Appendix: Special Analysis Methods - Revision history

Lisa Hacker at 19:15, 15 September 2023

2023-09-15T19:15:49Z

← Older revision		Revision as of 19:15, 15 September 2023
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	{{Template:~~LDABOOK_SUB~~\|Appendix C\|Special Analysis Methods}}		{{Template:LDABOOK\|Appendix C\|Special Analysis Methods}}

	==Grouped Data Analysis==		==Grouped Data Analysis==

Kate Racaza: Replaced content with '{{Template:LDABOOK_SUB|Appendix C|Special Analysis Methods}} ==Grouped Data Analysis== {{:Grouped_Data_Parameter_Estimation}} == ReliaSoft Ranking Method == {{:ReliaSoft's…'

2012-08-10T04:07:34Z

Replaced content with '{{Template:LDABOOK_SUB|Appendix C|Special Analysis Methods}} ==Grouped Data Analysis== {{:Grouped_Data_Parameter_Estimation}} == ReliaSoft Ranking Method == {{:ReliaSoft's…'

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Richard House: /* The Mathematics */

2012-08-08T03:47:04Z

The Mathematics

← Older revision		Revision as of 03:47, 8 August 2012
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	with:		with:

	::<math>N=30,\text{ }J=10</math>		::<math>\begin{align}
			N=30,\text{ }J=10
			\end{align}</math>

	One <math>Z</math> is obtained for the group, to represent the probability of 10 failures occurring out of 30.		One <math>Z</math> is obtained for the group, to represent the probability of 10 failures occurring out of 30.
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	where:		where:

	::<math>N=30,\text{ }J=20</math>		::<math>\begin{align}
			N=30,\text{ }J=20
			\end{align}</math>

	This represents the probability of 20 failures out of 30.		This represents the probability of 20 failures out of 30.
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	where:		where:

	::<math>N=30,\text{ }J=30</math>		::<math>\begin{align}
			N=30,\text{ }J=30
			\end{align}</math>

	This represents the probability of 30 failures out of 30.		This represents the probability of 30 failures out of 30.

Richard House: /* ReliaSoft's Ranking Method (RRM) */

2012-08-08T01:20:20Z

ReliaSoft's Ranking Method (RRM)

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	Consider the following test data:		Consider the following test data:

	{\|~~style~~= align="center" border="1"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|-		\|-
	\|colspan="4" style="text-align:center"\|Table B.1- The Test Data		\|colspan="4" style="text-align:center"\|Table B.1- The Test Data
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	{\|~~style~~= align="center" border="1"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|colspan="4" style="text-align:center"\|Table B.2- The Union of Exact Times-to-Failure with the "Midpoint" of the Interval Failures		\|colspan="4" style="text-align:center"\|Table B.2- The Union of Exact Times-to-Failure with the "Midpoint" of the Interval Failures
	\|-		\|-
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	This transforms our data into the format in Table B.3.		This transforms our data into the format in Table B.3.

	{\|~~style~~= align="center" border="1"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|colspan="5" style="text-align:center"\|Table B.3- The Union of Exact Times-to-Failure with the "Midpoint" of the Interval Failures, Based upon the Parameters <math>\beta</math> and <math>\eta</math>.		\|colspan="5" style="text-align:center"\|Table B.3- The Union of Exact Times-to-Failure with the "Midpoint" of the Interval Failures, Based upon the Parameters <math>\beta</math> and <math>\eta</math>.
	\|-		\|-
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	Now we arrange the data as in Table B.4.		Now we arrange the data as in Table B.4.

	{\|~~style~~= align="center" border="1"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|colspan="2"\|Table B.4- The Union of Exact Times-to-Failure with the "Midpoint" of the Interval Failures, in Ascending Order.		\|colspan="2"\|Table B.4- The Union of Exact Times-to-Failure with the "Midpoint" of the Interval Failures, in Ascending Order.
	\|-align="center"		\|-align="center"
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	We now consider the left and right censored data, as in Table B.5.		We now consider the left and right censored data, as in Table B.5.

	{\|~~style~~= align="center" border="1"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|-		\|-
	\|colspan="7" style="text-align:center"\|Table B.5- Computation of Increments in a Matrix Format for Computing a Revised Mean Order Number.		\|colspan="7" style="text-align:center"\|Table B.5- Computation of Increments in a Matrix Format for Computing a Revised Mean Order Number.
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	Sum up the increments (horizontally in rows), as in Table B.6.		Sum up the increments (horizontally in rows), as in Table B.6.

	{\|~~style~~= align="center" border="1"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|-		\|-
	\|colspan="8" style="text-align:center"\|Table B.6- Increments Solved Numerically, Along with the Sum of Each Row.		\|colspan="8" style="text-align:center"\|Table B.6- Increments Solved Numerically, Along with the Sum of Each Row.
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	Compute new mean order numbers (MON), as shown Table B.7, utilizing the increments obtained in Table B.6, by adding the ''number of items'' plus the ''previous MON'' plus the current increment.		Compute new mean order numbers (MON), as shown Table B.7, utilizing the increments obtained in Table B.6, by adding the ''number of items'' plus the ''previous MON'' plus the current increment.

