Recurrent Event Data Analysis - Revision history

Kate Racaza at 18:27, 1 December 2015

2015-12-01T18:27:59Z

← Older revision		Revision as of 18:27, 1 December 2015
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	= Non-Parametric Recurrent Event Data Analysis = <!-- THIS SECTION HEADER IS LINKED FROM ANOTHER LOCATION IN THIS DOCUMENT. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->		= Non-Parametric Recurrent Event Data Analysis = <!-- THIS SECTION HEADER IS LINKED FROM ANOTHER LOCATION IN THIS DOCUMENT. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
			<div class="noprint">
	{{:Non-Parametric_Recurrent_Event_Data_Analysis}}		{{:Non-Parametric_Recurrent_Event_Data_Analysis}}
			</div>

	= Parametric Recurrent Event Data Analysis = <!-- THIS SECTION HEADER IS LINKED FROM ANOTHER LOCATION IN THIS DOCUMENT. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->		= Parametric Recurrent Event Data Analysis = <!-- THIS SECTION HEADER IS LINKED FROM ANOTHER LOCATION IN THIS DOCUMENT. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
	{{:Parametric_Recurrent_Event_Data_Analysis}}		{{:Parametric_Recurrent_Event_Data_Analysis}}

Richard House at 17:14, 5 September 2012

2012-09-05T17:14:09Z

← Older revision		Revision as of 17:14, 5 September 2012
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	Weibull++ provides both parametric and non-parametric approaches to analyze such data.		Weibull++ provides both parametric and non-parametric approaches to analyze such data.

	*The [[Recurrent_Event_Data_Analysis#Non-Parametric_Recurrent_Event_Data_Analysis\|non-parametric approach]] is based on the well-known Mean Cumulative Function (MCF). The Weibull++ module for this type of analysis builds upon the work of Dr. Wayne Nelson, who has written extensively on the calculation and applications of MCF [[Appendix: ~~Weibull References~~\|[31]]].		*The [[Recurrent_Event_Data_Analysis#Non-Parametric_Recurrent_Event_Data_Analysis\|non-parametric approach]] is based on the well-known Mean Cumulative Function (MCF). The Weibull++ module for this type of analysis builds upon the work of Dr. Wayne Nelson, who has written extensively on the calculation and applications of MCF [[Appendix:_Life_Data_Analysis_References\|[31]]].

	*The [[Recurrent_Event_Data_Analysis#Parametric_Recurrent_Event_Data_Analysis\|parametric approach]] is based on the General Renewal Process (GRP) model, which is particularly useful in understanding the effects of the repairs on the age of a system. Traditionally, the commonly used models for analyzing repairable systems data are the perfect renewal processes (PRP), which corresponds to perfect repairs, and the nonhomogeneous Poisson processes (NHPP), which corresponds to minimal repairs. However, most repair activities may realistically not result in such extreme situations but in a complicated intermediate one (general repair or imperfect repair/maintenance), which are well treated with the GRP model.		*The [[Recurrent_Event_Data_Analysis#Parametric_Recurrent_Event_Data_Analysis\|parametric approach]] is based on the General Renewal Process (GRP) model, which is particularly useful in understanding the effects of the repairs on the age of a system. Traditionally, the commonly used models for analyzing repairable systems data are the perfect renewal processes (PRP), which corresponds to perfect repairs, and the nonhomogeneous Poisson processes (NHPP), which corresponds to minimal repairs. However, most repair activities may realistically not result in such extreme situations but in a complicated intermediate one (general repair or imperfect repair/maintenance), which are well treated with the GRP model.

Lisa Hacker at 07:14, 14 August 2012

2012-08-14T07:14:49Z

← Older revision		Revision as of 07:14, 14 August 2012
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	{{template:LDABOOK\|20\|Non-Parametric Recurrent Events Data Analysis}}		{{template:LDABOOK\|20\|Non-Parametric Recurrent Events Data Analysis}}
	Recurrent Event Data Analysis (RDA) is used in various applied fields such as reliability, medicine, social sciences, economics, business and criminology.		Recurrent Event Data Analysis (RDA) is used in various applied fields such as reliability, medicine, social sciences, economics, business and criminology. Whereas in life data analysis (LDA) it was assumed that events (failures) were independent and identically distributed (iid), there are many cases where events are dependent and not identically distributed (such as repairable system data) or where the analyst is interested in modeling the number of occurrences of events over time rather than the length of time prior to the first event, as in LDA.

