Arrhenius-Lognormal Model for Interval Data - Revision history

Kate Racaza at 18:18, 28 September 2015

2015-09-28T18:18:51Z

← Older revision		Revision as of 18:18, 28 September 2015
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	{{Reference Example\|{{Banner ALTA Reference Examples}}}}		{{Reference Example\|{{Banner ALTA Reference Examples}}}}

	This example ~~compares~~ the results for the Arrhenius-lognormal model with interval data.		This example validates the results for the Arrhenius-lognormal model with interval data in ALTA standard folios.

Kate Racaza at 14:58, 16 June 2014

2014-06-16T14:58:35Z

← Older revision		Revision as of 14:58, 16 June 2014
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	{{Reference_Example_Heading2}}		{{Reference_Example_Heading2}}

	The data set for a new-technology IC ~~Device~~ is given below.		The data set for a new-technology IC device is given below.

	{\| {{table}}		{\| {{table}}
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	The above equation is the general log-linear model in ALTA. In ALTA, the coefficients are denoted by <math>\alpha_{i}\,\!</math>. We can see <math>\beta_{0} = \alpha_{0}\,\!</math> and <math>\beta_{1} = \frac{\alpha_{1}}{11605}\,\!</math>.		The above equation is the general log-linear model in ALTA. In ALTA, the coefficients are denoted by <math>\alpha_{i}\,\!</math>. We can see that <math>\beta_{0} = \alpha_{0}\,\!</math> and <math>\beta_{1} = \frac{\alpha_{1}}{11605}\,\!</math>.

	In fact, the above model can also be expressed using the traditional Arrhenius model:		In fact, the above model can also be expressed using the traditional Arrhenius model:
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	In the book, the following results are provided:		In the book, the following results are provided:

	* ML estimations for the model parameters are: <math>\sigma\,\!</math> = 0.52, <math>\beta_{0}\,\!</math> = -10.2, <math>\beta_{1}\,\!</math> = 0.83, (<math>\alpha_{1}\,\!</math> = <math>\beta_{1}\times \,\!</math> 11605 = 9632.15).		* The ML estimations for the model parameters are: <math>\sigma\,\!</math> = 0.52, <math>\beta_{0}\,\!</math> = -10.2, <math>\beta_{1}\,\!</math> = 0.83, (<math>\alpha_{1}\,\!</math> = <math>\beta_{1}\times \,\!</math> 11605 = 9632.15).
	* The standard deviation of each parameter: <math>std(\sigma)\,\!</math> = 0.06,   <math>std(\beta_{0})\,\!</math> = 1.5,  <math>std(\beta_{1})\,\!</math> = 0.07. Therefore, their variances are: <math>Var(\sigma)\,\!</math> = 0.0036,  <math>Var(\beta_{0})\,\!</math> = 2.25,  <math>Var(\beta_{1})\,\!</math> = 0.0049. In terms of <math>\alpha_{1}\,\!</math> , the variance is <math>Var(\alpha_{1})\,\!</math> = 11605<sup>2</sup>  and  <math>Var(\beta_{1})\,\!</math> = 659912.5.		* The standard deviation of each parameter: <math>std(\sigma)\,\!</math> = 0.06,   <math>std(\beta_{0})\,\!</math> = 1.5,  <math>std(\beta_{1})\,\!</math> = 0.07. Therefore, their variances are: <math>Var(\sigma)\,\!</math> = 0.0036,  <math>Var(\beta_{0})\,\!</math> = 2.25,  <math>Var(\beta_{1})\,\!</math> = 0.0049. In terms of <math>\alpha_{1}\,\!</math> , the variance is <math>Var(\alpha_{1})\,\!</math> = 11605<sup>2</sup>  and  <math>Var(\beta_{1})\,\!</math> = 659912.5.
	* The 95% two-sided confidence bounds are: for <math>\sigma\,\!</math> it is [0.42, 0.64]; for <math>\beta_{0}\,\!</math> it is [-13.2, -7.2]; for <math>\beta_{1}\,\!</math> it is [0.68, 0.97]. In terms of <math>\alpha_{1}\,\!</math>, the bounds are [7891.14, 11256.85].		* The 95% two-sided confidence bounds are: for <math>\sigma\,\!</math> it is [0.42, 0.64]; for <math>\beta_{0}\,\!</math> it is [-13.2, -7.2]; for <math>\beta_{1}\,\!</math> it is [0.68, 0.97]. In terms of <math>\alpha_{1}\,\!</math>, the bounds are [7891.14, 11256.85].
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	The picture above also shows ~~the standard deviations~~ of ~~the parameters~~. These ~~are: <math>std(\sigma)\,\!</math> = 0.06,   <math>std(\beta_{0})\,\!</math> = 1.5,  <math>std(\beta_{1})\,\!</math> = 0.07. Therefore, their variances~~ are: <math>Var(\sigma)\,\!</math> = 0.~~0036~~,  <math>Var(\~~beta_~~{0})\,\!</math> = 2.25,  <math>Var(\~~beta_~~{1})\,\!</math> = 0.~~0049~~. In terms of <math>\~~alpha_~~{1}\,\!</math> , the variance is <math>Var(\~~alpha_~~{1})\~~,\!</math> = 11605<sup>2</sup>  and  <math>~~Var(\~~beta_~~{1})\,\!</math> = ~~659912~~.5.		The picture above also shows variance of each parameter. These are: <math>Var(\sigma)\,\!</math> = 0.003303,  <math>Var(\alpha_{0})\,\!</math> = 2.331652,  <math>Var(\alpha_{1})\,\!</math> = 721392.210588. In terms of <math>\beta_{1}\,\!</math> , the variance is <math>Var(\beta_{1}) = \frac{Var(\alpha_{1})}{11605^{2}}\,\!</math> = 0.0053565.

