Template:T-h lognormal - Revision history

Richard House: Redirected page to Temperature-Humidity Relationship#T-H Lognormal

2012-08-15T05:39:37Z

Redirected page to Temperature-Humidity Relationship#T-H Lognormal

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

2012-02-16T23:30:54Z

Maximum Likelihood Estimation Method

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	::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\~~mathop{}_~~{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>		::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>

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	• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.		• <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.~~
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	•		•<math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval.
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	The solution (parameter estimates) will be found by solving for <math>{{\widehat{\sigma }}_{{{T}'}}},</math> <math>\widehat{A},</math> <math>\widehat{\phi },</math> <math>\widehat{b}</math> so that <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0,</math> <math>\tfrac{\partial \Lambda }{\partial A}=0,</math> <math>\tfrac{\partial \Lambda }{\partial \phi }=0</math> and <math>\tfrac{\partial \Lambda }{\partial b}=0</math> .		The solution (parameter estimates) will be found by solving for <math>{{\widehat{\sigma }}_{{{T}'}}},</math> <math>\widehat{A},</math> <math>\widehat{\phi },</math> <math>\widehat{b}</math> so that <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0,</math> <math>\tfrac{\partial \Lambda }{\partial A}=0,</math> <math>\tfrac{\partial \Lambda }{\partial \phi }=0</math> and <math>\tfrac{\partial \Lambda }{\partial b}=0</math> .
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Athanasios Gerokostopoulos: /* Reliable Life */

2012-02-16T23:29:39Z

Reliable Life

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	and:		and:
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	::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\~~mathop{}_~~{-\infty }^{z({T}',V,U)}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>		::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}',V,U)}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>

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	====T-H Lognormal Failure Rate====		====T-H Lognormal Failure Rate====
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Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

2012-02-16T23:29:22Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 23:29, 16 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}{{\phi }_{pdf}}\left( \frac{\ln \left( {{T}_{i}} \right)-\ln (A)-\tfrac{\phi }{{{V}_{i}}}-\tfrac{b}{{{U}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] \\		& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}{{\phi }_{pdf}}\left( \frac{\ln \left( {{T}_{i}} \right)-\ln (A)-\tfrac{\phi }{{{V}_{i}}}-\tfrac{b}{{{U}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-\ln (A)-\tfrac{\phi }{{{V}_{i}}}-\tfrac{b}{{{U}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \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 })]
	~~& &~~ \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-\ln (A)-\tfrac{\phi }{{{V}_{i}}}-\tfrac{b}{{{U}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \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>		\end{align}</math>

Athanasios Gerokostopoulos: /* T-H Lognormal Failure Rate */

2012-02-16T23:28:32Z

T-H Lognormal Failure Rate

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	::<math>\lambda (T,V,U)=\frac{f(T,V,U)}{R(T,V,U)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (A)-\tfrac{\phi }{V}-\tfrac{b}{U}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\~~mathop{}_~~{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (A)-\tfrac{\phi }{V}-\tfrac{b}{U}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}</math>		::<math>\lambda (T,V,U)=\frac{f(T,V,U)}{R(T,V,U)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (A)-\tfrac{\phi }{V}-\tfrac{b}{U}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\int_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (A)-\tfrac{\phi }{V}-\tfrac{b}{U}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}</math>

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	===Parameter Estimation===		===Parameter Estimation===
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Athanasios Gerokostopoulos: /* T-H Lognormal Reliability */

2012-02-16T23:27:58Z

T-H Lognormal Reliability

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	::<math>R(T,V,U)=\~~mathop{}_~~{T}^{\infty }f(t,V,U)dt</math>		::<math>R(T,V,U)=\int_{T}^{\infty }f(t,V,U)dt</math>

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	or:		or:
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	::<math>R(T,V,U)=\~~mathop{}_~~{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (A)-\tfrac{\phi }{V}-\tfrac{b}{U}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt</math>		::<math>R(T,V,U)=\int_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (A)-\tfrac{\phi }{V}-\tfrac{b}{U}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt</math>

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	====Reliable Life====		====Reliable Life====
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Athanasios Gerokostopoulos: /* The Mode */

2012-02-16T23:27:26Z

The Mode

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	====The Mode====		====The Mode====
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	• The mode of the T-H lognormal model is given by:		• The mode of the T-H lognormal model is given by:

	<math>\begin{align}		::<math>\begin{align}
	& \tilde{T}= & {{e}^{{{\overline{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}} \\		& \tilde{T}= & {{e}^{{{\overline{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}}=\ & {{e}^{\ln (A)+\tfrac{\phi }{V}+\tfrac{b}{U}-\sigma _{{{T}'}}^{2}}}
	~~& =~~ & {{e}^{\ln (A)+\tfrac{\phi }{V}+\tfrac{b}{U}-\sigma _{{{T}'}}^{2}}}
	\end{align}</math>		\end{align}</math>


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	====T-H Lognormal Reliability====		====T-H Lognormal Reliability====
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Athanasios Gerokostopoulos: /* The Standard Deviation */

2012-02-16T23:25:31Z

The Standard Deviation

← Older revision		Revision as of 23:25, 16 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	& {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} \\		& {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} =\ & \sqrt{\left( {{e}^{2\left( \ln (A)+\tfrac{\phi }{V}+\tfrac{b}{U} \right)+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)}
	~~& =~~ & \sqrt{\left( {{e}^{2\left( \ln (A)+\tfrac{\phi }{V}+\tfrac{b}{U} \right)+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)}
	\end{align}</math>		\end{align}</math>

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	====The Mode====		====The Mode====
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Athanasios Gerokostopoulos: /* The Mean */

2012-02-16T22:11:07Z

The Mean

← Older revision		Revision as of 22:11, 16 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	& \bar{T}= & {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} \\		& \bar{T}= & {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} =\ {{e}^{\ln (A)+\tfrac{\phi }{V}+\tfrac{b}{U}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}
	~~& = &~~ {{e}^{\ln (A)+\tfrac{\phi }{V}+\tfrac{b}{U}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}
	\end{align}</math>		\end{align}</math>

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	====The Standard Deviation====		====The Standard Deviation====
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Athanasios Gerokostopoulos: /* T-H Lognormal */

2012-02-16T22:10:19Z

T-H Lognormal

← Older revision		Revision as of 22:10, 16 February 2012
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	• <math>\overline{{{T}'}}=</math> mean of the natural logarithms of the times-to-failure.		• <math>\overline{{{T}'}}=</math> mean of the natural logarithms of the times-to-failure.
			<br>
	• <math>{{\sigma }_{{{T}'}}}=</math> standard deviation of the natural logarithms of the times-to-failure.		• <math>{{\sigma }_{{{T}'}}}=</math> standard deviation of the natural logarithms of the times-to-failure.

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	The median of the lognormal distribution is given by:		The median of the lognormal distribution is given by:
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	The T-H lognormal model <math>pdf</math> can be obtained first by setting <math>\breve{T} =L(V,U)</math> ~~in Eqn. (Temp-Hum)~~. <br>		The T-H lognormal model <math>pdf</math> can be obtained first by setting <math>\breve{T} =L(V,U)</math>. <br>
	Therefore:		Therefore:

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	Substituting ~~Eqn. (TH-logn-mean)~~ into ~~Eqn. (TH-logn-~~pdf) yields the T-H lognormal model <math>pdf</math> or:		Substituting the above equation into the lognormal <math>pdf</math> yields the T-H lognormal model <math>pdf</math> or:

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