Template:Gen-eyring log par est: Difference between revisions

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(Created page with '====Parameter Estimation==== <br> Substituting the generalized Eyring model into the lognormal log-likelihood equation yields: <br> ::<math>\begin{align} & \ln (L)= & \Lambda…')
 
 
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====Parameter Estimation====
#REDIRECT [[Eyring_Relationship#Generalized_Eyring-Lognormal]]
 
<br>
Substituting the generalized Eyring model into the lognormal log-likelihood equation yields:
 
<br>
::<math>\begin{align}
  & \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln [\frac{\varphi (z(t))}{\sigma _{T}^{\prime }t}]\overset{S}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\ln \left( 1-\Phi (z(t_{i}^{\prime })) \right) \\
&  & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]
\end{align}</math>
 
<br>
:where:
 
<br>
::<math>z_{Ri}^{\prime \prime }=\frac{\ln t_{Ri}^{\prime \prime }-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}+\ln (V_{i}^{\prime \prime })}{\sigma _{T}^{\prime }}</math>
 
<br>
::<math>z_{Li}^{\prime \prime }=\frac{\ln t_{Ri}^{\prime \prime }-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}+\ln (V_{i}^{\prime \prime })}{\sigma _{T}^{\prime }}</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,B,C,D</math>  are 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 non-thermal 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.
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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.
• <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.
<br>
• <math>N_{i}^{\prime \prime }</math>  is the number of intervals in the  <math>{{i}^{th}}</math>  group of data intervals.
• <math>T_{Li}^{\prime \prime }</math>  is the beginning of the  <math>{{i}^{th}}</math>  interval.
• <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>B,</math>  <math>C,</math> and  <math>D</math>  so that  <math>\tfrac{\partial \Lambda }{\partial A}=0,</math>  <math>\tfrac{\partial \Lambda }{\partial B}=0,</math>  <math>\tfrac{\partial \Lambda }{\partial D}=0</math>  and  <math>\tfrac{\partial \Lambda }{\partial D}=0</math> .
<br>

Latest revision as of 00:52, 17 August 2012

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