Weibull-Bayesian with Prior Information on Beta - Revision history

Kate Racaza at 16:21, 28 September 2015

2015-09-28T16:21:42Z

← Older revision		Revision as of 16:21, 28 September 2015
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	{{Reference Example}}		{{Reference Example}}

	This example ~~compares~~ the Weibull-Bayesian ~~calculation~~.		This example validates the Weibull-Bayesian calculations in Weibull++ standard folios.

Kate Racaza at 16:28, 9 June 2014

2014-06-09T16:28:25Z

← Older revision		Revision as of 16:28, 9 June 2014
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	{{Reference_Example_Heading1}}		{{Reference_Example_Heading1}}

	The data from Example 14.1 on page 348 in the book ''Statistical Methods for Reliability Data'' by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998 is used.		The data set from Example 14.1 on page 348 in the book ''Statistical Methods for Reliability Data'' by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998 is used.


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	{{Reference_Example_Heading4}}		{{Reference_Example_Heading4}}

	In Weibull++, the prior distribution is set for <math>\beta\,\!</math> directly. Based on the information of <math>\sigma\,\!</math>, we know <math>\beta_{0.005}\,\!</math> = 2 and <math>\beta_{0.995}\,\!</math> = 5. Therefore, we can get the prior lognormal distribution for <math>\beta\,\!</math>. ~~It is~~ Log-Mean = 1.15129 and Log-Std = 0.17786.		In Weibull++, the prior distribution is set for <math>\beta\,\!</math> directly. Based on the information of <math>\sigma\,\!</math>, we know <math>\beta_{0.005}\,\!</math> = 2 and <math>\beta_{0.995}\,\!</math> = 5. Therefore, we can use the Quick Parameter Estimator (QPE) to get the prior lognormal distribution for <math>\beta\,\!</math>. The results are Log-Mean = 1.15129 and Log-Std = 0.17786, as shown next.

	[[Image:WeiBays_QPE.png\|center\|550px\|]]		[[Image:WeiBays_QPE.png\|center\|550px\|]]
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	[[Image:WeiBays_F5000.png\|center\|550px\|]]		[[Image:WeiBays_F5000.png\|center\|550px\|]]

	The results in Weibull++ are very close, but not exactly the same, as the results in the book. The differences are mainly caused by the fact that the prior lognormal distribution is for <math>\sigma\,\!</math> in the book while it is for <math>\beta\,\!</math> in Weibull++.		The results in Weibull++ are very close but not exactly the same as the results in the book. The differences are mainly caused by the fact that the prior lognormal distribution is for <math>\sigma\,\!</math> in the book while it is for <math>\beta\,\!</math> in Weibull++.

Kate Racaza at 20:00, 5 June 2014

2014-06-05T20:00:39Z

← Older revision		Revision as of 20:00, 5 June 2014
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	In Weibull++, the prior distribution is set for <math>\beta\,\!</math> directly. Based on the information of <math>\sigma\,\!</math>, we know <math>\beta_{0.005}\,\!</math> = 2 and <math>\beta_{0.995}\,\!</math> = 5. Therefore, we can get the prior lognormal distribution for <math>\beta\,\!</math>. It is Log-Mean = 1.15129 and Log-Std = 0.17786.		In Weibull++, the prior distribution is set for <math>\beta\,\!</math> directly. Based on the information of <math>\sigma\,\!</math>, we know <math>\beta_{0.005}\,\!</math> = 2 and <math>\beta_{0.995}\,\!</math> = 5. Therefore, we can get the prior lognormal distribution for <math>\beta\,\!</math>. It is Log-Mean = 1.15129 and Log-Std = 0.17786.

	[[Image:x.png\|center\|550px\|]]		[[Image:WeiBays_QPE.png\|center\|550px\|]]

	Applying this prior distribution for Wei-Bayesian, we have the following results:		Applying this prior distribution for Wei-Bayesian, we have the following results:
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	* The 95% two-sided Bayesian confidence interval for <math>t_{0.05}\,\!</math> (B5% life) is [1623, 3452].		* The 95% two-sided Bayesian confidence interval for <math>t_{0.05}\,\!</math> (B5% life) is [1623, 3452].

	[[Image:x.png\|center\|550px\|]]		[[Image:WeiBays_B5.png\|center\|550px\|]]

	* The 95% two-sided Bayesian confidence interval for <math>t_{0.10}\,\!</math> (B10% life) is [2030, 4763].		* The 95% two-sided Bayesian confidence interval for <math>t_{0.10}\,\!</math> (B10% life) is [2030, 4763].

	[[Image:x.png\|center\|550px\|]]		[[Image:WeiBays_B10.png\|center\|550px\|]]

	* The 95% two-sided Bayesian confidence interval for F(2000) (probability of failure at time of 2000) is [0.014, 0.095].		* The 95% two-sided Bayesian confidence interval for F(2000) (probability of failure at time of 2000) is [0.014, 0.095].

