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==T-H Weibull==
#REDIRECT [[Temperature-Humidity_Relationship#T-H_Weibull]]
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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:
 
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::<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>
 
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===T-H Weibull Statistical Properties Summary===
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====Mean or MTTF====
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The mean,  <math>\overline{T}</math>  (also called  <math>MTTF</math> ), of the T-H Weibull model is given by:
 
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::<math>\overline{T}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}\cdot \Gamma \left( \frac{1}{\beta }+1 \right)</math>
 
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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> .
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====Median====
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The median, <math>\breve{T},</math> of the T-H Weibull model is given by:
 
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::<math>\breve{T}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>
 
====Mode====
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The mode,  <math>\tilde{T},</math>  of the T-H Weibull model is given by:
 
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::<math>\tilde{T}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}</math>
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====Standard Deviation====
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The standard deviation,  <math>{{\sigma }_{T}},</math>  of the T-H Weibull model is given by:
 
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::<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>
 
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====T-H Weibull Reliability Function====
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The T-H Weibull reliability function is given by:
 
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::<math>R(T,V,U)={{e}^{-{{\left( \tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta }}}}</math>
 
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====Conditional Reliability Function====
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The T-H Weibull conditional reliability function at a specified stress level is given by:
 
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::<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>
 
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or:
 
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::<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>
 
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====Reliable Life====
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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:
 
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::<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>
 
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====T-H Weibull Failure Rate Function====
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The T-H Weibull failure rate function,  <math>\lambda (T,V,U)</math> , is given by:
 
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::<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>
 
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===Parameter Estimation===
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====Maximum Likelihood Estimation Method====
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Substituting the T-H model into the Weibull log-likelihood function yields:
 
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::<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>
 
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where:
 
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::<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>
 
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::<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>
 
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and:
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• <math>{{F}_{e}}</math>  is the number of groups of exact times-to-failure data points.
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• <math>{{N}_{i}}</math>  is the number of times-to-failure data points in the  <math>{{i}^{th}}</math>  time-to-failure data group.
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• <math>\beta </math>  is the Weibull shape parameter (unknown, the first of four parameters to be estimated).
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• ..  is the T-H parameter (unknown, the second of four parameters to be estimated).
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• <math>\phi </math>  is the second T-H parameter (unknown, the third of four parameters to be estimated).
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• <math>b</math>  is the third T-H parameter (unknown, the fourth of four parameters to be estimated).
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• <math>{{V}_{i}}</math>  is the temperature level of the  <math>{{i}^{th}}</math>  group.
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• <math>{{U}_{i}}</math>  is the relative humidity level of the  <math>{{i}^{th}}</math>  group.
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• <math>{{T}_{i}}</math>  is the exact failure time of the  <math>{{i}^{th}}</math>  group.
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• <math>S</math>  is the number of groups of suspension data points.
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• <math>N_{i}^{\prime }</math>  is the number of suspensions in the  <math>{{i}^{th}}</math>  group of suspension data points.
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• <math>T_{i}^{\prime }</math>  is the running time of the  <math>{{i}^{th}}</math>  suspension data group.
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• <math>FI</math>  is the number of interval data groups.
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• <math>N_{i}^{\prime \prime }</math>  is the number of intervals in the  <math>{{i}^{th}}</math>  group of data intervals.
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• <math>T_{Li}^{\prime \prime }</math>  is the beginning of the  interval.
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• <math>T_{Ri}^{\prime \prime }</math>  is the ending of the  <math>{{i}^{th}}</math>  interval.
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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> .
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{{Example:T-H}}

Latest revision as of 05:33, 15 August 2012

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