The Mixed Weibull Distribution - Revision history

Nicolette Young: /* Reliability Bathtub Curves */

2015-12-22T18:12:31Z

Reliability Bathtub Curves

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	A reliability bathtub curve is nothing more than the graph of the failure rate versus time, over the life of the product. In general, the life stages of the product consist of early, chance and wear-out. Weibull++ allows you to plot this by simply selecting the failure rate plot, as shown next.		A reliability bathtub curve is nothing more than the graph of the failure rate versus time, over the life of the product. In general, the life stages of the product consist of early, chance and wear-out. Weibull++ allows you to plot this by simply selecting the failure rate plot, as shown next.

	[[Image:WBfailure rate vs time.png\|center\|~~550px~~\| ]]		[[Image:WBfailure rate vs time.png\|center\|600px\| ]]

	===Determination of the Burn-in Period===		===Determination of the Burn-in Period===

Nicolette Young: /* Mixed Weibull Confidence Bounds */

2015-12-22T18:12:20Z

Mixed Weibull Confidence Bounds

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	This is graphically illustrated in the following figure. In this plot it can be seen that there are no data points between the last point of the first subpopulation and the first point of the second subpopulation, thus the uncertainty is high, as described by the mathematical model.		This is graphically illustrated in the following figure. In this plot it can be seen that there are no data points between the last point of the first subpopulation and the first point of the second subpopulation, thus the uncertainty is high, as described by the mathematical model.

	[[Image:WBprobabilityweibullplot.png\|center\|~~550px~~\| ]]		[[Image:WBprobabilityweibullplot.png\|center\|600px\| ]]

	Beta binomial bounds can be used instead in these cases, especially when estimations are to be obtained close to these regions.		Beta binomial bounds can be used instead in these cases, especially when estimations are to be obtained close to these regions.

Richard House at 22:47, 25 September 2012

2012-09-25T22:47:04Z

Show changes

Richard House at 17:51, 5 September 2012

2012-09-05T17:51:28Z

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	{{template:LDABOOK\|11\|The Mixed Weibull Distribution}}		{{template:LDABOOK\|11\|The Mixed Weibull Distribution}}
	The mixed Weibull distribution (also known as a ''multimodal Weibull'') is used to model data that do not fall on a straight line on a Weibull probability plot. Data of this type, particularly if the data points follow an S-shape on the probability plot, may be indicative of more than one failure mode at work in the population of failure times. Field data from a given mixed population may frequently represent multiple failure modes. The necessity of determining the life regions where these failure modes occur is apparent when it is realized that the times-to-failure for each mode may follow a distinct Weibull distribution, thus requiring individual mathematical treatment. Another reason is that each failure mode may require a different design change to improve the component's reliability [[Appendix: ~~Weibull References~~\|[19]]].		The mixed Weibull distribution (also known as a ''multimodal Weibull'') is used to model data that do not fall on a straight line on a Weibull probability plot. Data of this type, particularly if the data points follow an S-shape on the probability plot, may be indicative of more than one failure mode at work in the population of failure times. Field data from a given mixed population may frequently represent multiple failure modes. The necessity of determining the life regions where these failure modes occur is apparent when it is realized that the times-to-failure for each mode may follow a distinct Weibull distribution, thus requiring individual mathematical treatment. Another reason is that each failure mode may require a different design change to improve the component's reliability, as discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]].

	A decreasing failure rate is usually encountered during the early life period of components when the substandard components fail and are removed from the population. The failure rate continues to decrease until all such substandard components fail and are removed. This corresponds to a decreasing failure rate. The Weibull distribution having <math>\beta <1\,\!</math> is often used to depict this life characteristic.		A decreasing failure rate is usually encountered during the early life period of components when the substandard components fail and are removed from the population. The failure rate continues to decrease until all such substandard components fail and are removed. This corresponds to a decreasing failure rate. The Weibull distribution having <math>\beta <1\,\!</math> is often used to depict this life characteristic.

