Template:Ipl lognormal - Revision history

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

2012-08-15T23:18:45Z

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

Chris Kahn: /* Maximum Likelihood Estimation Method */

2012-03-09T18:28:50Z

Maximum Likelihood Estimation Method

Chris Kahn: /* The Mean */

2012-03-09T18:26:49Z

The Mean

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	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:		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:

Chris Kahn: /* IPL-Lognormal */

2012-03-09T18:26:24Z

IPL-Lognormal

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	The <math>pdf</math> for the Inverse Power Law relationship and the lognormal distribution is given next.		The <math>pdf</math> for the Inverse Power Law relationship and the lognormal distribution is given next.

	The <math>pdf</math>of the lognormal distribution is given by:		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>		::<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>

Chris Kahn: /* IPL-Lognormal */

2012-03-09T18:26:07Z

IPL-Lognormal

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	==IPL-Lognormal==		==IPL-Lognormal==

	The pdf for the Inverse Power Law relationship and the lognormal distribution is given next.		The <math>pdf</math> for the Inverse Power Law relationship and the lognormal distribution is given next.

	The pdf of the lognormal distribution is given by:		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>		::<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>
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	The IPL-lognormal model pdf can be obtained first by setting <math>\breve{T}=L(V)</math> in the lognormal <math>pdf</math>. Therefore:		The IPL-lognormal model <math>pdf</math> can be obtained first by setting <math>\breve{T}=L(V)</math> in the lognormal <math>pdf</math>. Therefore:

Athanasios Gerokostopoulos: /* Lognormal Failure Rate */

2012-02-16T18:08:38Z

Lognormal Failure Rate

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

	====Lognormal Failure Rate====		=====Lognormal Failure Rate=====

	The lognormal failure rate is given by:		The lognormal failure rate is given by:

Athanasios Gerokostopoulos: /* Reliable Life */

2012-02-16T18:08:21Z

Reliable Life

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

	====Reliable Life====		=====Reliable Life=====

	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:		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:

Athanasios Gerokostopoulos: /* IPL-Lognormal Reliability */

2012-02-16T18:08:17Z

IPL-Lognormal Reliability

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	::<math>\tilde{T}={{e}^{{\bar{T}}'-\sigma _{{{T}'}}^{2}}}={{e}^{-\ln (K)-n\ln (V)-\sigma _{{{T}'}}^{2}}}</math>		::<math>\tilde{T}={{e}^{{\bar{T}}'-\sigma _{{{T}'}}^{2}}}={{e}^{-\ln (K)-n\ln (V)-\sigma _{{{T}'}}^{2}}}</math>

	====IPL-Lognormal Reliability====		=====IPL-Lognormal Reliability=====

	The reliability for a mission of time T, starting at age 0, for the IPL-lognormal model is determined by:		The reliability for a mission of time T, starting at age 0, for the IPL-lognormal model is determined by:

Athanasios Gerokostopoulos: /* The Mode */

2012-02-16T18:08:03Z

The Mode

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

	====The Mode====		=====The Mode=====

	The mode of the IPL-lognormal model is given by:		The mode of the IPL-lognormal model is given by:

Athanasios Gerokostopoulos: /* The Standard Deviation */

2012-02-16T18:07:54Z

The Standard Deviation

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

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

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	* Fe is the number of groups of exact times-to-failure data points.		* Fe is the number of groups of exact times-to-failure data points.

	* Ni is the number of times-to-failure data points in the ~~ith~~ time-to-failure data group.		* Ni is the number of times-to-failure data points in the <math>{i}^{th}</math> time-to-failure data group.

	* <math>s_{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>s_{T'}</math> is the standard deviation of the natural logarithm of the times-to-failure (unknown, the first of three parameters to be estimated).
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	* <math>n</math> is the second IPL parameter (unknown, the third of three parameters to be estimated).		* <math>n</math> is the second IPL parameter (unknown, the third of three parameters to be estimated).

	* <math>Vi</math> is the stress level of the ~~ith~~ group.		* <math>Vi</math> is the stress level of the <math>{i}^{th}</math> group.

	* <math>Ti</math> is the exact failure time of the ~~ith~~ group.		* <math>Ti</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>S</math> is the number of groups of suspension data points.

	* <math>N'_i</math> is the number of suspensions in the ~~ith~~ group of suspension data points.		* <math>N'_i</math> is the number of suspensions in the <math>{i}^{th}</math> group of suspension data points.

	* <math>T^{'}_{i}</math> is the running time of the ~~ith~~ suspension data group.		* <math>T^{'}_{i}</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>FI</math> is the number of interval data groups.

	* is the number of intervals in the ~~ith~~ group of data intervals.		* is the number of intervals in the <math>{i}^{th}</math> group of data intervals.

	* is the beginning of the ~~ith~~ interval.		* is the beginning of the <math>{i}^{th}</math> interval.

	* is the ending of the ~~ith~~ interval.		* is the ending of the <math>{i}^{th}</math> interval.