Template:Erying-log - Revision history

Richard House: Redirected page to Eyring Relationship#Eyring-Lognormal

2012-08-16T23:31:25Z

Redirected page to Eyring Relationship#Eyring-Lognormal

← Older revision		Revision as of 23:31, 16 August 2012
Line 1:		Line 1:
	==Eyring-Lognormal==		#REDIRECT [[Eyring_Relationship#Eyring-Lognormal]]
	~~<br>~~
	~~The <math>pdf</math> of the lognormal distribution is given by:~~

	~~<br>~~
	~~::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>~~

	~~<br>~~
	~~where:~~

	~~<br>~~
	~~::<math>{T}'=\ln (T)</math>~~

	~~<br>~~
	~~::<math>T=\text{times-to-failure}</math>~~

	~~<br>~~
	~~and~~
	~~<br>~~
	~~• <math>\overline{{{T}'}}=</math> mean of the natural logarithms of the times-to-failure.~~

	~~• <math>{{\sigma }_{{{T}'}}}=</math> standard deviation of the natural logarithms of the times-to-failure.~~

	~~<br>~~
	~~The Eyring-lognormal model can be obtained first by setting <math>\breve{T}=L(V)</math>:~~

	~~<br>~~
	~~::<math>\breve{T}=L(V)=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}}</math>~~

	~~<br>~~
	~~or:~~
	~~<br>~~
	~~<br>~~
	~~::<math>{{e}^{{{\overline{T}}^{\prime }}}}=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}}</math>~~

	~~<br>~~
	~~Thus:~~
	~~<br>~~
	~~<br>~~
	~~::<math>{{\overline{T}}^{\prime }}=-\ln (V)-A+\frac{B}{V}</math>~~


	~~<br>~~
	~~Substituting this into the lognormal <math>pdf</math> yields the Eyring-lognormal model <math>pdf</math>:~~


	~~::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>~~

	~~<br>~~
	~~{{eyring-log stat prop sum}}~~

	~~===Parameter Estimation===~~

	~~{{eyring-log mle}}~~

Athanasios Gerokostopoulos: /* Eyring-Lognormal */

2012-02-27T23:51:35Z

Eyring-Lognormal

← Older revision		Revision as of 23:51, 27 February 2012
Line 43:		Line 43:
	<br>		<br>
	Substituting this into the lognormal <math>pdf</math> yields the Eyring-lognormal model <math>pdf</math>:		Substituting this into the lognormal <math>pdf</math> yields the Eyring-lognormal model <math>pdf</math>:


	::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>		::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>

Athanasios Gerokostopoulos: /* Eyring-Lognormal */

2012-02-27T23:50:51Z

Eyring-Lognormal

← Older revision		Revision as of 23:50, 27 February 2012
Line 29:		Line 29:

	<br>		<br>
	:or:		or:
	<br>		<br>
	<br>		<br>
Line 35:		Line 35:

	<br>		<br>
	:Thus:		Thus:
	<br>		<br>
	<br>		<br>

Athanasios Gerokostopoulos: /* Eyring-Lognormal */

2012-02-15T21:00:20Z

Eyring-Lognormal

← Older revision		Revision as of 21:00, 15 February 2012
Line 7:		Line 7:

	<br>		<br>
	:where:		where:

	<br>		<br>

Athanasios Gerokostopoulos: /* Eyring-Lognormal */

2012-02-14T23:07:53Z

Eyring-Lognormal

← Older revision		Revision as of 23:07, 14 February 2012
Line 42:		Line 42:

	<br>		<br>
	Substituting this into the lognormal <math>pdf</math> yields the Eyring-lognormal model <math>pdf</math>		Substituting this into the lognormal <math>pdf</math> yields the Eyring-lognormal model <math>pdf</math>:
	~~:or:~~

	::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>		::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>