	Whereas in life data analysis (LDA) it was assumed that events (failures) were independent and identically distributed (iid), there are many cases where events are dependent and not identically distributed (such as repairable system data) or where the analyst is interested in modeling the number of occurrences of events over time rather than the length of time prior to the first event, as in LDA.

	Weibull++ provides both parametric and non-parametric approaches to analyze such data.		Weibull++ provides both parametric and non-parametric approaches to analyze such data.

	*The non-parametric approach is based on the well-known Mean Cumulative Function (MCF). The Weibull++ module for this type of analysis builds upon the work of Dr. Wayne Nelson, who has written extensively on the calculation and applications of MCF [[Appendix: Weibull References\|[31]]].		*The [[Recurrent_Event_Data_Analysis#Non-Parametric_Recurrent_Event_Data_Analysis\|non-parametric approach]] is based on the well-known Mean Cumulative Function (MCF). The Weibull++ module for this type of analysis builds upon the work of Dr. Wayne Nelson, who has written extensively on the calculation and applications of MCF [[Appendix: Weibull References\|[31]]].

	*The parametric approach is based on the General Renewal Process (GRP) model, which is particularly useful in understanding the effects of the repairs on the age of a system. Traditionally, the commonly used models for analyzing repairable systems data are the perfect renewal processes (PRP), which corresponds to perfect repairs, and the nonhomogeneous Poisson processes (NHPP), which corresponds to minimal repairs. However, most repair activities may realistically not result in such extreme situations but in a complicated intermediate one (general repair or imperfect repair/maintenance), which are well treated with the GRP model.		*The [[Recurrent_Event_Data_Analysis#Parametric_Recurrent_Event_Data_Analysis\|parametric approach]] is based on the General Renewal Process (GRP) model, which is particularly useful in understanding the effects of the repairs on the age of a system. Traditionally, the commonly used models for analyzing repairable systems data are the perfect renewal processes (PRP), which corresponds to perfect repairs, and the nonhomogeneous Poisson processes (NHPP), which corresponds to minimal repairs. However, most repair activities may realistically not result in such extreme situations but in a complicated intermediate one (general repair or imperfect repair/maintenance), which are well treated with the GRP model.


	= Non-Parametric Recurrent Event Data Analysis =		= Non-Parametric Recurrent Event Data Analysis = <!-- THIS SECTION HEADER IS LINKED FROM ANOTHER LOCATION IN THIS DOCUMENT. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
	{{:Non-Parametric_Recurrent_Event_Data_Analysis}}		{{:Non-Parametric_Recurrent_Event_Data_Analysis}}


	= Parametric Recurrent Event Data Analysis =		= Parametric Recurrent Event Data Analysis = <!-- THIS SECTION HEADER IS LINKED FROM ANOTHER LOCATION IN THIS DOCUMENT. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
	{{:Parametric_Recurrent_Event_Data_Analysis}}		{{:Parametric_Recurrent_Event_Data_Analysis}}

Kate Racaza at 05:25, 10 August 2012

2012-08-10T05:25:17Z

Show changes

Lisa Hacker at 08:50, 9 August 2012

2012-08-09T08:50:02Z

← Older revision		Revision as of 08:50, 9 August 2012
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	{{template:LDABOOK\|20\|Non-Parametric Recurrent Events Data Analysis}}		{{template:LDABOOK\|20\|Non-Parametric Recurrent Events Data Analysis}}

	Recurrent Event Data Analysis (RDA) is used in various applied fields such as reliability, medicine, social sciences, economics, business and criminology.		Recurrent Event Data Analysis (RDA) is used in various applied fields such as reliability, medicine, social sciences, economics, business and criminology.

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	Weibull++ provides both parametric and non-parametric approaches to analyze such data.		Weibull++ provides both parametric and non-parametric approaches to analyze such data.

