Appendix C: Benchmark Examples - Revision history

Lisa Hacker at 19:18, 15 September 2023

2023-09-15T19:18:39Z

← Older revision		Revision as of 19:18, 15 September 2023
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	{{template:ALTABOOK ~~SUB~~\|Appendix C\|Benchmark Examples}}		{{template:ALTABOOK\|Appendix C\|Benchmark Examples}}
	In this section, five published examples are presented for comparison purposes. ReliaSoft's R&D validated the ALTA software with hundreds of data sets and methods. ALTA also cross-validates each provided solution by independently re-evaluating the second partial derivatives based on the estimated parameters each time a calculation is performed. These partials will be equal to zero when a solution is reached. Double precision is used throughout ALTA.		In this section, five published examples are presented for comparison purposes. ReliaSoft's R&D validated the ALTA software with hundreds of data sets and methods. ALTA also cross-validates each provided solution by independently re-evaluating the second partial derivatives based on the estimated parameters each time a calculation is performed. These partials will be equal to zero when a solution is reached. Double precision is used throughout ALTA.

Richard House at 22:38, 21 September 2012

2012-09-21T22:38:23Z

← Older revision		Revision as of 22:38, 21 September 2012
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	\widehat{B}=\ & 9920.195 \\		\widehat{B}=\ & 9920.195 \\
	\widehat{C}=\ & 9.69517\cdot {{10}^{-7}} \\		\widehat{C}=\ & 9.69517\cdot {{10}^{-7}} \\
	\end{align}</math>		\end{align}\,\!</math>

	==Computed Results for Example 1==		==Computed Results for Example 1==
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	\widehat{B}=\ & 9924.804 \\		\widehat{B}=\ & 9924.804 \\
	\widehat{C}=\ & 9.58978\cdot {{10}^{-7}} \\		\widehat{C}=\ & 9.58978\cdot {{10}^{-7}} \\
	\end{align}</math>		\end{align}\,\!</math>

	=Example 2=		=Example 2=
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	\widehat{K}=\ & 6.8742\cdot {{10}^{-29}} \\		\widehat{K}=\ & 6.8742\cdot {{10}^{-29}} \\
	\widehat{n}=\ & 17.72958 \\		\widehat{n}=\ & 17.72958 \\
	\end{align}</math>		\end{align}\,\!</math>

	*Published 95% confidence limits on <math>\beta \,\!</math> :		*Published 95% confidence limits on <math>\beta \,\!</math>:

	::<math>\begin{matrix}		::<math>\begin{matrix}
	\left\{ 0.653,0.923 \right\} \\		\left\{ 0.653,0.923 \right\} \\
	\end{matrix}</math>		\end{matrix}\,\!</math>

	==Computed Results for Example 2==		==Computed Results for Example 2==
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	\widehat{K}=\ & 6.8741\cdot {{10}^{-29}} \\		\widehat{K}=\ & 6.8741\cdot {{10}^{-29}} \\
	\widehat{n}=\ & 17.7296 \\		\widehat{n}=\ & 17.7296 \\
	\end{align}</math>		\end{align}\,\!</math>

	*ALTA computed 95% confidence limits on the parameters:		*ALTA computed 95% confidence limits on the parameters:

	::<math>\left\{ 0.6535,0.9228 \right\}\text{ for }\widehat{\beta }</math>		::<math>\left\{ 0.6535,0.9228 \right\}\text{ for }\widehat{\beta }\,\!</math>

	=Example 3=		=Example 3=
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	\widehat{K}=\ & 0.4350735 \\		\widehat{K}=\ & 0.4350735 \\
	\widehat{n}=\ & 13.8528 \\		\widehat{n}=\ & 13.8528 \\
	\end{align}</math>		\end{align}\,\!</math>

	*The 95% 2-sided confidence bounds on the parameters:		*The 95% 2-sided confidence bounds on the parameters:
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	& \left\{ 0.332906,0.568596 \right\}\text{ for }\widehat{K } \\		& \left\{ 0.332906,0.568596 \right\}\text{ for }\widehat{K } \\
	& \left\{ 11.43569,16.26991 \right\}\text{ for }\widehat{n }		& \left\{ 11.43569,16.26991 \right\}\text{ for }\widehat{n }
	\end{align}</math>		\end{align}\,\!</math>

