the exponential pdf becomes: ::f(t,U,V)=\frac{{{U}^{n}}}{C}{…'

Nicolette Young: Created page with '==T-NT Exponential==
By setting $m=L(U,V)$ as given in Eqn. (Temp-Volt), the exponential $pdf$ becomes:
:: $f(t,U,V)=\frac{{{U}^{n}}}{C}{…'$