	:*The non-parametric approach is based on the well-known Mean Cumulative Function (MCF). The Weibull++ module for this type of analysis builds upon the work of Dr. Wayne Nelson, who has written extensively on the calculation and applications of MCF [[Appendix: Weibull References\|[31]]].		*The non-parametric approach is based on the well-known Mean Cumulative Function (MCF). The Weibull++ module for this type of analysis builds upon the work of Dr. Wayne Nelson, who has written extensively on the calculation and applications of MCF [[Appendix: Weibull References\|[31]]].

	:*The parametric approach is based on the General Renewal Process (GRP) model, which is particularly useful in understanding the effects of the repairs on the age of a system. Traditionally, the commonly used models for analyzing repairable systems data are the perfect renewal processes (PRP), which corresponds to perfect repairs, and the nonhomogeneous Poisson processes (NHPP), which corresponds to minimal repairs. However, most repair activities may realistically not result in such extreme situations but in a complicated intermediate one (general repair or imperfect repair/maintenance), which are well treated with the GRP model.		*The parametric approach is based on the General Renewal Process (GRP) model, which is particularly useful in understanding the effects of the repairs on the age of a system. Traditionally, the commonly used models for analyzing repairable systems data are the perfect renewal processes (PRP), which corresponds to perfect repairs, and the nonhomogeneous Poisson processes (NHPP), which corresponds to minimal repairs. However, most repair activities may realistically not result in such extreme situations but in a complicated intermediate one (general repair or imperfect repair/maintenance), which are well treated with the GRP model.

	= Non-Parametric Recurrent Event Data Analysis =		= Non-Parametric Recurrent Event Data Analysis =

Richard House: /* Confidence Bounds */

2012-08-08T05:33:39Z

Confidence Bounds

← Older revision		Revision as of 05:33, 8 August 2012
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	== Confidence Bounds ==		== Confidence Bounds ==
	In general, in order to obtain the virtual age, the exact occurrence time of each event (failure) should be available (see equations for Type I and Type II models). However, the times are unknown until the corresponding events occur. For this reason, there are no closed-form expressions for total failure number and failure intensity, which are functions of failure times and virtual age. Therefore, in Weibull++, a Monte Carlo simulation is used to predict values of virtual time, failure number, MTBF and failure rate. The approximate confidence bounds obtained from simulation are provided. The uncertainty of model parameters is also considered in the bounds.		In general, in order to obtain the virtual age, the exact occurrence time of each event (failure) should be available (see equations for Type I and Type II models). However, the times are unknown until the corresponding events occur. For this reason, there are no closed-form expressions for total failure number and failure intensity, which are functions of failure times and virtual age. Therefore, in Weibull++, a Monte Carlo simulation is used to predict values of virtual time, failure number, MTBF and failure rate. The approximate confidence bounds obtained from simulation are provided. The uncertainty of model parameters is also considered in the bounds.


	=== Bounds on Cumulative Failure (Event) Numbers ===		=== Bounds on Cumulative Failure (Event) Numbers ===
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	::<math>{{\hat{m}}_{i}}{{(t)}_{U}}=\frac{1}{{{{\hat{\lambda }}}_{i}}{{(t)}_{L}}}</math>		::<math>{{\hat{m}}_{i}}{{(t)}_{U}}=\frac{1}{{{{\hat{\lambda }}}_{i}}{{(t)}_{L}}}</math>

	~~<br>~~

	::<math>{{\hat{m}}_{i}}{{(t)}_{L}}=\frac{1}{{{{\hat{\lambda }}}_{i}}{{(t)}_{U}}}</math>		::<math>{{\hat{m}}_{i}}{{(t)}_{L}}=\frac{1}{{{{\hat{\lambda }}}_{i}}{{(t)}_{U}}}</math>
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	::<math>R=\hat{R}{{e}^{\pm {{z}_{a}}\sqrt{Var(R)}/\hat{R}}}</math>		::<math>R=\hat{R}{{e}^{\pm {{z}_{a}}\sqrt{Var(R)}/\hat{R}}}</math>

	The smaller of the two upper bounds will be the final upper bound and the larger of the two lower bounds will be the final lower bound.		The smaller of the two upper bounds will be the final upper bound and the larger of the two lower bounds will be the final lower bound.

