Template:Example: Wei-Bayesian Log-normal Prior - Revision history

Lisa Hacker: Redirected page to Bayesian-Weibull Lognormal Prior Example

2012-08-13T01:33:35Z

Redirected page to Bayesian-Weibull Lognormal Prior Example

← Older revision		Revision as of 01:33, 13 August 2012
Line 1:		Line 1:
	[[~~Example: Wei-~~Bayesian ~~Log-normal Prior]]~~		#REDIRECT [[Bayesian-Weibull_Lognormal_Prior_Example]]
	~~A manufacturer has tested prototypes of a modified product. The test was terminated at 2000 hours, with only two failures observed from a sample size of eighteen.~~

	~~{\| border=1 cellspacing=1 align="center"~~
	\|-
	~~!Number of State~~
	~~!State of F or S~~
	~~!State End Time~~
	\|-
	~~\| 1 \|\| F \|\| 1180~~
	\|-
	~~\| 1 \|\| F \|\| 1842~~
	\|-
	~~\| 16 \|\| S \|\| 2000~~
	\|}

	Because of the lack of failure data in the prototype testing, the manufacturer decided to use information gathered from prior tests on this product to increase the confidence in the results of the prototype testing. This decision was made because failure analysis indicated that the failure mode of these two failures is the same as the one observed in previous tests. In other words, it is expected that the shape of the distribution hasn't changed, but hopefully the scale has, indicating longer life. The two-parameter Weibull distribution have been used to model all prior tests results. The list of the estimated <span class="texhtml">β</span> parameter is as follows:

	~~{\| border=1 cellspacing=1 align="center"~~
	\|-
	~~!Betas Obtained for Similar Mode~~
	\|-
	~~\| 1.7~~
	\|-
	~~\| 2.1~~
	\|-
	~~\| 2.4~~
	\|-
	~~\|3.1~~
	\|-
	~~\|3.5~~
	\|}

	~~'''Solution'''~~

	First, in order to fit the data to a Weibull-Bayesian model, a prior distribution for <span class="texhtml">β</span> needs to be determined. Based on the prior tests' <span class="texhtml">β</span> values, the prior distribution for <span class="texhtml">β</span> was found to be a lognormal distribution with <span class="texhtml">μ = 0.9064</span>, <span class="texhtml">σ = 0.3325</span> (obtained by entering the <span class="texhtml">β</span> values into a Weibull++ ''Standard Folio'' and analyzing it based on the RRX analysis method.)

	the test data is entered into a ''Standard Folio'', the '''B-W Lognormal Prior''' is selected under '' Distribution'' and the <span class="texhtml">β</span> prior distribution is entered after clicking the ''Calculate'' button.

	~~[[Image:Weibull Distribution Example 6 Data and Prior.png\|thumb\|center\|250px\| ]]~~

	Suppose that the reliability at 3000hr is the metric of interest in this example. This reliability can be obtained using the equation for median reliability, resulting in the median value of the posterior of the reliability at 3000hr. Using the ''QCP'', this value is calculated to be 76.97%. ( By default Weibull++ returns the median values of the posterior distribution. )

	The posterior <math>pdf</math> of the reliability function at 3000hrs can be obtained using the equation at the begining of this section. In Figure 6-10 the posterior <math>pdf</math> of the reliability at 3000hrs is plotted, with the corresponding median value as well as the 10th percentile value shown. The 10th percentile constitutes the 90 Lower 1-Sided bound on the reliability at 3000hrs, which is calculated to be 50.77%.

	~~[[Image:WB.8 pdf median.png\|center\|250px\| ]]~~

	Notice that the <math>pdf</math> plotted in the above Figure is of the reliability at 3000hrs, and not the <math>pdf</math> of the times-to-failure data. The <math>pdf</math> of the times-to-failure data can be obtained using the equation at the begining of this section and plotted using Weibull++, as shown next:

	~~[[Image:Weibull Distribution Example 6 pdf.png\|thumb\|center\|250px\| ]]~~

	~~<br>~~


	~~[[Category:Life_Data_Analysis_Reference~~]]