Richard House: /* Statistical Background */

2012-09-05T17:50:51Z

Statistical Background

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	::<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1}}}{N}{{R}_{1}}(t)+\frac{{{N}_{2}}}{N}{{R}_{2}}(t)+\frac{{{N}_{3}}}{N}{{R}_{3}}(t)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(t)</math>		::<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1}}}{N}{{R}_{1}}(t)+\frac{{{N}_{2}}}{N}{{R}_{2}}(t)+\frac{{{N}_{3}}}{N}{{R}_{3}}(t)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(t)</math>

	This expression can also be derived by applying Bayes's theorem [[Appendix: ~~Weibull References~~\|[20]]], which says that the reliability of a component drawn at random from a mixed population composed of <math>n\,\!</math> types of failure subpopulations is its reliability, <math>{{R}_{1}}(t)\,\!</math> , given that the component is from subpopulation 1, or <math>\tfrac{{{N}_{1}}}{N}\,\!</math> plus its reliability, <math>{{R}_{2}}(t)\,\!</math> , given that the component is from subpopulation 2, or <math>\tfrac{{{N}_{2}}}{N}\,\!</math> plus its reliability, <math>{{R}_{3}}(t)\,\!</math> , given that the component is from subpopulation 3, or <math>\tfrac{{{N}_{3}}}{N}\,\!</math> , and so on, plus its reliability, <math>{{R}_{n}}(t)\,\!</math> , given that the component is from subpopulation <math>n\,\!</math> , or <math>\tfrac{{{N}_{n}}}{N}\,\!</math> , and:		This expression can also be derived by applying Bayes's theorem, as discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[20]]], which says that the reliability of a component drawn at random from a mixed population composed of <math>n\,\!</math> types of failure subpopulations is its reliability, <math>{{R}_{1}}(t)\,\!</math> , given that the component is from subpopulation 1, or <math>\tfrac{{{N}_{1}}}{N}\,\!</math> plus its reliability, <math>{{R}_{2}}(t)\,\!</math> , given that the component is from subpopulation 2, or <math>\tfrac{{{N}_{2}}}{N}\,\!</math> plus its reliability, <math>{{R}_{3}}(t)\,\!</math> , given that the component is from subpopulation 3, or <math>\tfrac{{{N}_{3}}}{N}\,\!</math> , and so on, plus its reliability, <math>{{R}_{n}}(t)\,\!</math> , given that the component is from subpopulation <math>n\,\!</math> , or <math>\tfrac{{{N}_{n}}}{N}\,\!</math> , and:

	::<math>\underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}=1</math>		::<math>\underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}=1</math>

Richard House: /* Mixed Weibull Confidence Bounds */

2012-08-28T03:25:59Z

Mixed Weibull Confidence Bounds

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	==Mixed Weibull Confidence Bounds==		==Mixed Weibull Confidence Bounds==
	In Weibull++, two methods are available for estimating the confidence bounds for the mixed Weibull distribution. The first method is the beta binomial, described in [[Confidence Bounds]]. The second method is the Fisher matrix confidence bounds.		In Weibull++, two methods are available for estimating the confidence bounds for the mixed Weibull distribution. The first method is the beta binomial, described in [[Confidence Bounds]]. The second method is the Fisher matrix confidence bounds.
	For the Fisher matrix bounds, the methodology is the same as described in [[Confidence Bounds]]. The variance/covariance matrix for the mixed Weibull is a <math>(3\cdot S-1)\times (3\cdot S-1)</math> matrix, where <math>S</math> is the number of subpopulations. Bounds on the parameters, reliability and time are estimated using the same transformations and methods that were used for the [[The Weibull Distribution]] . Note, however, that in addition to the Weibull parameters, the bounds on the subpopulation portions are obtained as well. The bounds on the portions are estimated by:		For the Fisher matrix bounds, the methodology is the same as described in [[Confidence Bounds]]. The variance/covariance matrix for the mixed Weibull is a <math>(3\cdot S-1)\times (3\cdot S-1)</math> matrix, where <math>S\,\!</math> is the number of subpopulations. Bounds on the parameters, reliability and time are estimated using the same transformations and methods that were used for the [[The Weibull Distribution]] . Note, however, that in addition to the Weibull parameters, the bounds on the subpopulation portions are obtained as well. The bounds on the portions are estimated by:

