Template:Aae - Revision history

Lisa Hacker: Redirected page to Arrhenius Relationship#Arrhenius-Exponential

2012-08-15T23:57:53Z

Redirected page to Arrhenius Relationship#Arrhenius-Exponential

← Older revision		Revision as of 23:57, 15 August 2012
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	~~==Arrhenius-Exponential==~~		#REDIRECT [[Arrhenius_Relationship#Arrhenius-Exponential]]
	~~<br>~~
	~~The <math>pdf</math> of the 1-parameter exponential distribution is given by:~~

	~~<br>~~
	~~::<math>f(t)=\lambda {{e}^{-\lambda t}}</math>~~

	~~<br>~~
	~~It can be easily shown that the mean life for the 1-parameter exponential distribution (presented in detail~~ [[~~Distributions used in Accelerated Testing~~#~~The~~ Exponential ~~Distribution\|here~~]]~~) is given by:~~


	~~<br>~~
	~~::<math>\lambda =\frac{1}{m}</math>~~

	~~<br>~~
	~~thus:~~


	~~<br>~~
	~~::<math>f(t)=\frac{1}{m}{{e}^{-\tfrac{t}{m}}}</math>~~


	~~<br>~~
	~~The Arrhenius-exponential model <math>pdf</math> can then be obtained by setting <math>m=L(V)</math>:~~
	~~<br>~~
	~~Therefore:~~

	~~<br>~~
	~~::<math>m=L(V)=C{{e}^{\tfrac{B}{V}}}</math>~~

	~~<br>~~
	~~Substituting for <math>m</math> yields a <math>pdf</math> that is both a function of time and stress or:~~


	~~<br>~~
	~~::<math>f(t,V)=\frac{1}{C{{e}^{\tfrac{B}{V}}}}\cdot {{e}^{-\tfrac{1}{C{{e}^{\tfrac{B}{V}}}}\cdot t}}</math>~~

	~~{{aae stat prop sum}}~~

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

	~~{{aae mle}}~~

Athanasios Gerokostopoulos: /* Arrhenius-Exponential */

2012-02-14T22:31:45Z

Arrhenius-Exponential

← Older revision		Revision as of 22:31, 14 February 2012
Line 22:		Line 22:

	<br>		<br>
	The Arrhenius-exponential model <math>pdf</math> can then be obtained by setting <math>m=L(V)</math> ~~in the above equation.~~		The Arrhenius-exponential model <math>pdf</math> can then be obtained by setting <math>m=L(V)</math>:
	<br>		<br>
	Therefore:		Therefore:

Athanasios Gerokostopoulos: /* Arrhenius-Exponential */

2012-02-13T23:14:22Z

Arrhenius-Exponential

← Older revision		Revision as of 23:14, 13 February 2012
Line 7:		Line 7:

	<br>		<br>
	It can be easily shown that the mean life for the 1-parameter exponential distribution is given by:		It can be easily shown that the mean life for the 1-parameter exponential distribution (presented in detail [[Distributions used in Accelerated Testing#The Exponential Distribution\|here]]) is given by:

Athanasios Gerokostopoulos: /* Arrhenius-Exponential */

2012-02-13T22:26:43Z

Arrhenius-Exponential

← Older revision		Revision as of 22:26, 13 February 2012
Line 7:		Line 7:

	<br>		<br>
	It can be easily shown that the mean life for the 1-parameter exponential distribution ~~(presented in detail in Chapter 5)~~ is given by:		It can be easily shown that the mean life for the 1-parameter exponential distribution is given by:


Line 22:		Line 22:

	<br>		<br>
	The Arrhenius-exponential model <math>pdf</math> can then be obtained by setting <math>m=L(V)</math> in ~~Eqn. (arrhenius)~~.		The Arrhenius-exponential model <math>pdf</math> can then be obtained by setting <math>m=L(V)</math> in the above equation.
	<br>		<br>
	Therefore:		Therefore:
Line 30:		Line 30:

	<br>		<br>
	Substituting for <math>m</math> ~~in Eqn. (pdfexpm)~~ yields a <math>pdf</math> that is both a function of time and stress or:		Substituting for <math>m</math> yields a <math>pdf</math> that is both a function of time and stress or:

Nicolette Young: /* Maximum Likelihood Estimation Method */

2012-01-12T19:00:24Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 19:00, 12 January 2012
Line 40:		Line 40:
	===Parameter Estimation===		===Parameter Estimation===

	~~====Maximum Likelihood Estimation Method====~~		{{aae mle}}
	~~<br>~~
	~~The log-likelihood function for the exponential distribution is as shown next:~~

