Template:T-h weibull - Revision history

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

2012-08-15T05:33:28Z

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

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

2012-02-16T22:06:21Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 22:06, 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{\beta }{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}{{\left( \frac{{{T}_{i}}}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}} \right)}^{\beta }}}} \right] \\		& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}{{\left( \frac{{{T}_{i}}}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}} \right)}^{\beta }}}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}} \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{T_{i}^{\prime }}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}} \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>

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	• <math>\beta </math> is the Weibull shape parameter (unknown, the first of four parameters to be estimated).		• <math>\beta </math> is the Weibull shape parameter (unknown, the first of four parameters to be estimated).
	<br>		<br>
	• .. is the T-H parameter (unknown, the second of four parameters to be estimated).		• <math>A</math> is the T-H parameter (unknown, the second of four parameters to be estimated).
	<br>		<br>
	• <math>\phi </math> is the second T-H parameter (unknown, the third of four parameters to be estimated).		• <math>\phi </math> is the second T-H parameter (unknown, the third of four parameters to be estimated).
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	<br>		<br>
	• <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval.		• <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval.

	<br>		<br>
	The solution (parameter estimates) will be found by solving for the parameters <math>A,</math> <math>\phi ,</math> <math>b</math> and <math>\beta </math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0,</math> <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> .		The solution (parameter estimates) will be found by solving for the parameters <math>A,</math> <math>\phi ,</math> <math>b</math> and <math>\beta </math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0,</math> <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> .

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

2012-02-16T22:01:10Z

T-H Weibull Reliability Function

← Older revision		Revision as of 22:01, 16 February 2012
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	<br>		<br>
	::<math>R(T,V,U)={{e}^{-{{\left( \tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta }}}}</math>		::<math>R(T,V,U)={{e}^{-{{\left( \tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta }}}}</math>

			<br>

	====Conditional Reliability Function====		====Conditional Reliability Function====

Athanasios Gerokostopoulos: /* T-H Weibull */

2012-02-16T22:00:29Z

T-H Weibull

← Older revision		Revision as of 22:00, 16 February 2012
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	==T-H Weibull==		==T-H Weibull==
	<br>		<br>
	By setting <math>\eta =L(U,V)</math> ~~as given~~ in ~~Eqn. (Temp-Hum)~~, the T--H Weibull model's <math>pdf</math> is given by:		By setting <math>\eta =L(U,V)</math> in the Weibull <math>pdf</math>, the T--H Weibull model's <math>pdf</math> is given by:
	..
			<br>
			::<math>f(t,\,V,\,U)=\frac{\beta }{A}{{e}^{-\left( \tfrac{\varphi }{V}+\tfrac{b}{U} \right)}}{{\left( \frac{t}{A}{{e}^{-\left( \tfrac{\varphi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{A}{{e}^{-\left( \tfrac{\varphi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta }}}}</math>

	<br>		<br>

Nicolette Young: /* Example */

2012-02-14T17:44:07Z

Example

← Older revision		Revision as of 17:44, 14 February 2012
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	<br>		<br>

	~~===~~Example~~===~~		{{Example:T-H}}
	~~<br>~~
	~~The following data were collected after testing twelve electronic devices at different temperature and humidity conditions~~:
	~~<br>~~
	~~<br>~~
	~~[[Image:ex1chp9.gif\|thumb\|center\|300px\|]]~~
	~~<br>~~
	~~<br>~~
	~~Using ALTA, the following results were obtained:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& \widehat{\beta~~ }~~= & 5.87439512 \\~~
	~~& & \\~~
	~~& \widehat{A~~}~~= & 0.0000597 \\~~
	~~& & \\~~
	~~& \widehat{b}= & 0.2805985 \\~~
	~~& & \\~~
	~~& \widehat{\phi }= & 5630.329851~~
	~~\end{align}</math>~~

	~~<br>~~
	~~A probability plot for the entered data is shown next.~~

	~~<br>~~
	~~[[Image:ALTA9probabilityplot.gif\|thumb\|center\|300px\|Probability plots at the tested combinations.]]~~
	~~<br>~~
	~~<br>~~
	~~Note that three lines are plotted because there are three combinations of stresses, namely, (398K, 0.4), (378K, 0.8) and (378K, 0.4).~~

	~~<br>~~
	Given the use stress levels, time estimates can be obtained for specified probability. A Life vs. Stress plot can be obtained if one of the stresses is kept constant. For example, the following picture is a Life vs. Humidity plot at a constant temperature of 338K.
	~~<br>~~
	~~[[Image:ALTA9stressplot.gif\|thumb\|center\|300px\|Life vs. Humidity plot with temperature fixed at 338K.]]~~
	~~<br>~~

Nicolette Young: Created page with '==T-H Weibull==
By setting $\eta =L(U,V)$ as given in Eqn. (Temp-Hum), the T--H Weibull model's $pdf$ is given by: ..
===T-H Weibull Statist…'

2012-01-16T18:13:26Z

Created page with '==T-H Weibull== By setting <math>\eta =L(U,V)</math> as given in Eqn. (Temp-Hum), the T--H Weibull model's <math>pdf</math> is given by: .. ===T-H Weibull Statist…'

New page

==T-H Weibull==
 
By setting <math>\eta =L(U,V)</math> as given in Eqn. (Temp-Hum), the T--H Weibull model's <math>pdf</math> is given by:
..

 
===T-H Weibull Statistical Properties Summary===
 
====Mean or MTTF====
 
The mean, <math>\overline{T}</math> (also called <math>MTTF</math> ), of the T-H Weibull model is given by:

 
::<math>\overline{T}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}\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> of the T-H Weibull model is given by:

 
::<math>\breve{T}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>

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

 
::<math>\tilde{T}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}</math>
 

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

 
::<math>{{\sigma }_{T}}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}}}</math>

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

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

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

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

 
or:

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

 
====Reliable Life====
 
For the T-H 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}}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}{{\left\{ -\ln \left[ R\left( {{T}_{R}},V,U \right) \right] \right\}}^{\tfrac{1}{\beta }}}</math>

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

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

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

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

===Example===
 
The following data were collected after testing twelve electronic devices at different temperature and humidity conditions:
 
 
[[Image:ex1chp9.gif|thumb|center|300px|]]
 
 
Using ALTA, the following results were obtained:

 
::<math>\begin{align}
& \widehat{\beta }= & 5.87439512 \\
& & \\
& \widehat{A}= & 0.0000597 \\
& & \\
& \widehat{b}= & 0.2805985 \\
& & \\
& \widehat{\phi }= & 5630.329851
\end{align}</math>

 
A probability plot for the entered data is shown next.

 
[[Image:ALTA9probabilityplot.gif|thumb|center|300px|Probability plots at the tested combinations.]]
 
 
Note that three lines are plotted because there are three combinations of stresses, namely, (398K, 0.4), (378K, 0.8) and (378K, 0.4).

 
Given the use stress levels, time estimates can be obtained for specified probability. A Life vs. Stress plot can be obtained if one of the stresses is kept constant. For example, the following picture is a Life vs. Humidity plot at a constant temperature of 338K.
 
[[Image:ALTA9stressplot.gif|thumb|center|300px|Life vs. Humidity plot with temperature fixed at 338K.]]