Template:Probability Plotting - Revision history

Lisa Hacker: Redirected page to Parameter Estimation

2012-08-09T10:08:41Z

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	[[~~Category: For Deletion~~]]		#REDIRECT [[Parameter Estimation]]

Lisa Hacker: Replaced content with 'Category: For Deletion'

2012-07-20T12:24:20Z

Replaced content with 'Category: For Deletion'

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	~~===Probability Plotting===~~		[[Category: For Deletion]]
	The least mathematically intensive method for parameter estimation is the method of probability plotting. As the term implies, probability plotting involves a physical plot of the data on specially constructed '''probability plotting paper'''. This method is easily implemented by hand, given that one can obtain the appropriate probability plotting paper.
	~~==== Illustrating the Method for the 2-Parameter Weibull Distribution ====~~

	The method of probability plotting takes the ''cdf'' of the distribution and attempts to linearize it by employing a specially constructed paper. This is best illustrated using the 2-parameter Weibull distribution.

	~~In the case of the two-parameter Weibull distribution, the ''cdf'' (also the unreliability ''Q(t)'') is given by:~~

	~~::<math>F(t)=Q(t)=1-{e^{-\left(\tfrac{t}{\eta}\right)^{\beta}}}</math>~~

	~~<br>~~

	'''Linearizing the Weibull Unreliability Function''' <br>This function can then be linearized (i.e., put in the common form of <span class="texhtml">''y'' = ''m'x'' + ''b''</span> format) as follows''':'''

	~~::<math>\begin{align}~~
	~~Q(t)= & 1-{e^{-\left(\tfrac{t}{\eta}\right)^{\beta}}} \\~~
	~~\ln (1-Q(t))= & \ln \left[ {e^{-\left(\tfrac{t}{\eta}\right)^{\beta}}} \right] \\~~
	~~\ln (1-Q(t))=& -\left(\tfrac{t}{\eta}\right)^{\beta} \\~~
	~~\ln ( -\ln (1-Q(t)))= & \beta \left(\ln \left( \frac{t}{\eta }\right)\right) \\~~
	~~\ln \left( \ln \left( \frac{1}{1-Q(t)}\right) \right) = & \beta\ln{ t} -\beta(\eta ) \\~~
	~~\end{align}</math>~~

	~~<br>~~

	~~Then by setting~~

	~~::<math>y=\ln \left( \ln \left( \frac{1}{1-Q(t)} \right) \right)</math>~~

	~~and~~

	~~::<math>x=\ln \left( t \right)</math>~~

	~~the equation can then be rewritten as:~~

	~~::<math>y=\beta x-\beta \ln \left( \eta \right)</math>~~

	~~which is now a linear equation with a slope of~~

	~~::Slope<span class="texhtml"> = ''m'' = β</span>~~

	~~and an intercept of~~

	~~::Intercept<math>=b=-\beta \cdot ln(\eta)</math>.~~

	~~<br>~~

	~~'''Constructing the Paper''' <br>~~

	The next task is to construct the Weibull probability plotting paper with the appropriate ''y'' - and ''x''-axes. The ''x-axis'' trasnformation is simply logarithmic. The ''y-axis'', is a bit more complex requiring a double log reciprocal transformation, or

	~~<br>~~

	~~::<math>y=\ln \left(\ln \left( \frac{1}{1-Q(t)} ) \right) \right)</math>~~

	~~<br>where ''Q(t)'' is the unreliability.~~

	<br>Such papers have been created by different vendors and are called ''probability plotting papers''. Weibull.com has different plotting papers available for [http://www.weibull.com/GPaper/index.htm download].

	[[~~Image:WeibullPaper2C.png\|center\|250px]]~~

	~~To illustrate, consider the following probability plot on a slightly different type of Weibull probability paper.~~

	[~~[Image:different_weibull_paper.png\|center\|250px]]~~

	This paper is constructed based on the mentioned ''y'' - and ''x'' - transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.

