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| ==T-H Weibull==
| | #REDIRECT [[Temperature-Humidity_Relationship#T-H_Weibull]] |
| <br>
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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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| <br>
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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>
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| ====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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| <br>
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| ::<math>\begin{align}
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| & \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] \\
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| & & -\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 }]
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| \end{align}</math>
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| <br>
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| where:
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| <br>
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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}}
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