	The differences in the variance in ALTA and the results in the book are caused by the precision in the book since results are provided only up to the 2nd decimal.		The differences in the variance in ALTA and the results in the book are caused by the precision in the book since results are provided only up to the 2nd decimal.

Kate Racaza at 16:37, 13 June 2014

2014-06-13T16:37:19Z

← Older revision		Revision as of 16:37, 13 June 2014
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	{{Reference Example\|~~ALTA_Reference_Examples_Banner.png\|ALTA_Reference_Examples~~}}		{{Reference Example\|{{Banner ALTA Reference Examples}}}}

	This example compares the results for the Arrhenius-lognormal model with interval data.		This example compares the results for the Arrhenius-lognormal model with interval data.

Kate Racaza at 23:28, 12 June 2014

2014-06-12T23:28:15Z

← Older revision		Revision as of 23:28, 12 June 2014
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	{{Reference Example\|ALTA_Reference_Examples_Banner.png\|ALTA_Reference_Examples}}		{{Reference Example\|ALTA_Reference_Examples_Banner.png\|ALTA_Reference_Examples}}

	This example compares the results for the Arrhenius-~~Lognormal~~ model with interval data.		This example compares the results for the Arrhenius-lognormal model with interval data.

Kate Racaza at 15:25, 12 June 2014

2014-06-12T15:25:24Z

← Older revision		Revision as of 15:25, 12 June 2014
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	where ''T'' is the temperature; <math>\beta_{1}\,\!</math> is the activation energy; 11605 is calculated from the reciprocal of the Boltzmann constant. This function can be written in the following way:		where ''T'' is the temperature; <math>\beta_{1}\,\!</math> is the activation energy; and 11605 is calculated from the reciprocal of the Boltzmann constant. This function can be written in the following way:


	::<math>e^{\mu'} = e^{\alpha_{0}+ \frac {\alpha_{1}}{T}}\,\!</math>		::<math>e^{\mu'} = e^{\alpha_{0}+ \frac {\alpha_{1}}{T}}\,\!</math>
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	The above equation is the general log-linear model in ALTA. In ALTA, the coefficients are denoted by <math>\alpha_{i}\,\!</math>. We can see <math>\beta_{0} = \alpha_{0}\,\!</math> and <math>\beta_{1} = \frac{\alpha_{1}}{11605}\,\!</math>.		The above equation is the general log-linear model in ALTA. In ALTA, the coefficients are denoted by <math>\alpha_{i}\,\!</math>. We can see <math>\beta_{0} = \alpha_{0}\,\!</math> and <math>\beta_{1} = \frac{\alpha_{1}}{11605}\,\!</math>.

