Template:Alta a-e.e-e - Revision history

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

2012-08-16T22:58:14Z

Redirected page to Eyring Relationship#Eyring-Exponential

← Older revision		Revision as of 22:58, 16 August 2012
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	~~==Eyring-Exponential==~~		#REDIRECT [[Eyring_Relationship#Eyring-Exponential]]

	~~<br>~~
	~~The <math>pdf</math> of the 1-parameter exponential distribution is given by:~~

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

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

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

	~~<br>~~
	~~and substituting for <math>m</math> in the exponential <math>pdf</math> equation:~~

	~~<br>~~
	~~::<math>f(t,V)=V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}\cdot t}}</math>~~
	~~<br>~~
	~~{{eyring-ex stat prop sum}}~~

	~~===Parameter Estimation===~~
	~~<br>~~
	~~{{eyring-ex mle}}~~

Chris Kahn: /* Eyring-Exponential */

2012-03-09T17:55:31Z

Eyring-Exponential

← Older revision		Revision as of 17:55, 9 March 2012
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	<br>		<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:		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>		<br>

Athanasios Gerokostopoulos: /* Eyring-Exponential */

2012-02-27T23:40:58Z

Eyring-Exponential

← Older revision		Revision as of 23:40, 27 February 2012
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	<br>		<br>
	:thus:		thus:

	<br>		<br>

Athanasios Gerokostopoulos: /* Eyring-Exponential */

2012-02-14T22:33:46Z

Eyring-Exponential

← Older revision		Revision as of 22:33, 14 February 2012
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	<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 (presented in detail [[Distributions used in Accelerated Testing#The Exponential Distribution\|here]]) is given by:

	<br>		<br>
Line 20:		Line 20:

	<br>		<br>
	The Eyring-exponential model <math>pdf</math> can then be obtained by setting <math>m=L(V)</math> ~~in Eqn. (eyring)~~:		The Eyring-exponential model <math>pdf</math> can then be obtained by setting <math>m=L(V)</math>:

	<br>		<br>
Line 26:		Line 26:

	<br>		<br>
	and substituting for <math>m</math> in ~~Eqn. (pdfexpm2)~~:		and substituting for <math>m</math> in the exponential <math>pdf</math> equation:

	<br>		<br>

Nicolette Young: /* Maximum Likelihood Estimation Method */

2012-01-12T22:39:01Z

Maximum Likelihood Estimation Method

@@ Line 35: / Line 35: @@
 ===Parameter Estimation===
 <br>
-====Maximum Likelihood Estimation Method====
+{{eyring-ex mle}}

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

2012-01-12T22:33:14Z

Eyring-Exponential Statistical Properties Summary

@@ Line 31: / Line 31: @@
 ::<math>f(t,V)=V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}\cdot t}}</math>
 <br>
-===Eyring-Exponential Statistical Properties Summary===
+{{eyring-ex stat prop sum}}
 ===Parameter Estimation===

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

2012-01-12T22:32:02Z

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

New page

==Eyring-Exponential==

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

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

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

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

 
and substituting for <math>m</math> in Eqn. (pdfexpm2):

 
::<math>f(t,V)=V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}\cdot t}}</math>
 
===Eyring-Exponential Statistical Properties Summary===
====Mean or MTTF====
 
The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) for the Eyring-exponential is given by:

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

====Median====
 

The median, <math>\breve{T},</math>
for the Eyring-exponential model is given by:

 

::<math>\breve{T}=0.693\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>

 

====Mode====
 
The mode, <math>\tilde{T},</math>
for the Eyring-exponential model is <math>\tilde{T}=0.</math>
 
====Standard Deviation====
 
The standard deviation, <math>{{\sigma }_{T}}</math>, for the Eyring-exponential model is given by:
 
::<math>{{\sigma }_{T}}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
 
====Eyring-Exponential Reliability Function====
 
The Eyring-exponential reliability function is given by:

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

 
This function is the complement of the Eyring-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}V{{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}dT={{e}^{-T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>

====Conditional Reliability====
 
The conditional reliability function for the Eyring-exponential model 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}^{-t\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
 
====Reliable Life====
 
For the Eyring-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}^{-{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
 
::<math>\ln [R({{t}_{R}},V)]=-{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}</math>
 
:or:
 
::<math>{{t}_{R}}=-\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}\ln [R({{t}_{R}},V)]</math>

===Parameter Estimation===
 
====Maximum Likelihood Estimation Method====
 

The complete exponential log-likelihood function of the Eyring model is composed of two summation portions:

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

 
::<math>R_{Ri}^{\prime \prime }={{e}^{-T_{Ri}^{\prime \prime }{{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}}}}</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>{{V}_{i}}</math> is the stress level of the <math>{{i}^{th}}</math> group.
 
• <math>A</math> is the Eyring parameter (unknown, the first of two parameters to be estimated).
 
• <math>B</math> is the second Eyring parameter (unknown, the second of two parameters to be estimated).
 
• <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 the parameters <math>\widehat{A}</math> and <math>\widehat{B}</math> so that <math>\tfrac{\partial \Lambda }{\partial A}=0</math> and <math>\tfrac{\partial \Lambda }{\partial B}=0</math> where:

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

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

Template:Alta a-e.e-e - Revision history

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

Chris Kahn: /* Eyring-Exponential */

Athanasios Gerokostopoulos: /* Eyring-Exponential */

Athanasios Gerokostopoulos: /* Eyring-Exponential */

Nicolette Young: /* Maximum Likelihood Estimation Method */

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

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

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