Template:Weibull parameters probability plotting: Difference between revisions

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=== Probability Plotting ===
#REDIRECT [[The Weibull Distribution]]
 
One method of calculating the parameters of the Weibull distribution is by using probability plotting. To better illustrate this procedure, consider the following example from [[Appendix: Weibull References|Kececioglu [20]]].
 
 
'''Example 1:'''
{{Example: Weibull Probability Plot}}
 
 
'''Probability Plotting for the Location Parameter, <span class="texhtml">γ</span>'''
 
The third parameter of the Weibull distribution is utilized when the data do not fall on a straight line, but fall on either a concave up or down curve. The following statements can be made regarding the value of <span class="texhtml">γ:</span>
 
:''Case 1'': If the curve for MR versus <span class="texhtml">''t''<sub>''j''</sub></span> is concave down and the curve for MR versus <span class="texhtml">(''t''<sub>''j''</sub> − ''t''<sub>1</sub>)</span> is concave up, then there exists a <span class="texhtml">γ</span> such that <span class="texhtml">0 &lt; γ &lt; ''t''<sub>1</sub></span>, or <span class="texhtml">γ</span> has a positive value.
 
:''Case 2'': If the curves for MR versus <span class="texhtml">''t''<sub>''j''</sub></span> and MR versus <span class="texhtml">(''t''<sub>''j''</sub> − ''t''<sub>1</sub>)</span> are both concave up, then there exists a negative <span class="texhtml">γ</span> which will straighten out the curve of MR versus <span class="texhtml">''t''<sub>''j''</sub></span>.
 
:''Case 3'': If neither one of the previous two cases prevails, then either reject the Weibull  as one capable of representing the data, or proceed with the multiple population (mixed Weibull) analysis. To obtain the location parameter, <span class="texhtml">γ:</span>
 
::*Subtract the same arbitrary value, <span class="texhtml">γ</span>, from all the times to failure and replot the data.
::*If the initial curve is concave up, subtract a negative <span class="texhtml">γ</span> from each failure time.
::*If the initial curve is concave down, subtract a positive <span class="texhtml">γ</span> from each failure time.
::*Repeat until the data plots on an acceptable straight line.
::*The value of <span class="texhtml">γ</span> is the subtracted (positive or negative) value that places the points in an acceptable straight line.
 
 
The other two parameters are then obtained using the techniques previously described. Also, it is important to note that we used the term subtract a positive or negative gamma, where subtracting a negative gamma is equivalent to adding it. Note that when adjusting for gamma, the x-axis scale for the straight line becomes <span class="texhtml">(''t'' − γ).</span>
 
 
'''Example 2:'''
{{Example: 3P Weibull Distribution}}

Latest revision as of 08:25, 8 August 2012

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