	~~<br>~~
	~~::<math>\begin~~{~~align}~~
	& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime } \\
	~~& & \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>R_{Li}^{\prime \prime }={{e}^{-\lambda T_{Li}^{\prime \prime }}}</math>~~


	~~<br>~~
	~~::<math>R_{Ri}^{\prime \prime }={{e}^{-\lambda T_{Ri}^{\prime \prime }}}</math>~~

	~~<br>~~
	~~and:~~
	~~<br>~~
	~~• <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 in the <math>{{i}^{th}}</math> time-to-failure data group.~~

	~~• <math>\lambda </math> is the failure rate parameter (unknown).~~

	~~• <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 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.~~
	~~<br>~~
	~~Substituting the Arrhenius-exponential model into the log-likelihood function yields:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	& \Lambda = & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{e}^{-\tfrac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{T}_{i}}}} \right] \\
	& & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}T_{i}^{\prime }+\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}^{-\tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}}}</math>~~

	~~<br>~~
	~~::<math>R_{Ri}^{\prime \prime }={{e}^{-\tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}}}</math>~~


	~~<br>~~
	The solution (parameter estimates) will be found by solving for the parameters <math>\widehat{B},</math> <math>\widehat{C}</math> so that <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 B}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}-\frac{C}{{{V}_{i}}} \right)+\frac{1}{C}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}} \\
	& & \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime })C{{V}_{i}}{{e}^{\tfrac{B}~~{{{V}_{i}}}}}}~~
	~~\end{align}</math>~~


	~~<br>~~
	~~::<math>\begin{align}~~
	& \frac{\partial \Lambda }{\partial C}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}-1 \right)+\frac{1}{{{C}^{2}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{e}^{\tfrac{B}{{{V}_{i}}}}}} \\
	& & \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }){{C}^{2}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
	~~\end{align~~}~~</math>~~

	~~<br>~~

Nicolette Young: /* Arrhenius-Exponential Statistical Properties Summary */

2012-01-12T18:59:43Z

Arrhenius-Exponential Statistical Properties Summary

@@ Line 36: / Line 36: @@
 ::<math>f(t,V)=\frac{1}{C{{e}^{\tfrac{B}{V}}}}\cdot {{e}^{-\tfrac{1}{C{{e}^{\tfrac{B}{V}}}}\cdot t}}</math>
-===Arrhenius-Exponential Statistical Properties Summary===
+{{aae stat prop sum}}
 ===Parameter Estimation===

Nicolette Young: Created page with '==Arrhenius-Exponential==
The $pdf$ of the 1-parameter exponential distribution is given by:
:: $f(t)=\lambda {{e}^{-\lambda t}}$
It can b…'

2012-01-12T18:59:13Z

Created page with '==Arrhenius-Exponential== The <math>pdf</math> of the 1-parameter exponential distribution is given by: ::<math>f(t)=\lambda {{e}^{-\lambda t}}</math> It can b…'

New page

==Arrhenius-Exponential==
 
The <math>pdf</math> of the 1-parameter exponential distribution is given by:

 
::<math>f(t)=\lambda {{e}^{-\lambda t}}</math>

 
It can be easily shown that the mean life for the 1-parameter exponential distribution (presented in detail in Chapter 5) is given by:

 
::<math>\lambda =\frac{1}{m}</math>

 
thus:

 
::<math>f(t)=\frac{1}{m}{{e}^{-\tfrac{t}{m}}}</math>

 
The Arrhenius-exponential model <math>pdf</math> can then be obtained by setting <math>m=L(V)</math> in Eqn. (arrhenius).
 
Therefore:

 
::<math>m=L(V)=C{{e}^{\tfrac{B}{V}}}</math>

 
Substituting for <math>m</math> in Eqn. (pdfexpm) yields a <math>pdf</math> that is both a function of time and stress or:

 
::<math>f(t,V)=\frac{1}{C{{e}^{\tfrac{B}{V}}}}\cdot {{e}^{-\tfrac{1}{C{{e}^{\tfrac{B}{V}}}}\cdot t}}</math>

===Arrhenius-Exponential Statistical Properties Summary===
 
====Mean or MTTF====
 
The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) of the Arrhenius-exponential is given by,

 
::<math>\begin{align}
& \overline{T}= & \mathop{}_{0}^{\infty }t\cdot f(t,V)dt=\mathop{}_{0}^{\infty }t\cdot \frac{1}{C{{e}^{\tfrac{B}{V}}}}{{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V}}}}}}dt \\
& = & C{{e}^{\tfrac{B}{V}}}
\end{align}</math>