	Then, given the ''y'' and ''x'' value for each point, the points can easily be put on the plot. Once the points have been placed on the plot, the best possible straight line is drawn through these points. Once the line has been drawn, the slope of the line can be obtained (some probability papers include a slope indicator to simplify this calculation). This is the parameter <span class="texhtml">β,</span> which is the value of the slope. To determine the scale parameter, <span class="texhtml">η</span> (also called the ''characteristic life''), one reads the time from the ''x-axis'' corresponding to ''Q(t)=63.2%''.

	~~Note that at~~

	:~~:<math>\begin{align}~~
	~~Q(t=\eta)= & 1-{{e}^{-{{\left( \tfrac{t}{\eta } \right)}^{\beta }}}} \\~~
	~~= & 1-{{e}^{-1}} \\~~
	~~= & 0.632 \\~~
	~~= & 63.2%~~
	~~\end{align}</math>~~

	<br>Thus, if we enter the ''y'' axis at ''Q(t)=63.2%,'' the corresponding value of ''t'' will be equal to <span class="texhtml">η.</span> Thus, using this simple methodology, the parameters of the Weibull distribution can be estimated.

	~~'''Determining the ''x'' and ''y'' Position of the Plot Points''' <br>~~

	The points on the plot represent our data or, more specifically, our times-to-failure data. If, for example, we tested four units that failed at 10, 20, 30 and 40 hours, then we would use these times as our ''x'' values or time values.

	Determining the appropriate ''y'' plotting positions, or the unreliability values, is a little more complex. To determine the ''y'' plotting positions, we must first determine a value indicating the corresponding unreliability for that failure. In other words, we need to obtain the cumulative percent failed for each time-to-failure. For example, the cumulative percent failed by 10 hours may be 25%, by 20 hours 50%, and so forth. This is a simple method illustrating the idea. The problem with this simple method is the fact that the 100% point is not defined on most probability plots; thus, an alternative and more robust approach must be used. The most widely used method of determining this value is the method of obtaining the ''median rank'' for each failure. This is discussed next.

	~~{{MedianRanks}}~~

	~~====Beta and F Distributions Approach====~~

	A more straightforward and easier method of estimating median ranks is by applying two transformations to the cumulative binomial equation, first to the beta distribution and then to the F distribution, resulting in [[Appendix D: Weibull References\|[12, 13]]],

	~~::<math>\begin{array}{*{35}{l}}~~
	~~MR & = & \tfrac{1}{1+\tfrac{N-j+1}{j}{{F}_{0.50;m;n}}} \\~~
	~~m & = & 2(N-j+1) \\~~
	~~n & = & 2j \\~~
	~~\end{array}</math>~~

	~~where <span class="texhtml">''F''<sub>0.50;''m'';''n''</sub></span> denotes the ''F'' distribution at the ''0.50'' point, with ''m'' and ''n'' degrees of freedom, for failure ''j'' out of ''N'' units.~~

	~~{{Benard's approximation}}~~

	~~{{Kaplan-Meier}}~~

	~~{{Analysis RC Suspended Data}}~~

Nicolette Young: /* Illustrating the Method for the 2-Parameter Weibull Distribution */

2012-04-25T17:19:26Z

Illustrating the Method for the 2-Parameter Weibull Distribution

← Older revision		Revision as of 17:19, 25 April 2012
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	<br>Such papers have been created by different vendors and are called ''probability plotting papers''. Weibull.com has different plotting papers available for [http://www.weibull.com/GPaper/index.htm download].		<br>Such papers have been created by different vendors and are called ''probability plotting papers''. Weibull.com has different plotting papers available for [http://www.weibull.com/GPaper/index.htm download].

	[[Image:WeibullPaper2C.png\|center\|~~200px~~]]		[[Image:WeibullPaper2C.png\|center\|250px]]

	To illustrate, consider the following probability plot on a slightly different type of Weibull probability paper.		To illustrate, consider the following probability plot on a slightly different type of Weibull probability paper.

	[[Image:different_weibull_paper.png\|center\|~~200px~~]]		[[Image:different_weibull_paper.png\|center\|250px]]

	This paper is constructed based on the mentioned ''y'' - and ''x'' - transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.		This paper is constructed based on the mentioned ''y'' - and ''x'' - transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.

Nicolette Young: /* Illustrating the Method for the 2-Parameter Weibull Distribution */

2012-04-25T17:16:27Z

Illustrating the Method for the 2-Parameter Weibull Distribution

← Older revision		Revision as of 17:16, 25 April 2012
Line 57:		Line 57:
	<br>Such papers have been created by different vendors and are called ''probability plotting papers''. Weibull.com has different plotting papers available for [http://www.weibull.com/GPaper/index.htm download].		<br>Such papers have been created by different vendors and are called ''probability plotting papers''. Weibull.com has different plotting papers available for [http://www.weibull.com/GPaper/index.htm download].

	[[Image:WeibullPaper2C.png\|center\|~~400px~~]]		[[Image:WeibullPaper2C.png\|center\|200px]]

	To illustrate, consider the following probability plot on a slightly different type of Weibull probability paper.		To illustrate, consider the following probability plot on a slightly different type of Weibull probability paper.

	[[Image:different_weibull_paper.png\|center\|~~400px~~]]		[[Image:different_weibull_paper.png\|center\|200px]]

	This paper is constructed based on the mentioned ''y'' - and ''x'' - transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.		This paper is constructed based on the mentioned ''y'' - and ''x'' - transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.

Lisa Hacker: /* Illustrating the Method for the 2-Parameter Weibull Distribution */

2012-03-30T21:51:09Z

Illustrating the Method for the 2-Parameter Weibull Distribution

← Older revision		Revision as of 21:51, 30 March 2012
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	To illustrate, consider the following probability plot on a slightly different type of Weibull probability paper.		To illustrate, consider the following probability plot on a slightly different type of Weibull probability paper.

	[[Image:different_weibull_paper.png\|400px]]		[[Image:different_weibull_paper.png\|center\|400px]]

	This paper is constructed based on the mentioned ''y'' - and ''x'' - transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.		This paper is constructed based on the mentioned ''y'' - and ''x'' - transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.

Harry Guo: /* Beta and F Distributions Approach */

2012-03-16T22:33:33Z

Beta and F Distributions Approach

← Older revision		Revision as of 22:33, 16 March 2012
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	where <span class="texhtml">''F''<sub>0.50;''m'';''n''</sub></span> denotes the ''F'' distribution at the ''0.50'' point, with ''m'' and ''n'' degrees of freedom, for failure ''j'' out of ''N'' units.		where <span class="texhtml">''F''<sub>0.50;''m'';''n''</sub></span> denotes the ''F'' distribution at the ''0.50'' point, with ''m'' and ''n'' degrees of freedom, for failure ''j'' out of ''N'' units.

	~~<br>~~{{Benard's approximation}}		{{Benard's approximation}}

	{{Kaplan-Meier}}		{{Kaplan-Meier}}

	{{Analysis RC Suspended Data}}		{{Analysis RC Suspended Data}}

Kate Racaza: /* Illustrating the Method for the 2-Parameter Weibull Distribution */