	In fact, the above model also ~~can~~ be expressed using the traditional Arrhenius model:		In fact, the above model can also be expressed using the traditional Arrhenius model:

	::<math>e^{\mu'} = e^{\alpha_{0}+ \frac {\alpha_{1}}{T}} = C \times e^{\frac{B}{T}}\,\!</math>		::<math>e^{\mu'} = e^{\alpha_{0}+ \frac {\alpha_{1}}{T}} = C \times e^{\frac{B}{T}}\,\!</math>
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	* The following picture shows the ML estimations for the model parameters in ALTA and the log-likelihood value (LK Value).		* The following picture shows the ML estimations for the model parameters in ALTA and the log-likelihood value (LK Value).

	[[Image: Arrhenius Log results.png\|center\|~~550px~~]]		[[Image: Arrhenius Log results.png\|center\|650px]]


	The picture above also shows the standard deviations of the parameters. These are: <math>std(\sigma)\,\!</math> = 0.06,   <math>std(\beta_{0})\,\!</math> = 1.5,  <math>std(\beta_{1})\,\!</math> = 0.07. Therefore, their variances are: <math>Var(\sigma)\,\!</math> = 0.0036,  <math>Var(\beta_{0})\,\!</math> = 2.25,  <math>Var(\beta_{1})\,\!</math> = 0.0049. In terms of <math>\alpha_{1}\,\!</math> , the variance is <math>Var(\alpha_{1})\,\!</math> = 11605<sup>2</sup>  and  <math>Var(\beta_{1})\,\!</math> = 659912.5.		The picture above also shows the standard deviations of the parameters. These are: <math>std(\sigma)\,\!</math> = 0.06,   <math>std(\beta_{0})\,\!</math> = 1.5,  <math>std(\beta_{1})\,\!</math> = 0.07. Therefore, their variances are: <math>Var(\sigma)\,\!</math> = 0.0036,  <math>Var(\beta_{0})\,\!</math> = 2.25,  <math>Var(\beta_{1})\,\!</math> = 0.0049. In terms of <math>\alpha_{1}\,\!</math> , the variance is <math>Var(\alpha_{1})\,\!</math> = 11605<sup>2</sup>  and  <math>Var(\beta_{1})\,\!</math> = 659912.5.

	The ~~difference of~~ the variance in ALTA and the results in the book are caused by the precision in the book since results are provided only up to the 2nd decimal.		The differences in the variance in ALTA and the results in the book are caused by the precision in the book since results are provided only up to the 2nd decimal.

	* The 95% two-sided confidence bounds on the parameters are shown next:		* The 95% two-sided confidence bounds on the parameters are shown next:

Kate Racaza at 20:08, 11 June 2014

2014-06-11T20:08:34Z

← Older revision		Revision as of 20:08, 11 June 2014
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	{{Reference_Example_Heading4\|ALTA}}		{{Reference_Example_Heading4\|ALTA}}

			* The following picture shows the ML estimations for the model parameters in ALTA and the log-likelihood value (LK Value).

			[[Image: Arrhenius Log results.png\|center\|550px]]


			The picture above also shows the standard deviations of the parameters. These are: <math>std(\sigma)\,\!</math> = 0.06,   <math>std(\beta_{0})\,\!</math> = 1.5,  <math>std(\beta_{1})\,\!</math> = 0.07. Therefore, their variances are: <math>Var(\sigma)\,\!</math> = 0.0036,  <math>Var(\beta_{0})\,\!</math> = 2.25,  <math>Var(\beta_{1})\,\!</math> = 0.0049. In terms of <math>\alpha_{1}\,\!</math> , the variance is <math>Var(\alpha_{1})\,\!</math> = 11605<sup>2</sup>  and  <math>Var(\beta_{1})\,\!</math> = 659912.5.

			The difference of the variance in ALTA and the results in the book are caused by the precision in the book since results are provided only up to the 2nd decimal.

			* The 95% two-sided confidence bounds on the parameters are shown next:

			[[Image: Arrhenius Log parameter bounds.png\|center]]


			* The likelihood ratio common shape parameter test statistic is 4.7, as shown next.

			[[Image: Arrhenius Log LK test.png\|center]]


			As the results show, the values obtained by ALTA are very close to the results given in the book.