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

	where <math>Var(\widehat{\rho })</math> is obtained from the variance/covariance matrix.		where <math>Var(\widehat{\rho })\,\!</math> is obtained from the variance/covariance matrix.
	When using the Fisher matrix bounds method, problems can occur on the transition points of the distribution, and in particular on the Type 1 confidence bounds (bounds on time). The problems (i.e.. the departure from the expected monotonic behavior) occur when the transition region between two subpopulations becomes a "saddle" (i.e., the probability line is almost parallel to the time axis on a probability plot). In this case, the bounds on time approach infinity. This behavior is more frequently encountered with smaller sample sizes. The physical interpretation is that there is insufficient data to support any inferences when in this region.		When using the Fisher matrix bounds method, problems can occur on the transition points of the distribution, and in particular on the Type 1 confidence bounds (bounds on time). The problems (i.e.. the departure from the expected monotonic behavior) occur when the transition region between two subpopulations becomes a "saddle" (i.e., the probability line is almost parallel to the time axis on a probability plot). In this case, the bounds on time approach infinity. This behavior is more frequently encountered with smaller sample sizes. The physical interpretation is that there is insufficient data to support any inferences when in this region.

Richard House: /* MLE */

2012-08-28T03:25:13Z

MLE

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	'''About the Calculated Parameters'''		'''About the Calculated Parameters'''

	Weibull++ uses the numbers 1, 2, 3 and 4 (or first, second, third and fourth subpopulation) to identify each subpopulation. These are just designations for each subpopulation, and they are ordered based on the value of the scale parameter, <math>\eta </math> . Since the equation used is additive or:		Weibull++ uses the numbers 1, 2, 3 and 4 (or first, second, third and fourth subpopulation) to identify each subpopulation. These are just designations for each subpopulation, and they are ordered based on the value of the scale parameter, <math>\eta \,\!</math> . Since the equation used is additive or:

	::<math>{{R}_{1,..,S}}(T)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}{{e}^{-{{\left( \tfrac{t}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}}}}}</math>		::<math>{{R}_{1,..,S}}(T)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}{{e}^{-{{\left( \tfrac{t}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}}}}}</math>

	the order of the subpopulations which are given the designation 1, 2, 3, or 4 is of no consequence. For consistency, the application will always return the order of the results based on the magnitude of the scale parameter.		the order of the subpopulations which are given the designation 1, 2, 3, or 4 is of no consequence. For consistency, the application will always return the order of the results based on the magnitude of the scale parameter.


	==Mixed Weibull Confidence Bounds==		==Mixed Weibull Confidence Bounds==

Richard House: /* Regression Solution */

2012-08-28T03:24:48Z

Regression Solution

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	::<math>\underset{i=1}{\overset{S}{\mathop \sum }}\,{{\rho }_{i}}=1</math>		::<math>\underset{i=1}{\overset{S}{\mathop \sum }}\,{{\rho }_{i}}=1</math>

	to the parameters <math>\widehat{{{\rho }_{1,\text{ }}}}</math> <math>\widehat{{{\beta }_{1}}},</math> <math>\widehat{{{\eta }_{1}}},</math> <math>\widehat{{{\rho }_{2,\text{ }}}}\widehat{{{\beta }_{2}}},</math> <math>\widehat{{{\eta }_{2}}},...,</math> <math>\widehat{{{\rho }_{S,}}\text{ }}\widehat{{{\beta }_{S}}},</math> <math>\widehat{{{\eta }_{S}}},</math> utilizing the times-to-failure and their respective plotting positions. It is important to note that in the case of regression analysis, using a mixed Weibull model, the choice of regression axis, (i.e., <math>RRX</math> or <math>RRY,</math> is of no consequence since non-linear regression is utilized).		to the parameters <math>\widehat{{{\rho }_{1,\text{ }}}}\,\!</math> <math>\widehat{{{\beta }_{1}}},\,\!</math> <math>\widehat{{{\eta }_{1}}},\,\!</math> <math>\widehat{{{\rho }_{2,\text{ }}}}\widehat{{{\beta }_{2}}},\,\!</math> <math>\widehat{{{\eta }_{2}}},...,\,\!</math> <math>\widehat{{{\rho }_{S,}}\text{ }}\widehat{{{\beta }_{S}}},\,\!</math> <math>\widehat{{{\eta }_{S}}},\,\!</math> utilizing the times-to-failure and their respective plotting positions. It is important to note that in the case of regression analysis, using a mixed Weibull model, the choice of regression axis, (i.e., <math>RRX\,\!</math> or <math>RRY,\,\!</math> is of no consequence since non-linear regression is utilized).

