Template:Weibull-bayesian analysis - Revision history

Kate Racaza: Redirected page to Bayesian-Weibull Analysis

2012-08-10T03:18:05Z

Redirected page to Bayesian-Weibull Analysis

Kate Racaza: /* Weibull-Bayesian Analysis */

2012-05-30T11:50:45Z

Weibull-Bayesian Analysis

← Older revision		Revision as of 11:50, 30 May 2012
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	== Weibull~~-Bayesian~~ Analysis ==		== Bayesian-Weibull Analysis ==
	In this section, the Bayesian methods are presented for the two-parameter Weibull distribution. Bayesian concepts were introduced in Chapter [[Parameter Estimation]]. This model considers prior knowledge on the shape (<span class="texhtml">β</span>) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the one-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Weibull-Bayesian model in Weibull++ (which is actually a true "WeiBayes" model, unlike the one-parameter Weibull that is commonly referred to as such) offers an alternative to the one-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the two-parameter Weibull distribution and assuming the prior distributions of <span class="texhtml">β</span> and <span class="texhtml">η</span> are independent, we obtain the following posterior :		In this section, the Bayesian methods are presented for the two-parameter Weibull distribution. Bayesian concepts were introduced in Chapter [[Parameter Estimation]]. This model considers prior knowledge on the shape (<span class="texhtml">β</span>) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the one-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Weibull-Bayesian model in Weibull++ (which is actually a true "WeiBayes" model, unlike the one-parameter Weibull that is commonly referred to as such) offers an alternative to the one-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the two-parameter Weibull distribution and assuming the prior distributions of <span class="texhtml">β</span> and <span class="texhtml">η</span> are independent, we obtain the following posterior :

Nicolette Young: /* Weibull-Bayesian Analysis */

2012-04-25T18:19:32Z

Weibull-Bayesian Analysis

← Older revision		Revision as of 18:19, 25 April 2012
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	In other words, a distribution (the posterior ) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the ''Analysis'' page, under the ''Results As'' area, as shown next.		In other words, a distribution (the posterior ) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the ''Analysis'' page, under the ''Results As'' area, as shown next.

	[[Image:Weibull Distribution Wei-Bayesian for Choose Mean and Median.png\|thumb\|center\|~~250px~~\| ]]		[[Image:Weibull Distribution Wei-Bayesian for Choose Mean and Median.png\|thumb\|center\|200px\| ]]

	The expected value of <span class="texhtml">β</span> is obtained by:		The expected value of <span class="texhtml">β</span> is obtained by:

Nicolette Young: /* Weibull-Bayesian Analysis */

2012-04-25T18:19:08Z

Weibull-Bayesian Analysis

← Older revision		Revision as of 18:19, 25 April 2012
Line 14:		Line 14:
	In other words, a distribution (the posterior ) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the ''Analysis'' page, under the ''Results As'' area, as shown next.		In other words, a distribution (the posterior ) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the ''Analysis'' page, under the ''Results As'' area, as shown next.

	[[Image:Weibull Distribution Wei-Bayesian for Choose Mean and Median.png\|thumb\|center\|~~400px~~\| ]]		[[Image:Weibull Distribution Wei-Bayesian for Choose Mean and Median.png\|thumb\|center\|250px\| ]]

	The expected value of <span class="texhtml">β</span> is obtained by:		The expected value of <span class="texhtml">β</span> is obtained by:

Harry Guo: /* Weibull-Bayesian Analysis */

2012-03-01T23:05:51Z

Weibull-Bayesian Analysis

← Older revision		Revision as of 23:05, 1 March 2012
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	In other words, a distribution (the posterior ) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the ''Analysis'' page, under the ''Results As'' area, as shown next.		In other words, a distribution (the posterior ) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the ''Analysis'' page, under the ''Results As'' area, as shown next.