Athanasios Gerokostopoulos: /* Eyring-Lognormal */

2012-02-14T23:07:18Z

Eyring-Lognormal

← Older revision		Revision as of 23:07, 14 February 2012
Line 16:		Line 16:

	<br>		<br>
	:and:		and
	<br>		<br>
	• <math>\overline{{{T}'}}=</math> mean of the natural logarithms of the times-to-failure.		• <math>\overline{{{T}'}}=</math> mean of the natural logarithms of the times-to-failure.
Line 23:		Line 23:

	<br>		<br>
	The Eyring-lognormal model can be obtained first by setting <math>\breve{T}=L(V)</math> ~~in Eqn. (eyring). Therefore~~:		The Eyring-lognormal model can be obtained first by setting <math>\breve{T}=L(V)</math>:

	<br>		<br>
Line 42:		Line 42:

	<br>		<br>
	Substituting ~~Eqn. (eyr-logn-mean)~~ into ~~Eqn. (Eyr-logn-~~pdf) yields the Eyring-lognormal model <math>pdf</math>		Substituting this into the lognormal <math>pdf</math> yields the Eyring-lognormal model <math>pdf</math>
	:or:		:or:

Nicolette Young: /* Maximum Likelihood Estimation Method */

2012-01-12T23:06:55Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 23:06, 12 January 2012
Line 52:		Line 52:
	===Parameter Estimation===		===Parameter Estimation===

	~~====Maximum Likelihood Estimation Method====~~		{{eyring-log mle}}
	~~<br>~~
	~~The complete Eyring-lognormal 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[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}\phi \left( \frac{\ln \left( {{T}_{i}} \right)+\ln ({{V}_{i}})+A~~-~~\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] \\~~
	& & \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \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_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }+\ln {{V}_{i}}+A-\tfrac{B}{{{V}_{i}}}}{\sigma _{T}^{\prime }}</math>~~

	~~<br>~~
	~~::<math>z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }+\ln {{V}_{i}}+A-\tfrac{B}{{{V}_{i}}}}{\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>{{\sigma }_{{{T}'}}}</math> is the standard deviation of the natural logarithm of the times-to-failure (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>C</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.~~
	The solution (parameter estimates) will be found by solving for <math>{{\widehat{\sigma }}_{{{T}'}}},</math> <math>\widehat{A},</math> <math>\widehat{B}</math> so that <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0,</math> <math>\tfrac{\partial \Lambda }{\partial A}=0</math> and <math>\tfrac{\partial \Lambda }{\partial B}=0</math> :

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& \frac{\partial \Lambda }{\partial A}= & -\frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})+\ln ({{V}_{i}})+A-\frac{B}{{{V}_{i}}}) \\~~
	& & -\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \\
	& & \overset{FI}{\mathop{+\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))}
	~~\end{align}</math>~~


	~~<br>~~
	~~::<math>\begin{align}~~
	& \frac{\partial \Lambda }{\partial B}= & \frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\frac{1}{{{V}_{i}}}(\ln ({{T}_{i}})+\ln ({{V}_{i}})+A-\frac{B}{{{V}_{i}}}) \\
	& & +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{{{V}_{i}}}\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \\
	& & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }{{V}_{i}}(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\
	& \frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( \ln ({{T}_{i}})+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}} \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)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \\
	& & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{z_{Ri}^{\prime \prime }\varphi (z_{Ri}^{\prime \prime })-z_{Li}^{\prime \prime }\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))}
	~~\end{align}</math>~~

	~~<br>~~
	~~:and:~~

	~~<br>~~
	~~::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}</math>~~

	~~<br>~~
	~~::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}~~}}~~dt</math>~~

Nicolette Young: /* Eyring-Lognormal Statistical Properties Summary */

2012-01-12T23:00:55Z

Eyring-Lognormal Statistical Properties Summary

@@ Line 48: / Line 48: @@
 <br>
-===Eyring-Lognormal Statistical Properties Summary===
+{{eyring-log stat prop sum}}
 ===Parameter Estimation===