	{\|~~style~~= align="center" border="1"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|-		\|-
	\|colspan="4" style="text-align:center"\|Table B.7- Mean Order Numbers (MON)		\|colspan="4" style="text-align:center"\|Table B.7- Mean Order Numbers (MON)
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	Compute the median ranks based on these new MONs as shown in Table B.8.		Compute the median ranks based on these new MONs as shown in Table B.8.

	{\|~~style~~= align="center" border="1"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|colspan="3" style="text-align:center"\|Table B.8- Mean Order Numbers with Their Ranks for a Sample Size of 13 Units.		\|colspan="3" style="text-align:center"\|Table B.8- Mean Order Numbers with Their Ranks for a Sample Size of 13 Units.
	\|-		\|-
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	Compute new <math>\beta </math> and <math>\eta ,</math> using standard rank regression and based upon the data as shown in Table B.9.		Compute new <math>\beta </math> and <math>\eta ,</math> using standard rank regression and based upon the data as shown in Table B.9.
	<br>		<br>
	{\|~~style~~= align="center" border="1"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	!Time		!Time
	!Ranks		!Ranks

Richard House: /* Results */

2012-08-08T01:18:00Z

Results

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	Using Weibull++ with rank regression on X yields:		Using Weibull++ with rank regression on X yields:
	<br>		<br>
	{\|~~style~~= align="center" border="1"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|colspan="3" style="text-align:center;"\|Table B.10-The parameters after the first five iterations		\|colspan="3" style="text-align:center;"\|Table B.10-The parameters after the first five iterations
	\|-align="center"		\|-align="center"

Richard House: /* The Mathematics */

2012-08-08T01:15:24Z

The Mathematics

← Older revision		Revision as of 01:15, 8 August 2012
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	One <math>Z</math> is obtained for the group, to represent the probability of 10 failures occurring out of 30.		One <math>Z</math> is obtained for the group, to represent the probability of 10 failures occurring out of 30.


	For 10 failures at 200 hours, <math>Z</math> is estimated by using:		For 10 failures at 200 hours, <math>Z</math> is estimated by using:
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	This represents the probability of 20 failures out of 30.		This represents the probability of 20 failures out of 30.


	For 10 failures at 300 hours, <math>Z</math> is estimated by using:		For 10 failures at 300 hours, <math>Z</math> is estimated by using:

Lisa Hacker: /* ReliaSoft's Ranking Method (RRM) */

2012-08-02T06:27:18Z

ReliaSoft's Ranking Method (RRM)

← Older revision		Revision as of 06:27, 2 August 2012
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	The following step-by-step example illustrates the ReliaSoft ranking method (RRM), which is an iterative improvement on the standard ranking method (SRM). Although this method is illustrated by the use of the two-parameter Weibull distribution, it can be easily generalized for other models.		The following step-by-step example illustrates the ReliaSoft ranking method (RRM), which is an iterative improvement on the standard ranking method (SRM). Although this method is illustrated by the use of the two-parameter Weibull distribution, it can be easily generalized for other models.

	Consider the following test data~~, as shown in the following table.~~		Consider the following test data:

	{\|style= align="center" border="1"		{\|style= align="center" border="1"

Lisa Hacker: /* ReliaSoft's Ranking Method (RRM) */

2012-08-02T06:27:00Z

ReliaSoft's Ranking Method (RRM)

← Older revision		Revision as of 06:27, 2 August 2012
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	= ReliaSoft's Ranking Method (RRM) =		= ReliaSoft's Ranking Method (RRM) =
	In probability plotting or rank regression analysis of interval or left censored data, difficulties arise when attempting to estimate the exact time within the interval when the failure actually occurs, especially when an overlap on the intervals is present. In this case, the ''standard ranking method'' (SRM) is not applicable when dealing with interval data; thus, ReliaSoft has formulated a more sophisticated methodology to allow for more accurate probability plotting and regression analysis of data sets with interval or left censored data. This method utilizes the traditional rank regression method and iteratively improves upon the computed ranks by parametrically recomputing new ranks and the most probable failure time for interval data~~. A step-by-step example of this method follows~~.		In probability plotting or rank regression analysis of interval or left censored data, difficulties arise when attempting to estimate the exact time within the interval when the failure actually occurs, especially when an overlap on the intervals is present. In this case, the ''standard ranking method'' (SRM) is not applicable when dealing with interval data; thus, ReliaSoft has formulated a more sophisticated methodology to allow for more accurate probability plotting and regression analysis of data sets with interval or left censored data. This method utilizes the traditional rank regression method and iteratively improves upon the computed ranks by parametrically recomputing new ranks and the most probable failure time for interval data.

	~~This~~ example illustrates the ReliaSoft ranking method (RRM), which is an iterative improvement on the standard ranking method (SRM). Although this method is illustrated by the use of the two-parameter Weibull distribution, it can be easily generalized for other models.		The following step-by-step example illustrates the ReliaSoft ranking method (RRM), which is an iterative improvement on the standard ranking method (SRM). Although this method is illustrated by the use of the two-parameter Weibull distribution, it can be easily generalized for other models.