Kate Racaza: removed RS Copyright template

2012-07-06T11:53:18Z

removed RS Copyright template

← Older revision		Revision as of 11:53, 6 July 2012
Line 51:		Line 51:

	<br>		<br>
	~~{{RS Copyright}}~~

	[[Category:Life_Data_Analysis_Reference]]		[[Category:Life_Data_Analysis_Reference]]

Nicolette Young at 18:20, 25 April 2012

2012-04-25T18:20:11Z

← Older revision		Revision as of 18:20, 25 April 2012
Line 1:		Line 1:

	[[Example: Wei-Bayesian Log-normal Prior]]		[[Example: Wei-Bayesian Log-normal Prior]]
	A manufacturer has tested prototypes of a modified product. The test was terminated at 2000 hours, with only two failures observed from a sample size of eighteen.		A manufacturer has tested prototypes of a modified product. The test was terminated at 2000 hours, with only two failures observed from a sample size of eighteen.
Line 39:		Line 38:
	the test data is entered into a ''Standard Folio'', the '''B-W Lognormal Prior''' is selected under '' Distribution'' and the <span class="texhtml">β</span> prior distribution is entered after clicking the ''Calculate'' button.		the test data is entered into a ''Standard Folio'', the '''B-W Lognormal Prior''' is selected under '' Distribution'' and the <span class="texhtml">β</span> prior distribution is entered after clicking the ''Calculate'' button.

	[[Image:Weibull Distribution Example 6 Data and Prior.png\|thumb\|center\|~~500px~~\| ]]		[[Image:Weibull Distribution Example 6 Data and Prior.png\|thumb\|center\|250px\| ]]

	Suppose that the reliability at 3000hr is the metric of interest in this example. This reliability can be obtained using the equation for median reliability, resulting in the median value of the posterior of the reliability at 3000hr. Using the ''QCP'', this value is calculated to be 76.97%. ( By default Weibull++ returns the median values of the posterior distribution. )		Suppose that the reliability at 3000hr is the metric of interest in this example. This reliability can be obtained using the equation for median reliability, resulting in the median value of the posterior of the reliability at 3000hr. Using the ''QCP'', this value is calculated to be 76.97%. ( By default Weibull++ returns the median values of the posterior distribution. )
Line 45:		Line 44:
	The posterior <math>pdf</math> of the reliability function at 3000hrs can be obtained using the equation at the begining of this section. In Figure 6-10 the posterior <math>pdf</math> of the reliability at 3000hrs is plotted, with the corresponding median value as well as the 10th percentile value shown. The 10th percentile constitutes the 90 Lower 1-Sided bound on the reliability at 3000hrs, which is calculated to be 50.77%.		The posterior <math>pdf</math> of the reliability function at 3000hrs can be obtained using the equation at the begining of this section. In Figure 6-10 the posterior <math>pdf</math> of the reliability at 3000hrs is plotted, with the corresponding median value as well as the 10th percentile value shown. The 10th percentile constitutes the 90 Lower 1-Sided bound on the reliability at 3000hrs, which is calculated to be 50.77%.

	[[Image:WB.8 pdf median.png\|center\|~~400px~~\| ]]		[[Image:WB.8 pdf median.png\|center\|250px\| ]]

	Notice that the <math>pdf</math> plotted in the above Figure is of the reliability at 3000hrs, and not the <math>pdf</math> of the times-to-failure data. The <math>pdf</math> of the times-to-failure data can be obtained using the equation at the begining of this section and plotted using Weibull++, as shown next:		Notice that the <math>pdf</math> plotted in the above Figure is of the reliability at 3000hrs, and not the <math>pdf</math> of the times-to-failure data. The <math>pdf</math> of the times-to-failure data can be obtained using the equation at the begining of this section and plotted using Weibull++, as shown next:

	[[Image:Weibull Distribution Example 6 pdf.png\|thumb\|center\|~~400px~~\| ]]		[[Image:Weibull Distribution Example 6 pdf.png\|thumb\|center\|250px\| ]]

	<br>		<br>

Nicolette Young at 21:55, 2 April 2012

2012-04-02T21:55:09Z

← Older revision		Revision as of 21:55, 2 April 2012
Line 1:		Line 1:
	~~{\| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;"~~
	\|-
	~~\| valign="middle" align="left" bgcolor=EEEDF7\|[[Image:Weibull-Examples-banner.png\|400px\|center]]~~
	\|}
	[[Example: Wei-Bayesian Log-normal Prior]]		[[Example: Wei-Bayesian Log-normal Prior]]
	A manufacturer has tested prototypes of a modified product. The test was terminated at 2000 hours, with only two failures observed from a sample size of eighteen.		A manufacturer has tested prototypes of a modified product. The test was terminated at 2000 hours, with only two failures observed from a sample size of eighteen.