	*Percentile estimates at a stress of 0.87, with 95% 2-sided confidence bounds:		*Percentile estimates at a stress of 0.87, with 95% 2-sided confidence bounds:
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	\widehat{K}=\ & 0.4350548 \\		\widehat{K}=\ & 0.4350548 \\
	\widehat{n}=\ & 13.8529 \\		\widehat{n}=\ & 13.8529 \\
	\end{align}</math>		\end{align}\,\!</math>

	*The 95% 2-sided confidence bounds on the parameters:		*The 95% 2-sided confidence bounds on the parameters:
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	& \left\{ 0.330007,0.573542 \right\}\text{ for }\widehat{K } \\		& \left\{ 0.330007,0.573542 \right\}\text{ for }\widehat{K } \\
	& \left\{ 11.43510,16.27079 \right\}\text{ for }\widehat{n }		& \left\{ 11.43510,16.27079 \right\}\text{ for }\widehat{n }
	\end{align}</math>		\end{align}\,\!</math>

	*Percentile estimates at a stress of 0.87, with 95% 2-sided confidence bounds:		*Percentile estimates at a stress of 0.87, with 95% 2-sided confidence bounds:
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	\widehat{K}=\ & 1.14\cdot {{10}^{-12}}, \\		\widehat{K}=\ & 1.14\cdot {{10}^{-12}}, \\
	\widehat{n}=\ & 4.28. \\		\widehat{n}=\ & 4.28. \\
	\end{align}</math>		\end{align}\,\!</math>

	*The 95% 2-sided confidence bounds on the parameters:		*The 95% 2-sided confidence bounds on the parameters:
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	& \left\{ 3.123\cdot {{10}^{-15}},4.16\cdot {{10}^{-10}} \right\}\text{ for }\widehat{K } \\		& \left\{ 3.123\cdot {{10}^{-15}},4.16\cdot {{10}^{-10}} \right\}\text{ for }\widehat{K } \\
	& \left\{ 3.11,5.46 \right\}\text{ for }\widehat{n }		& \left\{ 3.11,5.46 \right\}\text{ for }\widehat{n }
	\end{align}</math>		\end{align}\,\!</math>

	==Computed Results for Example 4==		==Computed Results for Example 4==
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	\widehat{K}=\ & 1.15\cdot {{10}^{-12}} \\		\widehat{K}=\ & 1.15\cdot {{10}^{-12}} \\
	\widehat{n}=\ & 4.289 \\		\widehat{n}=\ & 4.289 \\
	\end{align}</math>		\end{align}\,\!</math>

	*ALTA computed 95% confidence limits on the parameters:		*ALTA computed 95% confidence limits on the parameters:
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	& \left\{ 3.227\cdot {{10}^{-15}},4.095\cdot {{10}^{-10}} \right\}\text{ for }\widehat{K } \\		& \left\{ 3.227\cdot {{10}^{-15}},4.095\cdot {{10}^{-10}} \right\}\text{ for }\widehat{K } \\
	& \left\{ 3.115,5.464 \right\}\text{ for }\widehat{n }		& \left\{ 3.115,5.464 \right\}\text{ for }\widehat{n }
	\end{align}</math>		\end{align}\,\!</math>

	=Example 5=		=Example 5=
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	\widehat{C}=\ & 4.513\cdot {{10}^{36}} \\		\widehat{C}=\ & 4.513\cdot {{10}^{36}} \\
	\widehat{n}=\ & 20.1 \\		\widehat{n}=\ & 20.1 \\
	\end{align}</math>		\end{align}\,\!</math>

	*The 95% 2-sided confidence bounds on the parameters:		*The 95% 2-sided confidence bounds on the parameters:
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	& \left\{ 1.265\cdot {{10}^{25}},1.609\cdot {{10}^{48}} \right\}\text{ for }\widehat{C } \\		& \left\{ 1.265\cdot {{10}^{25}},1.609\cdot {{10}^{48}} \right\}\text{ for }\widehat{C } \\
	& \left\{ 11.4,28.8 \right\}\text{ for }\widehat{n }		& \left\{ 11.4,28.8 \right\}\text{ for }\widehat{n }
	\end{align}</math>		\end{align}\,\!</math>

	==Computed Results for Example 5==		==Computed Results for Example 5==
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	\widehat{C}=\ & 4.772\cdot {{10}^{36}} \\		\widehat{C}=\ & 4.772\cdot {{10}^{36}} \\
	\widehat{n}=\ & 20.09 \\		\widehat{n}=\ & 20.09 \\
	\end{align}</math>		\end{align}\,\!</math>

	*ALTA computed 95% confidence limits on the parameters:		*ALTA computed 95% confidence limits on the parameters:
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	& \left\{ 1.268\cdot {{10}^{25}},1.796\cdot {{10}^{48}} \right\}\text{ for }\widehat{C } \\		& \left\{ 1.268\cdot {{10}^{25}},1.796\cdot {{10}^{48}} \right\}\text{ for }\widehat{C } \\
	& \left\{ 11.37,28.8 \right\}\text{ for }\widehat{n }		& \left\{ 11.37,28.8 \right\}\text{ for }\widehat{n }
	\end{align}</math>		\end{align}\,\!</math>

Richard House at 20:24, 4 September 2012

2012-09-04T20:24:47Z

← Older revision		Revision as of 20:24, 4 September 2012
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	=Example 1=		=Example 1=

	From Wayne Nelson [[~~Reference Appendix E~~: ~~References~~\|[28, p. 135]]].		From Wayne Nelson [[Appendix_E:_References\|[28, p. 135]]].

	==Published Results for Example 1==		==Published Results for Example 1==
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	=Example 2=		=Example 2=

	From Wayne Nelson [[~~Reference Appendix E~~: ~~References~~\|[28, p. 453]]], time to breakdown of a transformer oil, tested at 26kV, 28kV, 30kV, 32kV, 34kV, 36kV and 38kV.		From Wayne Nelson [[Appendix_E:_References\|[28, p. 453]]], time to breakdown of a transformer oil, tested at 26kV, 28kV, 30kV, 32kV, 34kV, 36kV and 38kV.

	==Published Results for Example 2==		==Published Results for Example 2==
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	=Example 3=		=Example 3=

	From Wayne Nelson [[~~Reference Appendix E~~: ~~References~~\|[28, p. 157]]], forty bearings were tested to failure at four different test loads. The data were analyzed using the inverse power law Weibull model.		From Wayne Nelson [[Appendix_E:_References\|[28, p. 157]]], forty bearings were tested to failure at four different test loads. The data were analyzed using the inverse power law Weibull model.

	==Published Results for Example 3==		==Published Results for Example 3==
	Nelson's [[~~Reference Appendix E~~: ~~References~~\|[28, p. 306]]] IPL-Weibull parameter estimates:		Nelson's [[Appendix_E:_References\|[28, p. 306]]] IPL-Weibull parameter estimates:

	::<math>\begin{align}		::<math>\begin{align}
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	=Example 4=		=Example 4=

	From Meeker and Escobar [[~~Reference Appendix E~~: ~~References~~\|[26, p. 504]]], Mylar-Polyurethane Insulating Structure data using the inverse power law lognormal model.		From Meeker and Escobar [[Appendix_E:_References\|[26, p. 504]]], Mylar-Polyurethane Insulating Structure data using the inverse power law lognormal model.

	==Published Results for Example 4==		==Published Results for Example 4==
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	=Example 5=		=Example 5=
	From Meeker and Escobar [[~~Reference Appendix E~~: ~~References~~\|[26, p. 515]]], Tantalum capacitor data using the combination (Temperature-NonThermal) Weibull model.		From Meeker and Escobar [[Appendix_E:_References\|[26, p. 515]]], Tantalum capacitor data using the combination (Temperature-NonThermal) Weibull model.

	==Published Results for Example 5==		==Published Results for Example 5==

Richard House at 20:23, 4 September 2012

2012-09-04T20:23:35Z

← Older revision		Revision as of 20:23, 4 September 2012
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	=Example 1=		=Example 1=

	From Wayne Nelson [[Reference Appendix D: References\|[28, p. 135]]].		From Wayne Nelson [[Reference Appendix E: References\|[28, p. 135]]].

	==Published Results for Example 1==		==Published Results for Example 1==
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	=Example 2=		=Example 2=

	From Wayne Nelson [[Reference Appendix D: References\|[28, p. 453]]], time to breakdown of a transformer oil, tested at 26kV, 28kV, 30kV, 32kV, 34kV, 36kV and 38kV.		From Wayne Nelson [[Reference Appendix E: References\|[28, p. 453]]], time to breakdown of a transformer oil, tested at 26kV, 28kV, 30kV, 32kV, 34kV, 36kV and 38kV.

	==Published Results for Example 2==		==Published Results for Example 2==
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	=Example 3=		=Example 3=

	From Wayne Nelson [[Reference Appendix D: References\|[28, p. 157]]], forty bearings were tested to failure at four different test loads. The data were analyzed using the inverse power law Weibull model.		From Wayne Nelson [[Reference Appendix E: References\|[28, p. 157]]], forty bearings were tested to failure at four different test loads. The data were analyzed using the inverse power law Weibull model.

	==Published Results for Example 3==		==Published Results for Example 3==
	Nelson's [[Reference Appendix D: References\|[28, p. 306]]] IPL-Weibull parameter estimates:		Nelson's [[Reference Appendix E: References\|[28, p. 306]]] IPL-Weibull parameter estimates:

	::<math>\begin{align}		::<math>\begin{align}
Line 133:		Line 133:
	=Example 4=		=Example 4=

	From Meeker and Escobar [[Reference Appendix D: References\|[26, p. 504]]], Mylar-Polyurethane Insulating Structure data using the inverse power law lognormal model.		From Meeker and Escobar [[Reference Appendix E: References\|[26, p. 504]]], Mylar-Polyurethane Insulating Structure data using the inverse power law lognormal model.

	==Published Results for Example 4==		==Published Results for Example 4==
Line 172:		Line 172:

	=Example 5=		=Example 5=
	From Meeker and Escobar [[Reference Appendix D: References\|[26, p. 515]]], Tantalum capacitor data using the combination (Temperature-NonThermal) Weibull model.		From Meeker and Escobar [[Reference Appendix E: References\|[26, p. 515]]], Tantalum capacitor data using the combination (Temperature-NonThermal) Weibull model.

	==Published Results for Example 5==		==Published Results for Example 5==

Richard House: /* Computed Results for Example 4 */

2012-08-21T04:34:27Z

Computed Results for Example 4

← Older revision		Revision as of 04:34, 21 August 2012
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	Use the inverse power law lognormal.		Use the inverse power law lognormal.

	• ALTA computed parameters are:		*ALTA computed parameters are:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	• ALTA computed 95% confidence limits on the parameters:		*ALTA computed 95% confidence limits on the parameters:

	::<math>\begin{align}		::<math>\begin{align}

Richard House: /* Computed Results for Example 5 */

2012-08-21T04:33:41Z

Computed Results for Example 5

← Older revision		Revision as of 04:33, 21 August 2012
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	Use the Temperature-NonThermal model and Weibull as the underlying life distribution.		Use the Temperature-NonThermal model and Weibull as the underlying life distribution.

	• ALTA computed parameters are:		*ALTA computed parameters are:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	• ALTA computed 95% confidence limits on the parameters:		*ALTA computed 95% confidence limits on the parameters:

	::<math>\begin{align}		::<math>\begin{align}

Richard House: /* Published Results for Example 5 */

2012-08-21T04:33:27Z

Published Results for Example 5

← Older revision		Revision as of 04:33, 21 August 2012
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	==Published Results for Example 5==		==Published Results for Example 5==
	• Published Results:		*Published Results:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	• The 95% 2-sided confidence bounds on the parameters:		*The 95% 2-sided confidence bounds on the parameters:

	::<math>\begin{align}		::<math>\begin{align}

Richard House: /* Published Results for Example 4 */

2012-08-21T04:33:10Z

Published Results for Example 4

← Older revision		Revision as of 04:33, 21 August 2012
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	==Published Results for Example 4==		==Published Results for Example 4==
	• Published Results:		*Published Results:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	• The 95% 2-sided confidence bounds on the parameters:		*The 95% 2-sided confidence bounds on the parameters:

	::<math>\begin{align}		::<math>\begin{align}

Richard House: /* Computed Results for Example 2 */

2012-08-21T04:32:47Z

Computed Results for Example 2

← Older revision		Revision as of 04:32, 21 August 2012
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	\end{align}</math>		\end{align}</math>

	• ALTA computed 95% confidence limits on the parameters:		*ALTA computed 95% confidence limits on the parameters:

	::<math>\left\{ 0.6535,0.9228 \right\}\text{ for }\widehat{\beta }</math>		::<math>\left\{ 0.6535,0.9228 \right\}\text{ for }\widehat{\beta }</math>

Richard House: /* Published Results for Example 2 */

2012-08-21T04:32:33Z

Published Results for Example 2

← Older revision		Revision as of 04:32, 21 August 2012
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	==Published Results for Example 2==		==Published Results for Example 2==
	• Published Results:		*Published Results:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	• Published 95% confidence limits on <math>\beta </math> :		*Published 95% confidence limits on <math>\beta \,\!</math> :

	::<math>\begin{matrix}		::<math>\begin{matrix}