Template:T-h exponential - Revision history

Richard House: Redirected page to Temperature-Humidity Relationship#T-H Exponential

2012-08-15T05:31:46Z

Redirected page to Temperature-Humidity Relationship#T-H Exponential

Chris Kahn: /* Maximum Likelihood Estimation Method */

2012-03-09T18:37:27Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 18:37, 9 March 2012
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	• <math>S</math> is the number of groups of suspension data points.		• <math>S</math> is the number of groups of suspension data points.
	<br>		<br>
	• .. is the number of suspensions in the <math>{{i}^{th}}</math> group 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.
	<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.

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

2012-02-16T21:56:20Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 21:56, 16 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{1}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}\cdot {{e}^{-\tfrac{{{T}_{i}}}{A}\cdot {{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}}} \right] \\		& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}\cdot {{e}^{-\tfrac{{{T}_{i}}}{A}\cdot {{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}+\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{T_{i}^{\prime }}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}+\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>

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

2012-02-16T21:49:29Z

T-H Exponential Reliability Function

← Older revision		Revision as of 21:49, 16 February 2012
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	<br>		<br>
	::<math>R(T,V,U)=1-Q(T,V,U)=1-\~~mathop{}_~~{0}^{T}f(T)dT</math>		::<math>R(T,V,U)=1-Q(T,V,U)=1-\int_{0}^{T}f(T)dT</math>

	<br>		<br>
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	<br>		<br>
	::<math>R(T,V,U)=1-\~~mathop{}_~~{0}^{T}\frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}{{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}dT={{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>		::<math>R(T,V,U)=1-\int_{0}^{T}\frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}{{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}dT={{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>

	<br>		<br>

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

Athanasios Gerokostopoulos: /* Standard Deviation */

2012-02-16T21:48:50Z

Standard Deviation

← Older revision		Revision as of 21:48, 16 February 2012
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	<br>		<br>
	::<math>{{\sigma }_{T}}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}</math>		::<math>{{\sigma }_{T}}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}</math>

			<br>

	====T-H Exponential Reliability Function====		====T-H Exponential Reliability Function====

Athanasios Gerokostopoulos: /* Mode */

2012-02-16T21:48:31Z

Mode

← Older revision		Revision as of 21:48, 16 February 2012
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	<br>		<br>
	::<math>\tilde{T}=0</math>		::<math>\tilde{T}=0</math>

			<br>

	====Standard Deviation====		====Standard Deviation====

Athanasios Gerokostopoulos: /* Mean or MTTF */

2012-02-16T21:48:10Z

Mean or MTTF

← Older revision		Revision as of 21:48, 16 February 2012
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	<br>		<br>
	::<math>\overline{T}=\~~mathop{}_~~{0}^{\infty }t\cdot f(t,V,U)dt</math>		::<math>\overline{T}=\int_{0}^{\infty }t\cdot f(t,V,U)dt</math>

	<br>		<br>
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \overline{T}= & \~~mathop{}_~~{0}^{\infty }t\cdot \frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}{{e}^{-\tfrac{t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}dt \\		& \overline{T}= & \int_{0}^{\infty }t\cdot \frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}{{e}^{-\tfrac{t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}dt =\ & A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}
	~~& =~~ & A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}
	\end{align}</math>		\end{align}</math>

	<br>		<br>

	====Median====		====Median====
	<br>		<br>

Athanasios Gerokostopoulos: /* T-H Exponential */

2012-02-16T21:47:23Z

T-H Exponential

← Older revision		Revision as of 21:47, 16 February 2012
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	==T-H Exponential==		==T-H Exponential==
	<br>		<br>
	By setting <math>m=L(U,V)</math> in ~~Eqn. (Temp-Hum)~~ the exponential <math>pdf</math> ~~becomes~~:		By setting <math>m=L(U,V)</math> in the exponential <math>pdf</math> we can obtain the T-H exponential <math>pdf</math>:

	<br>		<br>

Nicolette Young: Created page with '==T-H Exponential==
By setting $m=L(U,V)$ in Eqn. (Temp-Hum) the exponential $pdf$ becomes:
:: $f(t,V,U)=\frac{1}{A}{{e}^{-\left( \tfrac…'$

2012-01-16T18:18:08Z

Created page with '==T-H Exponential== By setting <math>m=L(U,V)</math> in Eqn. (Temp-Hum) the exponential <math>pdf</math> becomes: ::<math>f(t,V,U)=\frac{1}{A}{{e}^{-\left( \tfrac…'

New page

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

 
::<math>f(t,V,U)=\frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}\cdot {{e}^{-\tfrac{t}{A}\cdot {{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>

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

 
::<math>\overline{T}=\mathop{}_{0}^{\infty }t\cdot f(t,V,U)dt</math>

 
Substituting Eqn. (t-h exp pdf) yields:

 
::<math>\begin{align}
& \overline{T}= & \mathop{}_{0}^{\infty }t\cdot \frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}{{e}^{-\tfrac{t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}dt \\
& = & A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}
\end{align}</math>

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

 
::<math>\breve{T}=0.693\cdot A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}</math>

 

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

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

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

 
::<math>{{\sigma }_{T}}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}</math>

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

 
::<math>R(T,V,U)={{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>

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

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

 
and:

 
::<math>R(T,V,U)=1-\mathop{}_{0}^{T}\frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}{{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}dT={{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>

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

 
::<math>R(T,t,V,U)=\frac{R(T+t,V,U)}{R(T,V,U)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>

 
====Reliable Life====
 
For the T-H 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}},V,U)={{e}^{-\tfrac{{{t}_{R}}}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>

 
::<math>\ln [R({{t}_{R}},V,U)]=-\frac{{{t}_{R}}}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}</math>

 
or:

 
::<math>{{t}_{R}}=-A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}\ln [R({{t}_{R}},V,U)]</math>

 
===Parameter Estimation===
 
====Maximum Likelihood Estimation Method====
 
Substituting the T-H model 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{1}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}\cdot {{e}^{-\tfrac{{{T}_{i}}}{A}\cdot {{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}}} \right] \\
& & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}+\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 }}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{U_{i}^{\prime \prime }} \right)}}}}</math>

 
::<math>R_{Ri}^{\prime \prime }={{e}^{-\tfrac{T_{Ri}^{\prime \prime }}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{U_{i}^{\prime \prime }} \right)}}}}</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>A</math> is the T-H parameter (unknown, the first of three parameters to be estimated).
 
• <math>\phi </math> is the second T-H parameter (unknown, the second of three parameters to be estimated).
 
• <math>b</math> is the third T-H 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 relative humidity 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.
 
• .. 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>A,</math> <math>\phi </math> and <math>b</math> so that <math>\tfrac{\partial \Lambda }{\partial A}=0,</math> <math>\tfrac{\partial \Lambda }{\partial \phi }=0</math> and <math>\tfrac{\partial \Lambda }{\partial b}=0</math> .

{{t-h weibull}}

{{t-h lognormal}}