	==Example: Air Condition Unit==		==Example: Air Condition Unit==

	{{:Example:_Parametric_RDA_-_Air_Condition_Unit}}		{{:Example:_Parametric_RDA_-_Air_Condition_Unit}}

Richard House: /* Example: Mean Cumulative Function */

2012-08-08T05:32:31Z

Example: Mean Cumulative Function

← Older revision		Revision as of 05:32, 8 August 2012
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	\text{5} & \text{28} & \text{S} & \text{0} & {} & {} \\		\text{5} & \text{28} & \text{S} & \text{0} & {} & {} \\
	\end{matrix}</math></center>		\end{matrix}</math></center>


	Using the MCF variance equation, the following table of variance values can be obtained:		Using the MCF variance equation, the following table of variance values can be obtained:

			{\| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	{\| border="1" align="center"
	\|-		\|-
	! ID		! ID
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	\|		\|
	\|}		\|}


	Using the equation for the MCF bounds and <span class="texhtml">''K''<sub>5</sub> = 1.644</span> for a 95% confidence level, the confidence bounds can be obtained as follows:		Using the equation for the MCF bounds and <span class="texhtml">''K''<sub>5</sub> = 1.644</span> for a 95% confidence level, the confidence bounds can be obtained as follows:


	<center><math>\begin{matrix}		<center><math>\begin{matrix}
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	\text{5} & \text{28} & \text{S} & {} & {} & {} & {} \\		\text{5} & \text{28} & \text{S} & {} & {} & {} & {} \\
	\end{matrix}</math></center>		\end{matrix}</math></center>


	The analysis presented in this example can be performed automatically in Weibull++'s non-parametric RDA folio, as shown next.		The analysis presented in this example can be performed automatically in Weibull++'s non-parametric RDA folio, as shown next.


	[[Image:Recurrent Data Example 2 Data.png\|center\|550px]]		[[Image:Recurrent Data Example 2 Data.png\|center\|550px]]


	Note: In the folio above, the <span class="texhtml">''F''</span> refers to failures and <span class="texhtml">''E''</span> refers to suspensions (or censoring ages). The results, with calculated MCF values and upper and lower 95% confidence limits, are shown next along with the graphical plot.		Note: In the folio above, the <span class="texhtml">''F''</span> refers to failures and <span class="texhtml">''E''</span> refers to suspensions (or censoring ages). The results, with calculated MCF values and upper and lower 95% confidence limits, are shown next along with the graphical plot.


	[[Image:Recurrent Data Example 2 Result.png\|center\|550px]]		[[Image:Recurrent Data Example 2 Result.png\|center\|550px]]
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	[[Image:Recurrent Data Example 2 Plot.png\|center\|550px]]		[[Image:Recurrent Data Example 2 Plot.png\|center\|550px]]


	= Parametric Recurrent Event Data Analysis =		= Parametric Recurrent Event Data Analysis =

Richard House: /* The Mean Cumulative Function (MCF) */

2012-08-08T05:31:24Z

The Mean Cumulative Function (MCF)

← Older revision		Revision as of 05:31, 8 August 2012
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	== The Mean Cumulative Function (MCF) ==		== The Mean Cumulative Function (MCF) ==
	In a non-parametric analysis of recurrent event data, each population unit can be described by a cumulative history function for the cumulative number of recurrences. It is a staircase function that depicts the cumulative number of recurrences of a particular event, such as repairs over time. The figure below depicts a unit's cumulative history function.		In a non-parametric analysis of recurrent event data, each population unit can be described by a cumulative history function for the cumulative number of recurrences. It is a staircase function that depicts the cumulative number of recurrences of a particular event, such as repairs over time. The figure below depicts a unit's cumulative history function.


