Mixed Weibull Log-Likelihood Functions and their Partials - Revision history

Lisa Hacker: Redirected page to Appendix: Log-Likelihood Equations

2015-06-25T19:26:26Z

Redirected page to Appendix: Log-Likelihood Equations

← Older revision		Revision as of 19:26, 25 June 2015
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	~~=== Mixed Weibull Log-Likelihood Functions and their Partials===~~		#REDIRECT [[Appendix:_Log-Likelihood_Equations]]
	~~The log-likelihood function (without the constant) is composed of three summation portions:~~
	~~::<math>\begin{align}~~
	\frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( \ln ({{T}_{i}})-{\mu }' \right)}^{2}}}{\sigma _{{{T}'}}^{3}}-\frac{1}{{{\sigma }_{{{T}'}}}} \right) \\
	& +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)} \\
	& -\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\frac{\left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{\Phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}
	~~\end{align}</math>~~


	~~::<math>\begin{align}~~
	~~\ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left~~[ \underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}\frac{{{\beta }_{k}}}{{{\eta }_{k}}}{{\left( \frac{{{T}_{i}}}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}-1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right] \\
	~~& \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left~~[ ~~\underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}{{e}^{~~-~~{{\left( \tfrac{T_{i}^{\prime }}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right~~] \\
	& \text{ }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ \underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}\frac{{{\beta }_{k}}}{{{\eta }_{k}}}{{\left( \frac{T_{Li}^{\prime \prime }+T_{Ri}^{\prime \prime }}{2{{\eta }_{k}}} \right)}^{{{\beta }_{k}}-1}}{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }+T_{Ri}^{\prime \prime }}{2{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right]
	~~\end{align}</math>~~

	~~:where:~~
	~~::• <math>{{F}_{e}}</math> is the number of groups of times-to-failure data points~~
	~~::• <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group~~
	~~::• <math>Q</math> is the number of subpopulations~~
	~~::• <math>{{\rho }_{k}}</math> is the proportionality of the <math>{{k}^{th}}</math> subpopulation (unknown a priori, the first set of three sets of parameters to be found)~~
	~~::• <math>{{\beta }_{k}}</math> is the Weibull shape parameter of the <math>{{k}^{th}}</math> subpopulation (unknown a priori, the second set of three sets of parameters to be found)~~
	~~::• <math>{{\eta }_{k}}</math> is the Weibull scale parameter (unknown a priori, the third set of three sets of parameters to be found)~~
	~~::• <math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data~~
	~~::• <math>S</math> is the number of groups of suspension data points~~
	~~::• <math>N_{i}^{\prime }</math> is the number of suspensions in <math>{{i}^{th}}</math> group of suspension data points~~
	~~::• <math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group~~
	~~::• <math>FI</math> is the number of groups of interval data points~~
	~~::• <math>N_{i}^{\prime \prime }</math> is the number of intervals in <math>{{i}^{th}}</math> group of data intervals~~
	~~::• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval~~
	~~::• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval~~


	~~The solution will be found by solving for a group of parameters:~~

	::<math>\left( \widehat{{{\rho }_{1,}}}\widehat{{{\beta }_{1}}},\widehat{{{\eta }_{1}}},\widehat{{{\rho }_{2,}}}\widehat{{{\beta }_{2}}},\widehat{{{\eta }_{2}}},...,\widehat{{{\rho }_{Q,}}}\widehat{{{\beta }_{Q}}},\widehat{{{\eta }_{Q}}} \right)</math>

	~~:so that:~~

	~~::<math>\begin{align}~~
	~~\frac{\partial \Lambda }{\partial {{\rho }_{1}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{1}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{1}}}=0 \\~~
	~~\frac{\partial \Lambda }{\partial {{\rho }_{2}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{2}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{2}}}=0 \\~~
	~~\vdots \\~~
	~~\frac{\partial \Lambda }{\partial {{\rho }_{Q-1}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{Q-1}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{Q-1}}}=0 \\~~
	~~\frac{\partial \Lambda }{\partial {{\beta }_{Q}}}= & 0,\text{ and }\frac{\partial \Lambda }{\partial {{\eta }_{Q}}}=0~~
	~~\end{align}</math>~~
	~~<br>~~

Pantelis: Created page with '=== Mixed Weibull Log-Likelihood Functions and their Partials=== The log-likelihood function (without the constant) is composed of three summation portions: :: $\begin{ali…'$