Nicolette Young: Created page with '==Eyring-Lognormal==
The $pdf$ of the lognormal distribution is given by:
:: $f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}…'$

2012-01-12T23:00:04Z

Created page with '==Eyring-Lognormal== The <math>pdf</math> of the lognormal distribution is given by: ::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}…'

New page

==Eyring-Lognormal==
 
The <math>pdf</math> of the lognormal distribution is given by:

 
::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>

 
:where:

 
::<math>{T}'=\ln (T)</math>

 
::<math>T=\text{times-to-failure}</math>

 
:and:
 
• <math>\overline{{{T}'}}=</math> mean of the natural logarithms of the times-to-failure.

• <math>{{\sigma }_{{{T}'}}}=</math> standard deviation of the natural logarithms of the times-to-failure.

 
The Eyring-lognormal model can be obtained first by setting <math>\breve{T}=L(V)</math> in Eqn. (eyring). Therefore:

 
::<math>\breve{T}=L(V)=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}}</math>

 
:or:
 
 
::<math>{{e}^{{{\overline{T}}^{\prime }}}}=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}}</math>

 
:Thus:
 
 
::<math>{{\overline{T}}^{\prime }}=-\ln (V)-A+\frac{B}{V}</math>

 
Substituting Eqn. (eyr-logn-mean) into Eqn. (Eyr-logn-pdf) yields the Eyring-lognormal model <math>pdf</math>
:or:

::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>

 
===Eyring-Lognormal Statistical Properties Summary===

====The Mean====
 
• The mean life of the Eyring-lognormal model (mean of the times-to-failure), <math>\bar{T}</math> , is given by:
 

::<math>\begin{align}
& \bar{T}= & {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} \\
& = & {{e}^{-\ln (V)-A+\tfrac{B}{V}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}
\end{align}</math>

 
The mean of the natural logarithms of the times-to-failure, <math>{{\bar{T}}^{^{\prime }}}</math> , in terms of <math>\bar{T}</math> and <math>{{\sigma }_{T}}</math> is given by:
 

 
::<math>{{\bar{T}}^{\prime }}=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)</math>
 

====The Median====
 
The median of the Eyring-lognormal model is given by:

 
::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}</math>

 

====The Standard Deviation====
 
• The standard deviation of the Eyring-lognormal model (standard deviation of the times-to-failure), <math>{{\sigma }_{T}}</math> , is given by:

 
::<math>\begin{align}
& {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} \\
& = & \sqrt{\left( {{e}^{2\left( -\ln (V)-A+\tfrac{B}{V} \right)+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)}
\end{align}</math>

 
• The standard deviation of the natural logarithms of the times-to-failure, <math>{{\sigma }_{{{T}'}}}</math> , in terms of <math>\bar{T}</math> and <math>{{\sigma }_{T}}</math> is given by:

 
::<math>{{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}</math>

====The Mode====
 
• The mode of the Eyring-lognormal model is given by:

 
::<math>\begin{align}
& \tilde{T}= & {{e}^{{{\overline{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}} \\
& = & {{e}^{-\ln (V)-A+\tfrac{B}{V}-\sigma _{{{T}'}}^{2}}}
\end{align}</math>

====Eyring-Lognormal Reliability Function====
 
The reliability for a mission of time <math>T</math> , starting at age 0, for the Eyring-lognormal model is determined by:

 
::<math>R(T,V)=\mathop{}_{T}^{\infty }f(t,V)dt</math>

 
:or:

 
::<math>R(T,V)=\mathop{}_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt</math>

 
There is no closed form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability we will not discuss manual solution methods.

