Template:Ipl exponential - Revision history

Richard House: Redirected page to Inverse Power Law (IPL) Relationship#IPL-Exponential

2012-08-15T23:00:01Z

Redirected page to Inverse Power Law (IPL) Relationship#IPL-Exponential

← Older revision		Revision as of 23:00, 15 August 2012
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	~~==IPL-Exponential==~~		#REDIRECT [[Inverse_Power_Law_(IPL)_Relationship#IPL-Exponential]]
	~~<br>~~
	~~The IPL-exponential model can be derived by setting <math>m=L~~(~~V)</math> in the exponential <math>pdf</math>, yielding the following~~ IPL~~-exponential <math>pdf</math> :~~

	~~<br>~~
	~~::<math>f(t,V)=K{{V}^{n}}{{e}^{-K{{V}^{n}}t}}</math>~~

	~~<br>~~
	~~Note that this is a 2-parameter model. The failure rate (the parameter of the exponential distribution~~) ~~of the model is simply <math>\lambda =K{{V}^{n}},</math> and is only a function of stress.~~
	~~<br>~~
	~~[[Image:ALTA8.4.png\|center\|500px\|~~IPL-~~exponential failure rate function at different stress levels.~~]]
	~~<br>~~

	~~<br>~~
	~~{{ipl ex stat prop sum}}~~

	~~{{ipl ex rel function}}~~

	~~===Parameter Estimation===~~
	~~<br>~~
	~~{{ipl ex mle}}~~

Chris Kahn: /* IPL-Exponential */

2012-06-25T10:40:40Z

IPL-Exponential

← Older revision		Revision as of 10:40, 25 June 2012
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	Note that this is a 2-parameter model. The failure rate (the parameter of the exponential distribution) of the model is simply <math>\lambda =K{{V}^{n}},</math> and is only a function of stress.		Note that this is a 2-parameter model. The failure rate (the parameter of the exponential distribution) of the model is simply <math>\lambda =K{{V}^{n}},</math> and is only a function of stress.
	<br>		<br>
	[[Image:ALTA8.4.png\|center\|~~250px~~\|IPL-exponential failure rate function at different stress levels.]]		[[Image:ALTA8.4.png\|center\|500px\|IPL-exponential failure rate function at different stress levels.]]
	<br>		<br>

Chuck Smith: /* IPL-Exponential */

2012-04-26T15:49:27Z

IPL-Exponential

← Older revision		Revision as of 15:49, 26 April 2012
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	Note that this is a 2-parameter model. The failure rate (the parameter of the exponential distribution) of the model is simply <math>\lambda =K{{V}^{n}},</math> and is only a function of stress.		Note that this is a 2-parameter model. The failure rate (the parameter of the exponential distribution) of the model is simply <math>\lambda =K{{V}^{n}},</math> and is only a function of stress.
	<br>		<br>
	[[Image:ALTA8.4.png\|center\|~~500px~~\|IPL-exponential failure rate function at different stress levels.]]		[[Image:ALTA8.4.png\|center\|250px\|IPL-exponential failure rate function at different stress levels.]]
	<br>		<br>

Nicolette Young: /* IPL-Exponential */

2012-03-20T17:51:46Z

IPL-Exponential

← Older revision		Revision as of 17:51, 20 March 2012
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	Note that this is a 2-parameter model. The failure rate (the parameter of the exponential distribution) of the model is simply <math>\lambda =K{{V}^{n}},</math> and is only a function of stress.		Note that this is a 2-parameter model. The failure rate (the parameter of the exponential distribution) of the model is simply <math>\lambda =K{{V}^{n}},</math> and is only a function of stress.
	<br>		<br>
	[[Image:ALTA8.4.~~gif\|thumb~~\|center\|~~300px~~\|IPL-exponential failure rate function at different stress levels.]]		[[Image:ALTA8.4.png\|center\|500px\|IPL-exponential failure rate function at different stress levels.]]
	<br>		<br>

Athanasios Gerokostopoulos: /* IPL-Exponential */

2012-02-15T17:23:04Z

IPL-Exponential

← Older revision		Revision as of 17:23, 15 February 2012
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	==IPL-Exponential==		==IPL-Exponential==
	<br>		<br>
	The IPL-exponential model can be derived by setting <math>m=L(V)</math> in ~~Eqn. (inverse)~~, yielding the following IPL-exponential <math>pdf</math> :		The IPL-exponential model can be derived by setting <math>m=L(V)</math> in the exponential <math>pdf</math>, yielding the following IPL-exponential <math>pdf</math> :

	<br>		<br>

Nicolette Young: /* Maximum Likelihood Parameter Estimation */

2012-01-13T18:00:12Z

Maximum Likelihood Parameter Estimation

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 ===Parameter Estimation===
 <br>
-====Maximum Likelihood Parameter Estimation====
+{{ipl ex mle}}