	[[Image:~~time~~-~~failedID~~.png\|thumb\|center\|400px\| ]]		[[Image:Weibull Distribution Wei-Bayesian for Choose Mean and Median.png\|thumb\|center\|400px\| ]]

	The expected value of <span class="texhtml">β</span> is obtained by:		The expected value of <span class="texhtml">β</span> is obtained by:

Harry Guo: /* Posterior Distributions for Functions of Parameters */

2012-02-10T23:40:11Z

Posterior Distributions for Functions of Parameters

← Older revision		Revision as of 23:40, 10 February 2012
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	The posterior distribution of the failure time is given by:		The posterior distribution of the failure time is given by:

	::<math> f(T\|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }f(T,\beta ,\eta )f(\beta ,\eta \|Data)d\eta d\beta </math> ~~EQNREF WeibBayesPDF~~		::<math> f(T\|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }f(T,\beta ,\eta )f(\beta ,\eta \|Data)d\eta d\beta </math>

	where:		where:

Harry Guo: /* Posterior Distributions for Functions of Parameters */

2012-02-10T23:00:19Z

Posterior Distributions for Functions of Parameters

← Older revision		Revision as of 23:00, 10 February 2012
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	=== Posterior Distributions for Functions of Parameters ===		=== Posterior Distributions for Functions of Parameters ===
	As explained in [[Parameter Estimation ~~\|Chapter 4~~]], in Bayesian analysis, all the functions of the parameters are distributed. In other words, a posterior distribution is obtained for functions such as reliability and failure rate, instead of point estimate as in classical statistics. Therefore, in order to obtain a point estimate for these functions, a point on the posterior distributions needs to be calculated. Again, the expected value (mean) or median value are used.		As explained in Chpater [[Parameter Estimation]], in Bayesian analysis, all the functions of the parameters are distributed. In other words, a posterior distribution is obtained for functions such as reliability and failure rate, instead of point estimate as in classical statistics. Therefore, in order to obtain a point estimate for these functions, a point on the posterior distributions needs to be calculated. Again, the expected value (mean) or median value are used.

	'''<math>pdf</math> of the Times-to-Failure'''		'''<math>pdf</math> of the Times-to-Failure'''

Harry Guo: /* Weibull-Bayesian Analysis */

2012-02-10T22:59:55Z

Weibull-Bayesian Analysis

← Older revision		Revision as of 22:59, 10 February 2012
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	== Weibull-Bayesian Analysis ==		== Weibull-Bayesian Analysis ==
	In this section, the Bayesian methods are presented for the two-parameter Weibull distribution. Bayesian concepts were introduced in [[Parameter Estimation ~~\|Chapter 4~~]]. This model considers prior knowledge on the shape (<span class="texhtml">β</span>) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the one-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Weibull-Bayesian model in Weibull++ (which is actually a true "WeiBayes" model, unlike the one-parameter Weibull that is commonly referred to as such) offers an alternative to the one-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the two-parameter Weibull distribution and assuming the prior distributions of <span class="texhtml">β</span> and <span class="texhtml">η</span> are independent, we obtain the following posterior :		In this section, the Bayesian methods are presented for the two-parameter Weibull distribution. Bayesian concepts were introduced in Chapter [[Parameter Estimation]]. This model considers prior knowledge on the shape (<span class="texhtml">β</span>) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the one-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Weibull-Bayesian model in Weibull++ (which is actually a true "WeiBayes" model, unlike the one-parameter Weibull that is commonly referred to as such) offers an alternative to the one-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the two-parameter Weibull distribution and assuming the prior distributions of <span class="texhtml">β</span> and <span class="texhtml">η</span> are independent, we obtain the following posterior :

	::<math> f(\beta ,\eta \|Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } </math>		::<math> f(\beta ,\eta \|Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } </math>

Harry Guo: /* Weibull-Bayesian Analysis */

2012-02-09T23:42:35Z

Weibull-Bayesian Analysis

Show changes

Harry Guo: /* Confidence Bounds on Time */

2012-02-09T21:56:13Z

Confidence Bounds on Time

← Older revision		Revision as of 21:56, 9 February 2012
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	Using the same method for one-sided bounds, <span class="texhtml">''R''<sub>''U''</sub>(''T'')</span>and <span class="texhtml">''R''<sub>''L''</sub>(''T'')</span> can be computed.		Using the same method for one-sided bounds, <span class="texhtml">''R''<sub>''U''</sub>(''T'')</span>and <span class="texhtml">''R''<sub>''L''</sub>(''T'')</span> can be computed.

	=== ~~Confidence~~ Bounds on Time ===		=== Bounds on Time ===
	Following the same procedure described for bounds on Reliability, the bounds of time can be calculated, given . The Bayesian one-sided upper bound estimate for <span class="texhtml">''T''(''R'')</span> is given by:		Following the same procedure described for bounds on Reliability, the bounds of time can be calculated, given . The Bayesian one-sided upper bound estimate for <span class="texhtml">''T''(''R'')</span> is given by: