Weibull Distribution - Hazard Plotting
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Weibull Distribution - Hazard Plotting |
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Hazard Plotting is tedious and computer routines are often employed. Commercial software is available for these calculations, such as the Reliability & Maintenance Analyst. The following section describes hazard plotting for the Weibull distribution using the Reliability & Maintenance Analyst. The manual method is located here. Hazard plotting supports the 2-parameter and 3-parameter Weibull distribution, and is an excellent method for determining goodness-of-fit. To determine the goodness-of-fit click the "Plot" button. If the plotted points form a straight line, the distribution provides a good time to fail model for the data. The "R-Squared" value is a measure of how well the data forms a straight line. An R-Squared value of 1.0 indicates a perfectly straight line. R-Squared is also known as the coefficient of determination. To estimate the parameters of the Weibull distribution using hazard plotting, follow these steps:
The figure below shows the Weibull hazard plotting screen using the data in the file "Demo2.dat".
The default for this operation is a location parameter of zero. In the Location Parameter frame, the software will determine the best value for the location parameter if "Software Estimate" is selected. The user is also given the option of entering a location parameter in the text box below the "User Entered" option. Be cautious when using a non-zero location parameter. A positive location parameter indicates a zero probability of failure for time less than the value of the location parameter, and a negative location parameter means that the population had failures before testing began. Unless there is a sound engineering for one of these conditions, it is best to use a location parameter equal to zero. Clicking the "Plot" button gives a hazard plot If the plotted points do not follow a straight line, the Weibull distribution with the estimated parameters does not provide an adequate time to fail model. The untransformed option plots the cumulative distribution function against time. This is useful for determining the probability of failure at a given time. The title of the graphs can be changed by editing the text in the Graph Title frame. To check the spelling of the title, click the "Spell Check" button. To predict reliability or time-to-fail using the estimated parameters use the Predicting Module.
The Weibull cumulative hazard function is H(x) = -ln[1-F(x)] Replacing F(x) and rearranging gives a linear expression.
By plotting lnH(x) versus lnx the resulting slope (censored points are not plotted) provides an estimate of b . The y-intercept of this plot is an estimate of blnq. Thus, q is estimated from the expression
where, y0 is the y-intercept of the hazard plot. The hazard function h(x) is estimated from the inverse of the reverse rank of the ordered failures; and the cumulative hazard function, H(x), is the cumulative of the values of h(x). An alternative to plotting lnH(x) versus lnx is to directly plot H(x) versus x on specialized Weibull hazard paper. The advantage of hazard paper is that logarithmic transformations do not have to be computed. Computers have made this technique obsolete. Example: Time to Fail
Solution: The Table below is constructed to obtain the necessary plotting data.
The final two columns of this table can be plotted. This plot is shown in the figure below. The slope of the best-fit straight line through the data (found by linear regression) is equal to 2.34 and provides an estimate of b. The y-intercept of the best-fit straight line through the data is -13.004. The estimated scale parameter for the Weibull distribution is
Click here to download this solution in Microsoft Excel.
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