2012-03-16T17:45:51Z

Illustrating the Method for the 2-Parameter Weibull Distribution

← Older revision		Revision as of 17:45, 16 March 2012
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	Then, given the ''y'' and ''x'' value for each point, the points can easily be put on the plot. Once the points have been placed on the plot, the best possible straight line is drawn through these points. Once the line has been drawn, the slope of the line can be obtained (some probability papers include a slope indicator to simplify this calculation). This is the parameter <span class="texhtml">β,</span> which is the value of the slope. To determine the scale parameter, <span class="texhtml">η</span> (also called the ''characteristic life''), one reads the time from the ''x-axis'' corresponding to ''Q(t)=63.2%''.		Then, given the ''y'' and ''x'' value for each point, the points can easily be put on the plot. Once the points have been placed on the plot, the best possible straight line is drawn through these points. Once the line has been drawn, the slope of the line can be obtained (some probability papers include a slope indicator to simplify this calculation). This is the parameter <span class="texhtml">β,</span> which is the value of the slope. To determine the scale parameter, <span class="texhtml">η</span> (also called the ''characteristic life''), one reads the time from the ''x-axis'' corresponding to ''Q(t)=63.2%''.

	Note that ~~from before~~ at		Note that at

	::<math>\begin{align}		::<math>\begin{align}
Line 82:		Line 82:
	The points on the plot represent our data or, more specifically, our times-to-failure data. If, for example, we tested four units that failed at 10, 20, 30 and 40 hours, then we would use these times as our ''x'' values or time values.		The points on the plot represent our data or, more specifically, our times-to-failure data. If, for example, we tested four units that failed at 10, 20, 30 and 40 hours, then we would use these times as our ''x'' values or time values.

	Determining the appropriate ''y'' plotting positions, or the unreliability values, is a little more complex. To determine the ''y'' plotting positions, we must first determine a value indicating the corresponding unreliability for that failure. In other words, we need to obtain the cumulative percent failed for each time-to-failure. ~~In this~~ example~~, and by 10 hours~~, the cumulative percent failed is 25%, by 20 hours 50%, and so forth. This is a simple method illustrating the idea. The problem with this simple method is the fact that the 100% point is not defined on most probability plots, ~~thus~~ an alternative and more robust approach must be used. The most widely used method of determining this value is the method of obtaining the ''median rank'' for each failure. This is discussed next.		Determining the appropriate ''y'' plotting positions, or the unreliability values, is a little more complex. To determine the ''y'' plotting positions, we must first determine a value indicating the corresponding unreliability for that failure. In other words, we need to obtain the cumulative percent failed for each time-to-failure. For example, the cumulative percent failed by 10 hours may be 25%, by 20 hours 50%, and so forth. This is a simple method illustrating the idea. The problem with this simple method is the fact that the 100% point is not defined on most probability plots; thus, an alternative and more robust approach must be used. The most widely used method of determining this value is the method of obtaining the ''median rank'' for each failure. This is discussed next.

	{{MedianRanks}}		{{MedianRanks}}

Kate Racaza: /* Illustrating the Method for the 2-Parameter Weibull Distribution */

2012-03-16T17:41:44Z

Illustrating the Method for the 2-Parameter Weibull Distribution

← Older revision		Revision as of 17:41, 16 March 2012
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	::<math>x=\ln \left( t \right)</math>		::<math>x=\ln \left( t \right)</math>

	the equation can then be rewritten as		the equation can then be rewritten as:

	::<math>y=\beta x-\beta \ln \left( \eta \right)</math>		::<math>y=\beta x-\beta \ln \left( \eta \right)</math>
Line 63:		Line 63:
	[[Image:different_weibull_paper.png\|400px]]		[[Image:different_weibull_paper.png\|400px]]

	This paper is constructed based on the mentioned ''y'' - and ''x''-transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.		This paper is constructed based on the mentioned ''y'' - and ''x'' - transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.