 
====Median====
 
The median, <math>breve{T},</math>
of the Arrhenius-exponential model is given by:

 
::<math>\breve{T}=0.693\cdot C{{e}^{\tfrac{B}{V}}}</math>

 

====Mode====
 
The mode, <math>\tilde{T},</math>
of the Arrhenius-exponential model is given by:

 
::<math>\tilde{T}=0</math>

 

====Standard Deviation====
The standard deviation, <math>{{\sigma }_{T}}</math> , of the Arrhenius-exponential model is given by:

 
::<math>{{\sigma }_{T}}=C{{e}^{\tfrac{B}{V}}}</math>

====Arrhenius-Exponential Reliability Function====
 

The Arrhenius-exponential reliability function is given by:

 
::<math>R(T,V)={{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}</math>

 
This function is the complement of the Arrhenius-exponential cumulative distribution function or:

 
::<math>R(T,V)=1-Q(T,V)=1-\mathop{}_{0}^{T}f(T,V)dT</math>

 
and:

 
::<math>R(T,V)=1-\mathop{}_{0}^{T}\frac{1}{C{{e}^{\tfrac{B}{V}}}}{{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}dT={{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}</math>

 

====Conditional Reliability====
The Arrhenius-exponential conditional reliability function is given by,

::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V}}}}}}</math>

====Reliable Life====
 
For the Arrhenius-exponential model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}},</math> is given by:

 
::<math>R({{t}_{R}},V)={{e}^{-\tfrac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}}}</math>

 
::<math>\ln [R({{t}_{R}},V)]=-\frac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}</math>

 
or:

 
::<math>{{t}_{R}}=-C{{e}^{\tfrac{B}{V}}}\ln [R({{t}_{R}},V)]</math>

===Parameter Estimation===

====Maximum Likelihood Estimation Method====
 
The log-likelihood function for the exponential distribution is as shown next:

 
::<math>\begin{align}
& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime } \\
& & \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>R_{Li}^{\prime \prime }={{e}^{-\lambda T_{Li}^{\prime \prime }}}</math>

 
::<math>R_{Ri}^{\prime \prime }={{e}^{-\lambda T_{Ri}^{\prime \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 in the <math>{{i}^{th}}</math> time-to-failure data group.

• <math>\lambda </math> is the failure rate parameter (unknown).

• <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 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.
 
Substituting the Arrhenius-exponential model into the log-likelihood function yields:

 
::<math>\begin{align}
& \Lambda = & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{e}^{-\tfrac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{T}_{i}}}} \right] \\
& & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}T_{i}^{\prime }+\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>

 
where:

 
::<math>R_{Li}^{\prime \prime }={{e}^{-\tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}}}</math>

 
::<math>R_{Ri}^{\prime \prime }={{e}^{-\tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}}}</math>

 
The solution (parameter estimates) will be found by solving for the parameters <math>\widehat{B},</math> <math>\widehat{C}</math> so that <math>\tfrac{\partial \Lambda }{\partial B}=0</math> and <math>\tfrac{\partial \Lambda }{\partial C}=0</math> , where:

 
::<math>\begin{align}
& \frac{\partial \Lambda }{\partial B}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}-\frac{C}{{{V}_{i}}} \right)+\frac{1}{C}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}} \\
& & \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime })C{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
\end{align}</math>

 
::<math>\begin{align}
& \frac{\partial \Lambda }{\partial C}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}-1 \right)+\frac{1}{{{C}^{2}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{e}^{\tfrac{B}{{{V}_{i}}}}}} \\
& & \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }){{C}^{2}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
\end{align}</math>

Template:Aae - Revision history

Lisa Hacker: Redirected page to Arrhenius Relationship#Arrhenius-Exponential

Athanasios Gerokostopoulos: /* Arrhenius-Exponential */

Athanasios Gerokostopoulos: /* Arrhenius-Exponential */

Athanasios Gerokostopoulos: /* Arrhenius-Exponential */

Nicolette Young: /* Maximum Likelihood Estimation Method */

Nicolette Young: /* Arrhenius-Exponential Statistical Properties Summary */

Nicolette Young: Created page with '==Arrhenius-Exponential== The pdf of the 1-parameter exponential distribution is given by: ::f(t)=\lambda {{e}^{-\lambda t}} It can b…'

Nicolette Young: Created page with '==Arrhenius-Exponential==
The $pdf$ of the 1-parameter exponential distribution is given by:
:: $f(t)=\lambda {{e}^{-\lambda t}}$
It can b…'