Nicolette Young: /* IPL-Exponential Reliability Function */

2012-01-13T17:59:19Z

IPL-Exponential Reliability Function

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 {{ipl ex stat prop sum}}
-===IPL-Exponential Reliability Function===
+{{ipl ex rel function}}
 ===Parameter Estimation===

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

2012-01-13T17:57:42Z

IPL-Exponential Statistical Properties Summary

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 <br>
-===IPL-Exponential Statistical Properties Summary===
+{{ipl ex stat prop sum}}
 ===IPL-Exponential Reliability Function===
 <br>

Nicolette Young: Created page with '==IPL-Exponential==
The IPL-exponential model can be derived by setting $m=L(V)$ in Eqn. (inverse), yielding the following IPL-exponential $pdf$ : …'

2012-01-13T17:54:57Z

Created page with '==IPL-Exponential== The IPL-exponential model can be derived by setting <math>m=L(V)</math> in Eqn. (inverse), yielding the following IPL-exponential <math>pdf</math> : …'

New page

==IPL-Exponential==
 
The IPL-exponential model can be derived by setting <math>m=L(V)</math> in Eqn. (inverse), yielding the following IPL-exponential <math>pdf</math> :

 
::<math>f(t,V)=K{{V}^{n}}{{e}^{-K{{V}^{n}}t}}</math>

 
Note that this is a 2-parameter model. The failure rate (the parameter of the exponential distribution) of the model is simply <math>\lambda =K{{V}^{n}},</math> and is only a function of stress.
 
[[Image:ALTA8.4.gif|thumb|center|300px|IPL-exponential failure rate function at different stress levels.]]
 

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

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

 
Note that the MTTF is a function of stress only and is simply equal to the IPL relationship (which is the original assumption), when using the exponential distribution.
 
====Median====
 
The median, <math>\breve{T},</math> for the IPL-exponential model is given by:

 
::<math>\breve{T}=0.693\frac{1}{K{{V}^{n}}}</math>

 

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

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

 

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

 
::<math>{{\sigma }_{T}}=\frac{1}{K{{V}^{n}}}</math>

 
===IPL-Exponential Reliability Function===
 
The IPL-exponential reliability function is given by:

 
::<math>R(T,V)={{e}^{-TK{{V}^{n}}}}</math>

 
This function is the complement of the IPL-exponential cumulative distribution function:

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

 
:or:

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

 
====Conditional Reliability====
 
The conditional reliability function for the IPL-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}^{-K{{V}^{n}}t}}</math>

 
====Reliable Life====
 
For the IPL-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}^{-K{{V}^{n}}{{t}_{R}}}}</math>

 
::<math>\ln [R({{t}_{R}},V)]=-K{{V}^{n}}{{t}_{R}}</math>

 
:or:

 
::<math>{{t}_{R}}=-\frac{1}{K{{V}^{n}}}\ln [R({{t}_{R}},V)]</math>

 

===Parameter Estimation===
 
====Maximum Likelihood Parameter Estimation====
 
Substituting the inverse power law relationship into the exponential log-likelihood equation yields:

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

 
::<math>R_{Ri}^{\prime \prime }={{e}^{-T_{Ri}^{\prime \prime }KV_{i}^{n}}}</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>K</math> is the IPL parameter (unknown, the first of two parameters to be estimated).
 
• <math>n</math> is the second IPL 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{K},</math> <math>\widehat{n}</math> so that <math>\tfrac{\partial \Lambda }{\partial K}=0</math> and <math>\tfrac{\partial \Lambda }{\partial n}=0</math> , where:

 
::<math>\begin{align}
& \frac{\partial \Lambda }{\partial K}= & \frac{1}{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}-\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}V_{i}^{n}{{T}_{i}}-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }V_{i}^{n}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}^{n}}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}
\end{align}</math>

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

Template:Ipl exponential - Revision history

Richard House: Redirected page to Inverse Power Law (IPL) Relationship#IPL-Exponential

Chris Kahn: /* IPL-Exponential */

Chuck Smith: /* IPL-Exponential */

Nicolette Young: /* IPL-Exponential */

Athanasios Gerokostopoulos: /* IPL-Exponential */

Nicolette Young: /* Maximum Likelihood Parameter Estimation */

Nicolette Young: /* IPL-Exponential Reliability Function */

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

Nicolette Young: Created page with '==IPL-Exponential== The IPL-exponential model can be derived by setting m=L(V) in Eqn. (inverse), yielding the following IPL-exponential pdf : …'

Nicolette Young: Created page with '==IPL-Exponential==
The IPL-exponential model can be derived by setting $m=L(V)$ in Eqn. (inverse), yielding the following IPL-exponential $pdf$ : …'