Chuck Smith at 23:10, 21 March 2012

2012-03-21T23:10:03Z

← Older revision		Revision as of 23:10, 21 March 2012
Line 3:		Line 3:
	\| valign="middle" align="left" bgcolor=EEEDF7\|[[Image:Weibull-Examples-banner.png\|400px\|center]]		\| valign="middle" align="left" bgcolor=EEEDF7\|[[Image:Weibull-Examples-banner.png\|400px\|center]]
	\|}		\|}
			[[Example: Wei-Bayesian Log-normal Prior]]
	A manufacturer has tested prototypes of a modified product. The test was terminated at 2000 hours, with only two failures observed from a sample size of eighteen.		A manufacturer has tested prototypes of a modified product. The test was terminated at 2000 hours, with only two failures observed from a sample size of eighteen.

Chuck Smith at 23:08, 21 March 2012

2012-03-21T23:08:50Z

← Older revision		Revision as of 23:08, 21 March 2012
Line 1:		Line 1:
	~~'''~~Weibull-~~Bayesian with Lognormal Prior Example'''~~		{\| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;"
			\|-
			\| valign="middle" align="left" bgcolor=EEEDF7\|[[Image:Weibull-Examples-banner.png\|400px\|center]]
			\|}
	A manufacturer has tested prototypes of a modified product. The test was terminated at 2000 hours, with only two failures observed from a sample size of eighteen.		A manufacturer has tested prototypes of a modified product. The test was terminated at 2000 hours, with only two failures observed from a sample size of eighteen.

Nicolette Young at 16:52, 15 March 2012

2012-03-15T16:52:47Z

← Older revision		Revision as of 16:52, 15 March 2012
Line 45:		Line 45:
	The posterior <math>pdf</math> of the reliability function at 3000hrs can be obtained using the equation at the begining of this section. In Figure 6-10 the posterior <math>pdf</math> of the reliability at 3000hrs is plotted, with the corresponding median value as well as the 10th percentile value shown. The 10th percentile constitutes the 90 Lower 1-Sided bound on the reliability at 3000hrs, which is calculated to be 50.77%.		The posterior <math>pdf</math> of the reliability function at 3000hrs can be obtained using the equation at the begining of this section. In Figure 6-10 the posterior <math>pdf</math> of the reliability at 3000hrs is plotted, with the corresponding median value as well as the 10th percentile value shown. The 10th percentile constitutes the 90 Lower 1-Sided bound on the reliability at 3000hrs, which is calculated to be 50.77%.

	[[Image:~~lda6~~.10.~~gif\|thumb~~\|center\|~~300px~~\| ]]		[[Image:WB.8 pdf median.png\|center\|400px\| ]]

	Notice that the <math>pdf</math> plotted in the above Figure is of the reliability at 3000hrs, and not the <math>pdf</math> of the times-to-failure data. The <math>pdf</math> of the times-to-failure data can be obtained using the equation at the begining of this section and plotted using Weibull++, as shown next:		Notice that the <math>pdf</math> plotted in the above Figure is of the reliability at 3000hrs, and not the <math>pdf</math> of the times-to-failure data. The <math>pdf</math> of the times-to-failure data can be obtained using the equation at the begining of this section and plotted using Weibull++, as shown next:

Harry Guo at 23:15, 1 March 2012

2012-03-01T23:15:35Z

← Older revision		Revision as of 23:15, 1 March 2012
Line 47:		Line 47:
	[[Image:lda6.10.gif\|thumb\|center\|300px\| ]]		[[Image:lda6.10.gif\|thumb\|center\|300px\| ]]