====Reliable Life====
 
For the Eyring-lognormal model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}},</math> is estimated by first solving the reliability equation with respect to time, as follows:

 
::<math>T_{R}^{\prime }=-\ln (V)-A+\frac{B}{V}+z\cdot {{\sigma }_{{{T}'}}}</math>

 
:where:

 
::<math>z={{\Phi }^{-1}}\left[ F\left( T_{R}^{\prime },V \right) \right]</math>

 
:and:

 
::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}',V)}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>
 
 
Since <math>{T}'=\ln (T)</math> the reliable life, <math>{{t}_{R,}}</math> is given by:

 
::<math>{{t}_{R}}={{e}^{T_{R}^{\prime }}}</math>

====Eyring-Lognormal Failure Rate====

 
The Eyring-lognormal failure rate is given by:

::<math>\lambda (T,V)=\frac{f(T,V)}{R(T,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\mathop{}_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}</math>

===Parameter Estimation===

====Maximum Likelihood Estimation Method====
 
The complete Eyring-lognormal log-likelihood function is composed of two summation portions:

 
::<math>\begin{align}
& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}\phi \left( \frac{\ln \left( {{T}_{i}} \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] \\
& & \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \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>

 
:where:

 
::<math>z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }+\ln {{V}_{i}}+A-\tfrac{B}{{{V}_{i}}}}{\sigma _{T}^{\prime }}</math>

 
::<math>z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }+\ln {{V}_{i}}+A-\tfrac{B}{{{V}_{i}}}}{\sigma _{T}^{\prime }}</math>

 
:and:
 
• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.
 
• <math>{{N}_{i}}</math> is the number of times-to-failure data points in the <math>{{i}^{th}}</math> time-to-failure data group.
 
• <math>{{\sigma }_{{{T}'}}}</math> is the standard deviation of the natural logarithm of the times-to-failure (unknown, the first of three parameters to be estimated).
 
• <math>A</math> is the Eyring parameter (unknown, the second of three parameters to be estimated).
 
• <math>C</math> is the second Eyring 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{\sigma }}_{{{T}'}}},</math> <math>\widehat{A},</math> <math>\widehat{B}</math> so that <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0,</math> <math>\tfrac{\partial \Lambda }{\partial A}=0</math> and <math>\tfrac{\partial \Lambda }{\partial B}=0</math> :

 
::<math>\begin{align}
& \frac{\partial \Lambda }{\partial A}= & -\frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})+\ln ({{V}_{i}})+A-\frac{B}{{{V}_{i}}}) \\
& & -\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \\
& & \overset{FI}{\mathop{+\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))}
\end{align}</math>

 
::<math>\begin{align}
& \frac{\partial \Lambda }{\partial B}= & \frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\frac{1}{{{V}_{i}}}(\ln ({{T}_{i}})+\ln ({{V}_{i}})+A-\frac{B}{{{V}_{i}}}) \\
& & +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{{{V}_{i}}}\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \\
& & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }{{V}_{i}}(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\
& \frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( \ln ({{T}_{i}})+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}} \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)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln ({{V}_{i}})+A-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \\
& & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{z_{Ri}^{\prime \prime }\varphi (z_{Ri}^{\prime \prime })-z_{Li}^{\prime \prime }\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))}
\end{align}</math>

 
:and:

 
::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}</math>

 
::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>

Template:Erying-log - Revision history

Richard House: Redirected page to Eyring Relationship#Eyring-Lognormal

Athanasios Gerokostopoulos: /* Eyring-Lognormal */

Athanasios Gerokostopoulos: /* Eyring-Lognormal */

Athanasios Gerokostopoulos: /* Eyring-Lognormal */

Athanasios Gerokostopoulos: /* Eyring-Lognormal */

Athanasios Gerokostopoulos: /* Eyring-Lognormal */

Nicolette Young: /* Maximum Likelihood Estimation Method */

Nicolette Young: /* Eyring-Lognormal Statistical Properties Summary */

Nicolette Young: Created page with '==Eyring-Lognormal== The pdf of the lognormal distribution is given by: ::f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}…'

Nicolette Young: Created page with '==Eyring-Lognormal==
The $pdf$ of the lognormal distribution is given by:
:: $f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}…'$