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(Created page with '==T-H Exponential== <br> By setting <math>m=L(U,V)</math> in Eqn. (Temp-Hum) the exponential <math>pdf</math> becomes: <br> ::<math>f(t,V,U)=\frac{1}{A}{{e}^{-\left( \tfrac…')
 
 
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==T-H Exponential==
#REDIRECT [[Temperature-Humidity_Relationship#T-H_Exponential]]
<br>
By setting  <math>m=L(U,V)</math>  in Eqn. (Temp-Hum) the exponential  <math>pdf</math>  becomes:
 
<br>
::<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>
 
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===T-H Exponential Statistical Properties Summary===
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====Mean or MTTF====
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The mean,  <math>\overline{T},</math>  or Mean Time To Failure (MTTF) for the T-H exponential model is given by:
 
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::<math>\overline{T}=\mathop{}_{0}^{\infty }t\cdot f(t,V,U)dt</math>
 
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Substituting Eqn. (t-h exp pdf) yields:
 
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::<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>
 
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====Median====
<br>
The median,  <math>\breve{T},</math> for the T-H exponential model is given by:
 
<br>
::<math>\breve{T}=0.693\cdot A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}</math>
 
<br>
 
====Mode====
<br>
The mode,  <math>\tilde{T},</math>  for the T-H exponential model is given by:
 
<br>
::<math>\tilde{T}=0</math>
 
====Standard Deviation====
<br>
The standard deviation,  <math>{{\sigma }_{T}}</math> , for the T-H exponential model is given by:
 
<br>
::<math>{{\sigma }_{T}}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}</math>
 
====T-H Exponential Reliability Function====
<br>
The T-H exponential reliability function is given by:
 
<br>
::<math>R(T,V,U)={{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>
 
<br>
This function is the complement of the T-H exponential cumulative distribution function or:
 
<br>
::<math>R(T,V,U)=1-Q(T,V,U)=1-\mathop{}_{0}^{T}f(T)dT</math>
 
<br>
and:
 
 
<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>
 
<br>
====Conditional Reliability====
<br>
The conditional reliability function for the T-H exponential model is given by:
 
<br>
::<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>
 
<br>
====Reliable Life====
<br>
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:
 
<br>
::<math>R({{t}_{R}},V,U)={{e}^{-\tfrac{{{t}_{R}}}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>
 
<br>
::<math>\ln [R({{t}_{R}},V,U)]=-\frac{{{t}_{R}}}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}</math>
 
<br>
or:
 
<br>
::<math>{{t}_{R}}=-A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}\ln [R({{t}_{R}},V,U)]</math>
 
<br>
===Parameter Estimation===
<br>
====Maximum Likelihood Estimation Method====
<br>
Substituting the T-H model into the exponential log-likelihood equation yields:
 
<br>
::<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>
 
<br>
where:
 
<br>
::<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>
 
<br>
::<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>
 
<br>
and:
<br>
• <math>{{F}_{e}}</math>  is the number of groups of exact times-to-failure data points.
<br>
• <math>{{N}_{i}}</math>  is the number of times-to-failure data points in the  <math>{{i}^{th}}</math>  time-to-failure data group.
<br>
• <math>A</math>  is the T-H parameter (unknown, the first of three parameters to be estimated).
<br>
• <math>\phi </math>  is the second T-H parameter (unknown, the second of three parameters to be estimated).
<br>
• <math>b</math>  is the third T-H parameter (unknown, the third of three parameters to be estimated).
<br>
• <math>{{V}_{i}}</math>  is the temperature level of the  <math>{{i}^{th}}</math>  group.
<br>
• <math>{{U}_{i}}</math>  is the relative humidity level of the  <math>{{i}^{th}}</math>  group.
<br>
• <math>{{T}_{i}}</math>  is the exact failure time of the  <math>{{i}^{th}}</math>  group.
<br>
• <math>S</math>  is the number of groups of suspension data points.
<br>
• ..  is the number of suspensions in the  <math>{{i}^{th}}</math>  group of suspension data points.
<br>
• <math>T_{i}^{\prime }</math>  is the running time of the  <math>{{i}^{th}}</math>  suspension data group.
<br>
• <math>FI</math>  is the number of interval data groups.
<br>
• <math>N_{i}^{\prime \prime }</math>  is the number of intervals in the  <math>{{i}^{th}}</math>  group of data intervals.
<br>
• <math>T_{Li}^{\prime \prime }</math>  is the beginning of the  <math>{{i}^{th}}</math>  interval.
<br>
• <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>  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}}

Latest revision as of 05:31, 15 August 2012

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