	Notice that the <math>pdf</math> plotted in ~~Fig. 6-10~~ is of the reliability at 3000hrs, and not the <math>pdf</math> of the times-to-failure data. The <math>pdf</math> of the times-to-failure data can be obtained using the equation at the begining of this section and plotted using Weibull++, as shown next:		Notice that the <math>pdf</math> plotted in the above Figure is of the reliability at 3000hrs, and not the <math>pdf</math> of the times-to-failure data. The <math>pdf</math> of the times-to-failure data can be obtained using the equation at the begining of this section and plotted using Weibull++, as shown next:

	[[Image:Weibull Distribution Example 6 pdf.png\|thumb\|center\|400px\| ]]		[[Image:Weibull Distribution Example 6 pdf.png\|thumb\|center\|400px\| ]]

Harry Guo at 23:14, 1 March 2012

2012-03-01T23:14:47Z

← Older revision		Revision as of 23:14, 1 March 2012
Line 41:		Line 41:
	[[Image:Weibull Distribution Example 6 Data and Prior.png\|thumb\|center\|500px\| ]]		[[Image:Weibull Distribution Example 6 Data and Prior.png\|thumb\|center\|500px\| ]]

	Suppose that the reliability at 3000hr is the metric of interest in this example. This reliability can be obtained using the equation for median reliability, resulting in the median value of the posterior of the reliability at 3000hr. Using the ''QCP'', this value is calculated to be 76.97. ( By default Weibull++ returns the median values of the posterior distribution. )		Suppose that the reliability at 3000hr is the metric of interest in this example. This reliability can be obtained using the equation for median reliability, resulting in the median value of the posterior of the reliability at 3000hr. Using the ''QCP'', this value is calculated to be 76.97%. ( By default Weibull++ returns the median values of the posterior distribution. )

	The posterior <math>pdf</math> of the reliability function at 3000hrs can be obtained using the equation at the begining of this section. In Figure 6-10 the posterior <math>pdf</math> of the reliability at 3000hrs is plotted, with the corresponding median value as well as the 10th percentile value shown. The 10th percentile constitutes the 90 Lower 1-Sided bound on the reliability at 3000hrs, which is calculated to be 50.77.		The posterior <math>pdf</math> of the reliability function at 3000hrs can be obtained using the equation at the begining of this section. In Figure 6-10 the posterior <math>pdf</math> of the reliability at 3000hrs is plotted, with the corresponding median value as well as the 10th percentile value shown. The 10th percentile constitutes the 90 Lower 1-Sided bound on the reliability at 3000hrs, which is calculated to be 50.77%.

	[[Image:lda6.10.gif\|thumb\|center\|300px\| ]]		[[Image:lda6.10.gif\|thumb\|center\|300px\| ]]

Harry Guo at 23:13, 1 March 2012

2012-03-01T23:13:55Z

← Older revision		Revision as of 23:13, 1 March 2012
Line 39:		Line 39:
	the test data is entered into a ''Standard Folio'', the '''B-W Lognormal Prior''' is selected under '' Distribution'' and the <span class="texhtml">β</span> prior distribution is entered after clicking the ''Calculate'' button.		the test data is entered into a ''Standard Folio'', the '''B-W Lognormal Prior''' is selected under '' Distribution'' and the <span class="texhtml">β</span> prior distribution is entered after clicking the ''Calculate'' button.

	[[Image:Weibull Distribution Example 6 Data and Prior.png\|thumb\|center\|~~300px~~\| ]]		[[Image:Weibull Distribution Example 6 Data and Prior.png\|thumb\|center\|500px\| ]]

	Suppose that the reliability at 3000hr is the metric of interest in this example. This reliability can be obtained using the equation for median reliability, resulting in the median value of the posterior of the reliability at 3000hr. Using the ''QCP'', this value is calculated to be 76.97. ( By default Weibull++ returns the median values of the posterior distribution. )		Suppose that the reliability at 3000hr is the metric of interest in this example. This reliability can be obtained using the equation for median reliability, resulting in the median value of the posterior of the reliability at 3000hr. Using the ''QCP'', this value is calculated to be 76.97. ( By default Weibull++ returns the median values of the posterior distribution. )