2011-10-29T13:18:43Z

Created page with '=== Mixed Weibull Log-Likelihood Functions and their Partials=== The log-likelihood function (without the constant) is composed of three summation portions: ::<math>\begin{ali…'

New page

=== Mixed Weibull Log-Likelihood Functions and their Partials===
The log-likelihood function (without the constant) is composed of three summation portions:
::<math>\begin{align}
\frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( \ln ({{T}_{i}})-{\mu }' \right)}^{2}}}{\sigma _{{{T}'}}^{3}}-\frac{1}{{{\sigma }_{{{T}'}}}} \right) \\
& +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)} \\
& -\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\frac{\left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{\Phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}
\end{align}</math>

::<math>\begin{align}
\ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}\frac{{{\beta }_{k}}}{{{\eta }_{k}}}{{\left( \frac{{{T}_{i}}}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}-1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right] \\
& \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ \underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}{{e}^{-{{\left( \tfrac{T_{i}^{\prime }}{{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right] \\
& \text{ }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ \underset{k=1}{\overset{Q}{\mathop \sum }}\,{{\rho }_{k}}\frac{{{\beta }_{k}}}{{{\eta }_{k}}}{{\left( \frac{T_{Li}^{\prime \prime }+T_{Ri}^{\prime \prime }}{2{{\eta }_{k}}} \right)}^{{{\beta }_{k}}-1}}{{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }+T_{Ri}^{\prime \prime }}{2{{\eta }_{k}}} \right)}^{{{\beta }_{k}}}}}} \right]
\end{align}</math>

:where:
::• <math>{{F}_{e}}</math> is the number of groups of times-to-failure data points
::• <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group
::• <math>Q</math> is the number of subpopulations
::• <math>{{\rho }_{k}}</math> is the proportionality of the <math>{{k}^{th}}</math> subpopulation (unknown a priori, the first set of three sets of parameters to be found)
::• <math>{{\beta }_{k}}</math> is the Weibull shape parameter of the <math>{{k}^{th}}</math> subpopulation (unknown a priori, the second set of three sets of parameters to be found)
::• <math>{{\eta }_{k}}</math> is the Weibull scale parameter (unknown a priori, the third set of three sets of parameters to be found)
::• <math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data
::• <math>S</math> is the number of groups of suspension data points
::• <math>N_{i}^{\prime }</math> is the number of suspensions in <math>{{i}^{th}}</math> group of suspension data points
::• <math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group
::• <math>FI</math> is the number of groups of interval data points
::• <math>N_{i}^{\prime \prime }</math> is the number of intervals in <math>{{i}^{th}}</math> group of data intervals
::• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval
::• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval

The solution will be found by solving for a group of parameters:

::<math>\left( \widehat{{{\rho }_{1,}}}\widehat{{{\beta }_{1}}},\widehat{{{\eta }_{1}}},\widehat{{{\rho }_{2,}}}\widehat{{{\beta }_{2}}},\widehat{{{\eta }_{2}}},...,\widehat{{{\rho }_{Q,}}}\widehat{{{\beta }_{Q}}},\widehat{{{\eta }_{Q}}} \right)</math>

:so that:

::<math>\begin{align}
\frac{\partial \Lambda }{\partial {{\rho }_{1}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{1}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{1}}}=0 \\
\frac{\partial \Lambda }{\partial {{\rho }_{2}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{2}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{2}}}=0 \\
\vdots \\
\frac{\partial \Lambda }{\partial {{\rho }_{Q-1}}}= & 0,\frac{\partial \Lambda }{\partial {{\beta }_{Q-1}}}=0,\frac{\partial \Lambda }{\partial {{\eta }_{Q-1}}}=0 \\
\frac{\partial \Lambda }{\partial {{\beta }_{Q}}}= & 0,\text{ and }\frac{\partial \Lambda }{\partial {{\eta }_{Q}}}=0
\end{align}</math>
<br>

Mixed Weibull Log-Likelihood Functions and their Partials - Revision history

Lisa Hacker: Redirected page to Appendix: Log-Likelihood Equations

Pantelis: Created page with '=== Mixed Weibull Log-Likelihood Functions and their Partials=== The log-likelihood function (without the constant) is composed of three summation portions: ::\begin{ali…'

Pantelis: Created page with '=== Mixed Weibull Log-Likelihood Functions and their Partials=== The log-likelihood function (without the constant) is composed of three summation portions: :: $\begin{ali…'$