Kate Racaza at 19:44, 11 June 2014

2014-06-11T19:44:28Z

← Older revision		Revision as of 19:44, 11 June 2014
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			::<math>e^{\mu'} = e^{\alpha_{0}+ \frac {\alpha_{1}}{T}}\,\!</math>


			The above equation is the general log-linear model in ALTA. In ALTA, the coefficients are denoted by <math>\alpha_{i}\,\!</math>. We can see <math>\beta_{0} = \alpha_{0}\,\!</math> and <math>\beta_{1} = \frac{\alpha_{1}}{11605}\,\!</math>.

			In fact, the above model also can be expressed using the traditional Arrhenius model:

			::<math>e^{\mu'} = e^{\alpha_{0}+ \frac {\alpha_{1}}{T}} = C \times e^{\frac{B}{T}}\,\!</math>


			In the book, the following results are provided:

			* ML estimations for the model parameters are: <math>\sigma\,\!</math> = 0.52, <math>\beta_{0}\,\!</math> = -10.2, <math>\beta_{1}\,\!</math> = 0.83, (<math>\alpha_{1}\,\!</math> = <math>\beta_{1}\times \,\!</math> 11605 = 9632.15).
			* The standard deviation of each parameter: <math>std(\sigma)\,\!</math> = 0.06,   <math>std(\beta_{0})\,\!</math> = 1.5,  <math>std(\beta_{1})\,\!</math> = 0.07. Therefore, their variances are: <math>Var(\sigma)\,\!</math> = 0.0036,  <math>Var(\beta_{0})\,\!</math> = 2.25,  <math>Var(\beta_{1})\,\!</math> = 0.0049. In terms of <math>\alpha_{1}\,\!</math> , the variance is <math>Var(\alpha_{1})\,\!</math> = 11605<sup>2</sup>  and  <math>Var(\beta_{1})\,\!</math> = 659912.5.
			* The 95% two-sided confidence bounds are: for <math>\sigma\,\!</math> it is [0.42, 0.64]; for <math>\beta_{0}\,\!</math> it is [-13.2, -7.2]; for <math>\beta_{1}\,\!</math> it is [0.68, 0.97]. In terms of <math>\alpha_{1}\,\!</math>, the bounds are [7891.14, 11256.85].
			* The log-likelihood value is -88.26.
			* Common shape parameter test result: the likelihood ratio common shape parameter test statistic is 4.7. It is larger than the critical value 3.84. This indicates that there is some lack of fit in the constant <math>\sigma\,\!</math> assumption across all the stress levels.


	{{Reference_Example_Heading4\|ALTA}}		{{Reference_Example_Heading4\|ALTA}}

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2014-06-11T18:43:02Z

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{{Reference Example|ALTA_Reference_Examples_Banner.png|ALTA_Reference_Examples}}

This example compares the results for the Arrhenius-Lognormal model with interval data.

{{Reference_Example_Heading1}}

The data set is from Example 19.11 on page 508 in the book ''Statistical Methods for Reliability Data'' by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998.

{{Reference_Example_Heading2}}

The data set for a new-technology IC Device is given below.

{| {{table}}
! Number in Group||Last Inspected (Hr)||State F/S||Time to State||Temperature (K)
|-
| 50||788||S||1536||423.15
|-
| 50||788||S||1536||448.15
|-
| 50||96||S||96||473.15
|-
| 1||384||F||788||523.15
|-
| 3||788||F||1536||523.15
|-
| 5||1536||F||2304||523.15
|-
| 41||1536||S||2304||523.15
|-
| 4||192||F||384||573.15
|-
| 27||384||F||788||573.15
|-
| 16||788||F||1536||573.15
|-
| 3||788||S||1536||573.15
|-
|}

{{Reference_Example_Heading3}}

The following function is used for the Ln-Mean <math>\mu'\,\!</math> of the lognormal distribution:

::<math>\mu' = \beta_{0}+\beta_{1} \times \frac{11605}{T}\,\!</math>

where ''T'' is the temperature; <math>\beta_{1}\,\!</math> is the activation energy; 11605 is calculated from the reciprocal of the Boltzmann constant. This function can be written in the following way:

{{Reference_Example_Heading4|ALTA}}