	Then, given the ''y'' and ''x'' value for each point, the points can easily be put on the plot. Once the points have been placed on the plot, the best possible straight line is drawn through these points. Once the line has been drawn, the slope of the line can be obtained (some probability papers include a slope indicator to simplify this calculation). This is the parameter <span class="texhtml">β,</span> which is the value of the slope. To determine the scale parameter, <span class="texhtml">η</span> (also called the ''characteristic life''), one reads the time from the ''x-axis'' corresponding to ''Q(t)=63.2%''.		Then, given the ''y'' and ''x'' value for each point, the points can easily be put on the plot. Once the points have been placed on the plot, the best possible straight line is drawn through these points. Once the line has been drawn, the slope of the line can be obtained (some probability papers include a slope indicator to simplify this calculation). This is the parameter <span class="texhtml">β,</span> which is the value of the slope. To determine the scale parameter, <span class="texhtml">η</span> (also called the ''characteristic life''), one reads the time from the ''x-axis'' corresponding to ''Q(t)=63.2%''.

Kate Racaza at 17:39, 16 March 2012

2012-03-16T17:39:08Z

← Older revision		Revision as of 17:39, 16 March 2012
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	<br>		<br>

	'''Linearizing the Weibull Unreliability Function''' <br>This function can then be linearized (i.e., put in the common form of <span class="texhtml">''y'' = ''m'x'' + ''b~~'')''''' '''~~''</span>format as follows''':'''		'''Linearizing the Weibull Unreliability Function''' <br>This function can then be linearized (i.e., put in the common form of <span class="texhtml">''y'' = ''m'x'' + ''b''</span> format) as follows''':'''

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

Nicolette Young: /* Illustrating the Method for the 2-Parameter Weibull Distribution */

2012-03-13T16:00:07Z

Illustrating the Method for the 2-Parameter Weibull Distribution

← Older revision		Revision as of 16:00, 13 March 2012
Line 57:		Line 57:
	<br>Such papers have been created by different vendors and are called ''probability plotting papers''. Weibull.com has different plotting papers available for [http://www.weibull.com/GPaper/index.htm download].		<br>Such papers have been created by different vendors and are called ''probability plotting papers''. Weibull.com has different plotting papers available for [http://www.weibull.com/GPaper/index.htm download].

	[[Image:WeibullPaper2C.png~~\|thumb~~\|center\|~~300px~~]]		[[Image:WeibullPaper2C.png\|center\|400px]]

	To illustrate, consider the following probability plot on a slightly different type of Weibull probability paper.		To illustrate, consider the following probability plot on a slightly different type of Weibull probability paper.

	[[Image:different_weibull_paper.png\|~~thumb\|center\|300px~~]]		[[Image:different_weibull_paper.png\|400px]]

	This paper is constructed based on the mentioned ''y'' - and ''x''-transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.		This paper is constructed based on the mentioned ''y'' - and ''x''-transformations, where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each time-to-failure point we want to plot.
Line 84:		Line 84:
	Determining the appropriate ''y'' plotting positions, or the unreliability values, is a little more complex. To determine the ''y'' plotting positions, we must first determine a value indicating the corresponding unreliability for that failure. In other words, we need to obtain the cumulative percent failed for each time-to-failure. In this example, and by 10 hours, the cumulative percent failed is 25%, by 20 hours 50%, and so forth. This is a simple method illustrating the idea. The problem with this simple method is the fact that the 100% point is not defined on most probability plots, thus an alternative and more robust approach must be used. The most widely used method of determining this value is the method of obtaining the ''median rank'' for each failure. This is discussed next.		Determining the appropriate ''y'' plotting positions, or the unreliability values, is a little more complex. To determine the ''y'' plotting positions, we must first determine a value indicating the corresponding unreliability for that failure. In other words, we need to obtain the cumulative percent failed for each time-to-failure. In this example, and by 10 hours, the cumulative percent failed is 25%, by 20 hours 50%, and so forth. This is a simple method illustrating the idea. The problem with this simple method is the fact that the 100% point is not defined on most probability plots, thus an alternative and more robust approach must be used. The most widely used method of determining this value is the method of obtaining the ''median rank'' for each failure. This is discussed next.

	{{MedianRanks}}		{{MedianRanks}}

	====Beta and F Distributions Approach====		====Beta and F Distributions Approach====