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==Eyring-Weibull==
#REDIRECT [[Eyring_Relationship#Eyring-Weibull]]
 
<br>
The  <math>pdf</math>  for 2-parameter Weibull distribution is given by:
 
<br>
::<math>f(t)=\frac{\beta }{\eta }{{\left( \frac{t}{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{\eta } \right)}^{\beta }}}}</math>
 
<br>
The scale parameter (or characteristic life) of the Weibull distribution is  <math>\eta </math> . The Eyring-Weibull model  <math>pdf</math>  can then be obtained by setting  <math>\eta =L(V)</math>  in Eqn. (eyring):
 
<br>
::<math>\eta =L(V)=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
 
<br>
:or:
 
<br>
::<math>\frac{1}{\eta }=V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}</math>
 
<br>
Substituting for  <math>\eta </math>  into Eqn. (Eyrpdf):
 
<br>
::<math>f(t,V)=\beta \cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}{{\left( t\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta -1}}{{e}^{-{{\left( t\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}</math>
 
===Eyring-Weibull Statistical Properties Summary===
 
====Mean or MTTF====
 
The mean,  <math>\overline{T}</math>, or Mean Time To Failure (MTTF) for the Eyring-Weibull model is given by:
 
 
::<math>\overline{T}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}\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> .
 
<br>
 
====Median====
 
<br>
The median,  <math>\breve{T}</math>
for the Eyring-Weibull model is given by:
 
<br>
::<math>\breve{T}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>
 
====Mode====
 
<br>
The mode,  <math>\tilde{T},</math>
for the Eyring-Weibull model is given by:
 
<br>
::<math>\tilde{T}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}</math>
 
<br>
 
====Standard Deviation====
 
<br>
The standard deviation,  <math>{{\sigma }_{T}},</math> 
for the Eyring-Weibull model is given by:
 
<br>
::<math>{{\sigma }_{T}}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}}}</math>
 
<br>
====Eyring-Weibull Reliability Function====
 
<br>
The Eyring-Weibull reliability function is given by:
 
<br>
::<math>R(T,V)={{e}^{-{{\left( V\cdot T\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}</math>
<br>
====Conditional Reliability Function====
<br>
 
The Eyring-Weibull conditional reliability function at a specified stress level is given by:
<br>
::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left( \left( T+t \right)\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}}{{{e}^{-{{\left( V\cdot T\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}}</math>
<br>
:or:
 
<br>
::<math>R(T,t,V)={{e}^{-\left[ {{\left( \left( T+t \right)\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}-{{\left( V\cdot T\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }} \right]}}</math>
<br>
====Reliable Life====
<br>
 
For the Eyring-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:
 
<br>
::<math>{{t}_{R}}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}{{\left\{ -\ln \left[ R\left( {{T}_{R}},V \right) \right] \right\}}^{\tfrac{1}{\beta }}}</math>
 
 
====Eyring-Weibull Failure Rate Function====
<br>
The Eyring-Weibull failure rate function,  <math>\lambda (T)</math> , is given by:
<br>
 
::<math>\lambda \left( T,V \right)=\frac{f\left( T,V \right)}{R\left( T,V \right)}=\beta {{\left( T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta -1}}</math>
<br>
 
===Parameter Estimation===
<br>
====Maximum Likelihood Estimation Method====
<br>
 
The Eyring-Weibull log-likelihood function is composed of two summation portions:
 
<br>
::<math>\begin{align}
  & \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \beta \cdot {{V}_{i}}\cdot {{e}^{A-\tfrac{B}{{{V}_{i}}}}}{{\left( {{T}_{i}}{{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta -1}}{{e}^{-{{\left( {{T}_{i}}{{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}}} \right] \\
&  & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( {{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}}T_{i}^{\prime } \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>
 
<br>
:where:
 
 
<br>
::<math>R_{Li}^{\prime \prime }={{e}^{-{{\left( T_{Li}^{\prime \prime }{{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}}}</math>
 
 
<br>
::<math>R_{Ri}^{\prime \prime }={{e}^{-{{\left( T_{Ri}^{\prime \prime }{{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}}}</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>\beta </math>  is the Weibull shape parameter (unknown, the first of three parameters to be estimated).
<br>
• <math>A</math>  is the Eyring parameter (unknown, the second of three parameters to be estimated).
<br>
• <math>B</math>  is the second Eyring parameter (unknown, the third of three parameters to be estimated).
<br>
• <math>{{V}_{i}}</math>  is the stress 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>
• <math>N_{i}^{\prime }</math>  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 i <math>^{th}</math>  group of data intervals.
<br>
• <math>T_{Li}^{\prime \prime }</math>  is the beginning of the i <math>^{th}</math>  interval.
<br>
• <math>T_{Ri}^{\prime \prime }</math>  is the ending of the i <math>^{th}</math>  interval.
 
<br>
The solution (parameter estimates) will be found by solving for the parameters  <math>\beta ,</math>  <math>A</math>  and  <math>B</math>  so that  <math>\tfrac{\partial \Lambda }{\partial \beta }=0,</math>  <math>\tfrac{\partial \Lambda }{\partial A}=0</math>  and  <math>\tfrac{\partial \Lambda }{\partial B}=0</math> 
:where:
 
 
<br>
::<math>\begin{align}
& \frac{\partial \Lambda }{\partial A}= & \beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}-\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( {{T}_{i}}{{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }} \\
&  & -\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( T_{i}^{\prime }{{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }} \\
&  & \overset{FI}{\mathop{-\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta V_{i}^{\beta }{{e}^{A\beta -\tfrac{B\beta }{{{V}_{i}}}}}\left[ {{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} 
\end{align}</math>
 
 
<br>
::<math>\begin{align}
  & \frac{\partial \Lambda }{\partial B}= & -\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{{V}_{i}}}+\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{{V}_{i}}}{{\left( {{T}_{i}}{{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }} \\
&  & +\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\frac{1}{{{V}_{i}}}{{\left( T_{i}^{\prime }{{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }} \\
&  & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta V_{i}^{(\beta -1)}{{e}^{A\beta -\tfrac{B\beta }{{{V}_{i}}}}}\left[ {{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} 
\end{align}</math>
 
 
::<math>\frac{\partial \Lambda}{\partial \beta}=\frac{1}{\beta}\sum_{i=1}^{F_e} N_i\frac{1}{V_i}+\sum_{i=1}^{F_e} N_i ln\left(T_iV_i e^{A-\tfrac{B}{V_i}}\right)</math>
:<math>-\sum_{i=1}^{F_e} N_i\left(T_iV_i e^{A-\tfrac{B}{V_i}}\right)^\beta ln\left(T_iV)i e^{A-\tfrac{B}{V_i}}\right)</math>
:<math>-\sum_{i=1}^S N_i^'\left(T_i^'V_I e^{A-\tfrac{B}{V_i}}\right)^\beta ln\left(T_iV)i e^{A-\tfrac{B}{V_i}}\right)</math>
:<math>-\sum_{i=1}^{FI} N_i^{''}V_i e^{A-\tfrac{B}{V_i}}\frac{R_{Li}^{''} T_{Li}^{''}\left(ln(T_{Li}^' V_i)+A-\tfrac{B}{V_i}\right)-R_{Ri}^{''} T_{Ri}^{''}\left(ln(T_{Ri}^{''} V_i)+A-\tfrac{B}{V_i}\right)}{R_{L_i}^{''}-F_{Ri}^{''}}</math>

Latest revision as of 23:13, 16 August 2012

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