	===MLE===		===MLE===

Richard House: /* The Mixed Weibull Equations */

2012-08-28T03:23:57Z

The Mixed Weibull Equations

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	::<math>{{f}_{1,...,S}}(t)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{{{N}_{i}}{{\beta }_{i}}}{N{{\eta }_{i}}}{{\left( \frac{t}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}-1}}{{e}^{-{{(\tfrac{t}{{{\eta }_{i}}})}^{{{\beta }_{i}}}}}}</math>		::<math>{{f}_{1,...,S}}(t)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{{{N}_{i}}{{\beta }_{i}}}{N{{\eta }_{i}}}{{\left( \frac{t}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}-1}}{{e}^{-{{(\tfrac{t}{{{\eta }_{i}}})}^{{{\beta }_{i}}}}}}</math>

	where <math>S=2</math> , <math>S=3</math> , and <math>S=4</math> for 2, 3 and 4 subpopulations respectively. Weibull++ uses a non-linear regression method or direct maximum likelihood methods to estimate the parameters.		where <math>S=2\,\!</math> , <math>S=3\,\!</math> , and <math>S=4\,\!</math> for 2, 3 and 4 subpopulations respectively. Weibull++ uses a non-linear regression method or direct maximum likelihood methods to estimate the parameters.


	==Mixed Weibull Parameter Estimation==		==Mixed Weibull Parameter Estimation==

Richard House: /* Statistical Background */

2012-08-28T03:23:30Z

Statistical Background

← Older revision		Revision as of 03:23, 28 August 2012
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	==Statistical Background==		==Statistical Background==
	Consider a life test of identical components. The components were placed in a test at age <math>t=0</math> and were tested to failure, with their times-to-failure recorded. Further assume that the test covered the entire lifespan of the units, and different failure modes were observed over each region of life, namely early life (early failure mode), chance life (chance failure mode), and wear-out life (wear-out failure mode). Also, as items failed during the test, they were removed from the test, inspected and segregated into lots according to their failure mode. At the conclusion of the test, there will be <math>n</math> subpopulations of <math>{{N}_{1}},{{N}_{2}},{{N}_{3}},...,{{N}_{n}}</math> failed components. If the events of the test are now reconstructed, it may be theorized that at age <math>t=0</math> there were actually <math>n</math> separate subpopulations in the test, each with a different times-to-failure distribution and failure mode, even though at <math>t=0</math> the subpopulations were not physically distinguishable. The mixed Weibull methodology accomplishes this segregation based on the results of the life test.		Consider a life test of identical components. The components were placed in a test at age <math>t=0\,\!</math> and were tested to failure, with their times-to-failure recorded. Further assume that the test covered the entire lifespan of the units, and different failure modes were observed over each region of life, namely early life (early failure mode), chance life (chance failure mode), and wear-out life (wear-out failure mode). Also, as items failed during the test, they were removed from the test, inspected and segregated into lots according to their failure mode. At the conclusion of the test, there will be <math>n\,\!</math> subpopulations of <math>{{N}_{1}},{{N}_{2}},{{N}_{3}},...,{{N}_{n}}\,\!</math> failed components. If the events of the test are now reconstructed, it may be theorized that at age <math>t=0\,\!</math> there were actually <math>n\,\!</math> separate subpopulations in the test, each with a different times-to-failure distribution and failure mode, even though at <math>t=0\,\!</math> the subpopulations were not physically distinguishable. The mixed Weibull methodology accomplishes this segregation based on the results of the life test.