	[[Image:x.png\|center\|550px\|]]		[[Image:WeiBays_F2000.png\|center\|550px\|]]

	* The 95% two-sided Bayesian confidence interval for F(5000) (probability of failure at time of 5000) is [0.111, 0.903].		* The 95% two-sided Bayesian confidence interval for F(5000) (probability of failure at time of 5000) is [0.111, 0.903].

	[[Image:x.png\|center\|550px\|]]		[[Image:WeiBays_F5000.png\|center\|550px\|]]

	The results in Weibull++ are very close, but not exactly the same, as the results in the book. The differences are mainly caused by the fact that the prior lognormal distribution is for <math>\sigma\,\!</math> in the book while it is for <math>\beta\,\!</math> in Weibull++.		The results in Weibull++ are very close, but not exactly the same, as the results in the book. The differences are mainly caused by the fact that the prior lognormal distribution is for <math>\sigma\,\!</math> in the book while it is for <math>\beta\,\!</math> in Weibull++.

Kate Racaza: Created page with '{{Reference Example}} This example compares the Weibull-Bayesian calculation. {{Reference_Example_Heading1}} The data from Example 14.1 on page 348 in the book ''Statistical …'

2014-06-05T19:53:57Z

Created page with '{{Reference Example}} This example compares the Weibull-Bayesian calculation. {{Reference_Example_Heading1}} The data from Example 14.1 on page 348 in the book ''Statistical …'

New page

{{Reference Example}}

This example compares the Weibull-Bayesian calculation.

{{Reference_Example_Heading1}}

The data from Example 14.1 on page 348 in the book ''Statistical Methods for Reliability Data'' by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998 is used.

{{Reference_Example_Heading2}}

{| {{table}}
! Number in State
!State F or S
!Time to Failure
|-
| 288||S||50
|-
| 148||S||150
|-
| 1||F||230
|-
| 124||S||250
|-
| 1||F||334
|-
| 111||S||350
|-
| 1||F||423
|-
| 106||S||450
|-
| 99||S||550
|-
| 110||S||650
|-
| 114||S||750
|-
| 119||S||850
|-
| 127||S||950
|-
| 1||F||990
|-
| 1||F||1009
|-
| 123||S||1050
|-
| 93||S||1150
|-
| 47||S||1250
|-
| 41||S||1350
|-
| 27||S||1450
|-
| 1||F||1510
|-
| 11||S||1550
|-
| 6||S||1650
|-
| 1||S||1850
|-
| 2||S||2050
|-
|}

{{Reference_Example_Heading3}}

In the book, the prior distribution is set for <math>\sigma\,\!</math> with <math>\sigma = \frac{1}{\beta}\,\!</math>. The prior for <math>\sigma\,\!</math> is a lognormal distribution specified by <math>\sigma_{0.005}\,\!</math> = 0.2 and <math>\sigma_{0.995}\,\!</math> = 0.5. The following results are obtained using the Bayesian method:

* The 95% two-sided Bayesian confidence interval for <math>t_{0.05}\,\!</math> (B5% life) is [1613, 3236]. This result is given in Example 14.7 on page 357.
* The 95% two-sided Bayesian confidence interval for <math>t_{0.10}\,\!</math> (B10% life) is [2018, 4400]. This result is given in Example 14.7 on page 357.
* The 95% two-sided Bayesian confidence interval for F(2000) (probability of failure at time of 2000) is [0.015, 0.097]. This result is given in Example 14.8 on page 357.
* The 95% two-sided Bayesian confidence interval for F(5000) (probability of failure at time of 5000) is [0.132, 0.905]. This result is given in Example 14.8 on page 357.

{{Reference_Example_Heading4}}

In Weibull++, the prior distribution is set for <math>\beta\,\!</math> directly. Based on the information of <math>\sigma\,\!</math>, we know <math>\beta_{0.005}\,\!</math> = 2 and <math>\beta_{0.995}\,\!</math> = 5. Therefore, we can get the prior lognormal distribution for <math>\beta\,\!</math>. It is Log-Mean = 1.15129 and Log-Std = 0.17786.

[[Image:x.png|center|550px|]]

Applying this prior distribution for Wei-Bayesian, we have the following results:

* The 95% two-sided Bayesian confidence interval for <math>t_{0.05}\,\!</math> (B5% life) is [1623, 3452].

[[Image:x.png|center|550px|]]

* The 95% two-sided Bayesian confidence interval for <math>t_{0.10}\,\!</math> (B10% life) is [2030, 4763].

[[Image:x.png|center|550px|]]

* The 95% two-sided Bayesian confidence interval for F(2000) (probability of failure at time of 2000) is [0.014, 0.095].

[[Image:x.png|center|550px|]]

* The 95% two-sided Bayesian confidence interval for F(5000) (probability of failure at time of 5000) is [0.111, 0.903].

[[Image:x.png|center|550px|]]

The results in Weibull++ are very close, but not exactly the same, as the results in the book. The differences are mainly caused by the fact that the prior lognormal distribution is for <math>\sigma\,\!</math> in the book while it is for <math>\beta\,\!</math> in Weibull++.