	Consider the following test data, as shown in the following table.		Consider the following test data, as shown in the following table.

Lisa Hacker: /* ReliaSoft's Ranking Method (RRM) */

2012-08-02T06:26:09Z

ReliaSoft's Ranking Method (RRM)

← Older revision		Revision as of 06:26, 2 August 2012
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	In probability plotting or rank regression analysis of interval or left censored data, difficulties arise when attempting to estimate the exact time within the interval when the failure actually occurs, especially when an overlap on the intervals is present. In this case, the ''standard ranking method'' (SRM) is not applicable when dealing with interval data; thus, ReliaSoft has formulated a more sophisticated methodology to allow for more accurate probability plotting and regression analysis of data sets with interval or left censored data. This method utilizes the traditional rank regression method and iteratively improves upon the computed ranks by parametrically recomputing new ranks and the most probable failure time for interval data. A step-by-step example of this method follows.		In probability plotting or rank regression analysis of interval or left censored data, difficulties arise when attempting to estimate the exact time within the interval when the failure actually occurs, especially when an overlap on the intervals is present. In this case, the ''standard ranking method'' (SRM) is not applicable when dealing with interval data; thus, ReliaSoft has formulated a more sophisticated methodology to allow for more accurate probability plotting and regression analysis of data sets with interval or left censored data. This method utilizes the traditional rank regression method and iteratively improves upon the computed ranks by parametrically recomputing new ranks and the most probable failure time for interval data. A step-by-step example of this method follows.

	This ~~section~~ illustrates the ReliaSoft ranking method (RRM), which is an iterative improvement on the standard ranking method (SRM). Although this method is illustrated by the use of the two-parameter Weibull distribution, it can be easily generalized for other models.		This example illustrates the ReliaSoft ranking method (RRM), which is an iterative improvement on the standard ranking method (SRM). Although this method is illustrated by the use of the two-parameter Weibull distribution, it can be easily generalized for other models.

	Consider the following test data, as shown in the following table.		Consider the following test data, as shown in the following table.
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	\|}		\|}

	==== Initial Parameter Estimation====		=== Initial Parameter Estimation===
	As a preliminary step, we need to provide a crude estimate of the Weibull parameters for this data. To begin, we will extract the exact times-to-failure (10, 40, and 50) and the midpoints of the interval failures. The midpoints are 50 (for the interval of 20 to 80) and 47.5 (for the interval of 10 to 85). Next, we group together the items that have the same failure times, as shown in Table B.2.		As a preliminary step, we need to provide a crude estimate of the Weibull parameters for this data. To begin, we will extract the exact times-to-failure (10, 40, and 50) and the midpoints of the interval failures. The midpoints are 50 (for the interval of 20 to 80) and 47.5 (for the interval of 10 to 85). Next, we group together the items that have the same failure times, as shown in Table B.2.

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	Return and repeat the process from Step 1 until an acceptable convergence is reached on the parameters (i.e., the parameter values stabilize).		Return and repeat the process from Step 1 until an acceptable convergence is reached on the parameters (i.e., the parameter values stabilize).

	====Results====		===Results===
	The results of the first five iterations are shown in Table B.10.		The results of the first five iterations are shown in Table B.10.
	Using Weibull++ with rank regression on X yields:		Using Weibull++ with rank regression on X yields:

Lisa Hacker: /* Step-by-Step Example */

2012-08-02T06:25:34Z

Step-by-Step Example

← Older revision		Revision as of 06:25, 2 August 2012
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	In probability plotting or rank regression analysis of interval or left censored data, difficulties arise when attempting to estimate the exact time within the interval when the failure actually occurs, especially when an overlap on the intervals is present. In this case, the ''standard ranking method'' (SRM) is not applicable when dealing with interval data; thus, ReliaSoft has formulated a more sophisticated methodology to allow for more accurate probability plotting and regression analysis of data sets with interval or left censored data. This method utilizes the traditional rank regression method and iteratively improves upon the computed ranks by parametrically recomputing new ranks and the most probable failure time for interval data. A step-by-step example of this method follows.		In probability plotting or rank regression analysis of interval or left censored data, difficulties arise when attempting to estimate the exact time within the interval when the failure actually occurs, especially when an overlap on the intervals is present. In this case, the ''standard ranking method'' (SRM) is not applicable when dealing with interval data; thus, ReliaSoft has formulated a more sophisticated methodology to allow for more accurate probability plotting and regression analysis of data sets with interval or left censored data. This method utilizes the traditional rank regression method and iteratively improves upon the computed ranks by parametrically recomputing new ranks and the most probable failure time for interval data. A step-by-step example of this method follows.

	~~=== Step-by-Step Example===~~
	This section illustrates the ReliaSoft ranking method (RRM), which is an iterative improvement on the standard ranking method (SRM). Although this method is illustrated by the use of the two-parameter Weibull distribution, it can be easily generalized for other models.		This section illustrates the ReliaSoft ranking method (RRM), which is an iterative improvement on the standard ranking method (SRM). Although this method is illustrated by the use of the two-parameter Weibull distribution, it can be easily generalized for other models.