	If <math>N</math> identical components from a mixed population undertake a mission of <math>t</math> duration, starting the mission at age zero, then the number of components surviving this mission can be found from the following definition of reliability:		If <math>N\,\!</math> identical components from a mixed population undertake a mission of <math>t\,\!</math> duration, starting the mission at age zero, then the number of components surviving this mission can be found from the following definition of reliability:

	::<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1,2,3,..,{{n}_{S}}}}(t)}{N}</math>		::<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1,2,3,..,{{n}_{S}}}}(t)}{N}</math>
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	\end{align}</math>		\end{align}</math>

	The total number surviving by age <math>t</math> in the mixed population is the sum of the number surviving in all subpopulations or:		The total number surviving by age <math>t\,\!</math> in the mixed population is the sum of the number surviving in all subpopulations or:

	::<math>{{N}_{1,2,...,{{n}_{S}}}}(t)={{N}_{{{1}_{S}}}}(t)+{{N}_{{{2}_{S}}}}(t)+{{N}_{{{3}_{S}}}}(t)+\cdots +{{N}_{{{n}_{S}}}}(t)</math>		::<math>{{N}_{1,2,...,{{n}_{S}}}}(t)={{N}_{{{1}_{S}}}}(t)+{{N}_{{{2}_{S}}}}(t)+{{N}_{{{3}_{S}}}}(t)+\cdots +{{N}_{{{n}_{S}}}}(t)</math>
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	::<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1}}}{N}{{R}_{1}}(t)+\frac{{{N}_{2}}}{N}{{R}_{2}}(t)+\frac{{{N}_{3}}}{N}{{R}_{3}}(t)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(t)</math>		::<math>{{R}_{1,2,...,n}}(t)=\frac{{{N}_{1}}}{N}{{R}_{1}}(t)+\frac{{{N}_{2}}}{N}{{R}_{2}}(t)+\frac{{{N}_{3}}}{N}{{R}_{3}}(t)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(t)</math>

	This expression can also be derived by applying Bayes's theorem [[Appendix: Weibull References\|[20]]], which says that the reliability of a component drawn at random from a mixed population composed of <math>n</math> types of failure subpopulations is its reliability, <math>{{R}_{1}}(t)\,\!</math> , given that the component is from subpopulation 1, or <math>\tfrac{{{N}_{1}}}{N}</math> plus its reliability, <math>{{R}_{2}}(t)\,\!</math> , given that the component is from subpopulation 2, or <math>\tfrac{{{N}_{2}}}{N}</math> plus its reliability, <math>{{R}_{3}}(t)\,\!</math> , given that the component is from subpopulation 3, or <math>\tfrac{{{N}_{3}}}{N}</math> , and so on, plus its reliability, <math>{{R}_{n}}(t)\,\!</math> , given that the component is from subpopulation <math>n\,\!</math> , or <math>\tfrac{{{N}_{n}}}{N}</math> , and:		This expression can also be derived by applying Bayes's theorem [[Appendix: Weibull References\|[20]]], which says that the reliability of a component drawn at random from a mixed population composed of <math>n\,\!</math> types of failure subpopulations is its reliability, <math>{{R}_{1}}(t)\,\!</math> , given that the component is from subpopulation 1, or <math>\tfrac{{{N}_{1}}}{N}\,\!</math> plus its reliability, <math>{{R}_{2}}(t)\,\!</math> , given that the component is from subpopulation 2, or <math>\tfrac{{{N}_{2}}}{N}\,\!</math> plus its reliability, <math>{{R}_{3}}(t)\,\!</math> , given that the component is from subpopulation 3, or <math>\tfrac{{{N}_{3}}}{N}\,\!</math> , and so on, plus its reliability, <math>{{R}_{n}}(t)\,\!</math> , given that the component is from subpopulation <math>n\,\!</math> , or <math>\tfrac{{{N}_{n}}}{N}\,\!</math> , and:

	::<math>\underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}=1</math>		::<math>\underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}=1</math>
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	\end{align}</math>		\end{align}</math>

	The conditional reliability for a new mission of duration <math>t</math> , starting this mission at age <math>T</math> , or after having already operated a total of <math>T</math> hours, is given by:		The conditional reliability for a new mission of duration <math>t\,\!</math> , starting this mission at age <math>T\,\!</math> , or after having already operated a total of <math>T\,\!</math> hours, is given by:

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