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==Arrhenius-Weibull==
#REDIRECT [[Arrhenius_Relationship]]
<br>
The  <math>pdf</math>  for 2-parameter Weibull distribution is given by:
 
<br>
::{{weibull2pdf}}
 
<br>
The scale parameter (or characteristic life) of the Weibull distribution is  <math>\eta </math> .
 
<br>
The Arrhenius-Weibull model pdf can then be obtained by setting  <math>\eta =L(V)</math>  in Eqn. (arrhenius):
 
<br>
::<math>\eta =L(V)=C\cdot {{e}^{\tfrac{B}{V}}}</math>
 
 
<br>
and substituting for  <math>\eta </math>  in Eqn. (Weibullpdf):
 
<br>
::<math>f(t,V)=\frac{\beta }{C\cdot {{e}^{\tfrac{B}{V}}}}{{\left( \frac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}}}</math>
 
<br>
An illustration of the  <math>pdf</math>  for different stresses is shown in Fig. 6.  As expected, the  <math>pdf</math>  at lower stress levels is more stretched to the right, with a higher scale parameter, while its shape remains the same (the shape parameter is approximately 3 in Fig. 6). This behavior is observed when the parameter  <math>B</math>  of the Arrhenius model is positive.
 
<br>
[[Image:ALTA6.6.gif|thumb|center|300px|Behavior of the probability density function at different stresses and with the parameters held constant.]]
 
<br>
The advantage of using the Weibull distribution as the life distribution lies in its flexibility to assume different shapes. The Weibull distribution is presented in greater detail in Chapter 5.
 
{{aaw stat prob sum}}
 
===Parameter Estimation===
<br>
====Maximum Likelihood Estimation Method====
<br>
The Arrhenius-Weibull log-likelihood function is as follows:
 
<br>
::<math>\begin{align}
  & \Lambda = & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}} \right] \\
&  & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \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( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}}</math>
 
 
<br>
::<math>R_{Ri}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\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.
 
• ..  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 three parameters to be estimated).
 
• <math>B</math>  is the Arrhenius parameter (unknown, the second of three parameters to be estimated).
 
• <math>C</math>  is the second Arrhenius parameter (unknown, the third of three parameters to be estimated).
 
• <math>{{V}_{i}}</math>  is the stress 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 i <math>^{th}</math>  group of data intervals.
 
• <math>T_{Li}^{\prime \prime }</math>  is the beginning of the i <math>^{th}</math>  interval.
 
• <math>T_{Ri}^{\prime \prime }</math>  is the ending of the i <math>^{th}</math>  interval.
 
The solution (parameter estimates) will be found by solving for  <math>\widehat{\beta },</math>  <math>\widehat{B},</math>  <math>\widehat{C}</math>  so that  <math>\tfrac{\partial \Lambda }{\partial \beta }=0,</math>  <math>\tfrac{\partial \Lambda }{\partial B}=0</math>  and  <math>\tfrac{\partial \Lambda }{\partial C}=0</math> , where:
 
<br>
::<math>\begin{align}
  & \frac{\partial \Lambda }{\partial \beta }= & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
&  & -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
&  & -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
&  & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)R_{Li}^{\prime \prime }-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)R_{Ri}^{\prime \prime }}{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( \frac{{{T}_{i}}}{\widehat{C}{{e}^{\tfrac{\widehat{B}}{{{V}_{i}}}}}} \right)}^{\beta }}+\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\frac{1}{{{V}_{i}}}{{\left( \frac{T_{i}^{\prime }}{\widehat{C}{{e}^{\tfrac{\widehat{B}}{{{V}_{i}}}}}} \right)}^{\beta }} \\
&  & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{{{V}_{i}}}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)} 
\end{align}</math>
 
 
 
<br>
::<math>\begin{align}
  & \frac{\partial \Lambda }{\partial C}= & -\frac{\beta }{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}+\frac{\beta }{C}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }} \\
&  & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{C}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)} 
\end{align}</math>

Latest revision as of 07:26, 8 August 2012

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