Weibull Distribution - Weibull Plot

Weibull Plot

The fit of data to a Weibull distribution can be visually assessed using a Weibull Plot. The Weibull Plot is a plot of the empirical cumulative distribution function of data on special axes in a type of Q-Q plot. The axes are versus . The reason for this change of variables is the cumulative distribution function can be linearised:

\begin{align} F(x) &= 1-e^{-(x/\lambda)^k}\\ -\ln(1-F(x)) &= (x/\lambda)^k\\ \underbrace{\ln(-\ln(1-F(x)))}_{\textrm{'y'}} &= \underbrace{k\ln x}_{\textrm{'mx'}} - \underbrace{k\ln \lambda}_{\textrm{'c'}} \end{align}

which can be seen to be in the standard form of a straight line. Therefore if the data came from a Weibull distribution then a straight line is expected on a Weibull plot.

There are various approaches to obtaining the empirical distribution function from data: one method is to obtain the vertical coordinate for each point using where is the rank of the data point and is the number of data points.

Linear regression can also be used to numerically assess goodness of fit and estimate the parameters of the Weibull distribution. The gradient informs one directly about the shape parameter and the scale parameter can also be inferred.

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