Template:Generalized eyring-log - Revision history

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

2012-08-17T00:48:48Z

Redirected page to Eyring Relationship#Generalized Eyring-Lognormal

← Older revision		Revision as of 00:48, 17 August 2012
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	~~==Generalized Eyring~~-Lognormal==		#REDIRECT [[Eyring_Relationship#Generalized_Eyring-Lognormal]]
	~~<br>~~
	~~By setting <math>\sigma _{T}^{\prime }=L(V,U)</math> to the lognormal <math>pdf</math>, the generalized Erying lognormal model is given by:~~

	~~<br>~~
	~~::<math>f(t,V,U)=\frac{\varphi (z(t))}{\sigma _{T}^{\prime }t}</math>~~

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

	~~<br>~~
	~~::<math>z(t)=\frac{\ln t-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}+\ln (V)}{\sigma _{T}^{\prime }}</math>~~


	~~<br>~~
	~~{{gen-eyring log rf}}~~

	~~{{gen-eyring log par est}}~~

Athanasios Gerokostopoulos: /* Generalized Eyring-Lognormal */

2012-02-27T23:57:26Z

Generalized Eyring-Lognormal

← Older revision		Revision as of 23:57, 27 February 2012
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	::<math>z(t)=\frac{\ln t-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}+\ln (V)}{\sigma _{T}^{\prime }}</math>		::<math>z(t)=\frac{\ln t-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}+\ln (V)}{\sigma _{T}^{\prime }}</math>


			<br>
	{{gen-eyring log rf}}		{{gen-eyring log rf}}

	{{gen-eyring log par est}}		{{gen-eyring log par est}}

Athanasios Gerokostopoulos: /* Generalized Eyring-Lognormal */

2012-02-15T01:23:23Z

Generalized Eyring-Lognormal

← Older revision		Revision as of 01:23, 15 February 2012
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	===Generalized Eyring-Lognormal===		==Generalized Eyring-Lognormal==
	<br>		<br>
	By setting <math>\sigma _{T}^{\prime }=L(V,U)</math> to the lognormal <math>pdf</math>, the generalized Erying lognormal model is given by:		By setting <math>\sigma _{T}^{\prime }=L(V,U)</math> to the lognormal <math>pdf</math>, the generalized Erying lognormal model is given by:

Athanasios Gerokostopoulos: /* Generalized Eyring-Lognormal */

2012-02-15T00:14:40Z

Generalized Eyring-Lognormal

← Older revision		Revision as of 00:14, 15 February 2012
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	<br>		<br>
	:where:		where:

	<br>		<br>
	::<math></math>		::<math>z(t)=\frac{\ln t-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}+\ln (V)}{\sigma _{T}^{\prime }}</math>

	{{gen-eyring log rf}}		{{gen-eyring log rf}}

	{{gen-eyring log par est}}		{{gen-eyring log par est}}

Athanasios Gerokostopoulos: /* Generalized Eyring-Lognormal */

2012-02-15T00:08:44Z

Generalized Eyring-Lognormal

← Older revision		Revision as of 00:08, 15 February 2012
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	===Generalized Eyring-Lognormal===		===Generalized Eyring-Lognormal===
	<br>		<br>
	By setting <math>\sigma _{T}^{\prime }=L(V,U)</math> ~~from Eqn. (Gen-Eyr)~~, the generalized Erying lognormal model is given by:		By setting <math>\sigma _{T}^{\prime }=L(V,U)</math> to the lognormal <math>pdf</math>, the generalized Erying lognormal model is given by:

	<br>		<br>

Nicolette Young: /* Parameter Estimation */

2012-01-12T23:31:38Z

Parameter Estimation

← Older revision		Revision as of 23:31, 12 January 2012
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	{{gen-eyring log rf}}		{{gen-eyring log rf}}

	~~====Parameter Estimation====~~		{{gen-eyring log par est}}

	~~<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.~~
	~~<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.~~
	~~• <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>~~

Nicolette Young: /* Generalized Eyring-Lognormal Reliability Function */

2012-01-12T23:31:05Z

Generalized Eyring-Lognormal Reliability Function

← Older revision		Revision as of 23:31, 12 January 2012
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	::<math></math>		::<math></math>

	~~====Generalized Eyring~~-~~Lognormal Reliability Function====~~		{{gen-eyring log rf}}
	~~<br>~~

	~~The generalized Erying lognormal reliability function is given by:~~

	~~<br>~~

	~~::<math>R(T,V,U)=1-\Phi (z)</math>~~

	====Parameter Estimation====		====Parameter Estimation====

Nicolette Young: Created page with '===Generalized Eyring-Lognormal===
By setting $\sigma _{T}^{\prime }=L(V,U)$ from Eqn. (Gen-Eyr), the generalized Erying lognormal model is given by:
::
2012-01-12T23:12:10Z

Created page with '===Generalized Eyring-Lognormal=== By setting <math>\sigma _{T}^{\prime }=L(V,U)</math> from Eqn. (Gen-Eyr), the generalized Erying lognormal model is given by: ::<m…'

New page
===Generalized Eyring-Lognormal===
 
By setting <math>\sigma _{T}^{\prime }=L(V,U)</math> from Eqn. (Gen-Eyr), the generalized Erying lognormal model is given by:

 
::<math>f(t,V,U)=\frac{\varphi (z(t))}{\sigma _{T}^{\prime }t}</math>

 
:where:

 
::<math></math>

====Generalized Eyring-Lognormal Reliability Function====
 

The generalized Erying lognormal reliability function is given by:

 

::<math>R(T,V,U)=1-\Phi (z)</math>

====Parameter Estimation====

 
Substituting the generalized Eyring model into the lognormal log-likelihood equation yields:

 
::<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>

 
:where:

 
::<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>

 
::<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>

 
: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>A,B,C,D</math> are parameters to be estimated.
 
• <math>{{V}_{i}}</math> is the temperature level of the <math>{{i}^{th}}</math> group.
 
• <math>{{U}_{i}}</math> is the non-thermal 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 <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.
 
 
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> .