	[[Image:Lda11.1.png\|center\|400px]]		[[Image:Lda11.1.png\|center\|400px]]


	The non-parametric model for a population of units is described as the population of cumulative history functions (curves). It is the population of all staircase functions of every unit in the population. At age t, the units have a distribution of their cumulative number of events. That is, a fraction of the population has accumulated 0 recurrences, another fraction has accumulated 1 recurrence, another fraction has accumulated 2 recurrences, etc. This distribution differs at different ages <span class="texhtml">''t''</span> , and has a mean <span class="texhtml">''M''(''t'')</span> called the mean cumulative function (MCF). The <span class="texhtml">''M''(''t'')</span> is the point-wise average of all population cumulative history functions (see figure below).		The non-parametric model for a population of units is described as the population of cumulative history functions (curves). It is the population of all staircase functions of every unit in the population. At age t, the units have a distribution of their cumulative number of events. That is, a fraction of the population has accumulated 0 recurrences, another fraction has accumulated 1 recurrence, another fraction has accumulated 2 recurrences, etc. This distribution differs at different ages <span class="texhtml">''t''</span> , and has a mean <span class="texhtml">''M''(''t'')</span> called the mean cumulative function (MCF). The <span class="texhtml">''M''(''t'')</span> is the point-wise average of all population cumulative history functions (see figure below).


	[[Image:Lda11.2.png\|center\|400px]]		[[Image:Lda11.2.png\|center\|400px]]


	For the case of uncensored data, the mean cumulative function <math>M{{(t)}_{i}}\ </math> values at different recurrence ages <span class="texhtml">''t''<sub>''i''</sub></span> are estimated by calculating the average of the cumulative number of recurrences of events for each unit in the population at <span class="texhtml">''t''<sub>''i''</sub></span> . When the histories are censored, the following steps are applied.		For the case of uncensored data, the mean cumulative function <math>M{{(t)}_{i}}\ </math> values at different recurrence ages <span class="texhtml">''t''<sub>''i''</sub></span> are estimated by calculating the average of the cumulative number of recurrences of events for each unit in the population at <span class="texhtml">''t''<sub>''i''</sub></span> . When the histories are censored, the following steps are applied.


	'''1st Step - Order all ages:'''		'''1st Step - Order all ages:'''

	Order all recurrence and censoring ages from smallest to largest. If a recurrence age for a unit is the same as its censoring (suspension) age, then the recurrence age goes first. If multiple units have a common recurrence or censoring age, then these units could be put in a certain order or be sorted randomly.		Order all recurrence and censoring ages from smallest to largest. If a recurrence age for a unit is the same as its censoring (suspension) age, then the recurrence age goes first. If multiple units have a common recurrence or censoring age, then these units could be put in a certain order or be sorted randomly.


	'''2nd Step - Calculate the number, <span class="texhtml">''r''<sub>''i''</sub></span> , of units that passed through age <span class="texhtml">''t''<sub>''i''</sub></span> :'''		'''2nd Step - Calculate the number, <span class="texhtml">''r''<sub>''i''</sub></span> , of units that passed through age <span class="texhtml">''t''<sub>''i''</sub></span> :'''
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	<span class="texhtml">''N''</span> is the total number of units and <span class="texhtml">''r''<sub>1</sub> = ''N''</span> at the first observed age which could be a recurrence or suspension.		<span class="texhtml">''N''</span> is the total number of units and <span class="texhtml">''r''<sub>1</sub> = ''N''</span> at the first observed age which could be a recurrence or suspension.


	'''3rd Step - Calculate the MCF estimate, M*(t):'''		'''3rd Step - Calculate the MCF estimate, M*(t):'''

Kate Racaza: organized examples

2012-08-01T04:30:22Z

organized examples

Show changes

Kate Racaza at 04:37, 31 July 2012

2012-07-31T04:37:51Z

← Older revision		Revision as of 04:37, 31 July 2012
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	The smaller of the two upper bounds will be the final upper bound and the larger of the two lower bounds will be the final lower bound.		The smaller of the two upper bounds will be the final upper bound and the larger of the two lower bounds will be the final lower bound.

	<br>'''Example 4:''' {{Example: ~~Recurrent Events Data Parameteric Air~~-~~Condition Example~~}}		<br>'''Example 4: Air Condition Unit Example'''

			{{:Example:_Parametric_RDA_-_Air_Condition_Unit}}