WeibullFit

Performs a curve fit using the two-parameter Weibull distribution (lifetime analysis).

Syntax

WeibullFit(Sample, [ NumberOfRightCensoredValues = 0 ], [ Algorithm = WEIBULLFIT_ALGORITHM_LSE ], [ MLECorrectionFactor = TRUE ], [ MLEMaximumNumberOfIterations = 500n ] [ , MLETolerance = 1e-6 ])

The syntax of the WeibullFit function consists of the following parts:

Remarks

As a result of the curve fitting, the fitted parameters α and β of the two-parameter Weibull distribution are output as a list. The continuous probability density f(x) and distribution function F(x) of the Weibull distribution with the parameters α and β (shape parameters) are given by:

Note: in the literature, the characteristic lifetime T is often used as an alternative to the parameter α (and β is denoted by k). The following relation applies here:

The results of the list (i.e. the curve fitting parameters found) can be accessed using the following syntax: result.alpha or result.beta. With the help of the Distribution function can then be used to output the distribution (or a survival plot) of the fitted Weibull distribution (lifetime distribution).

The algorithms for Weibull curve fitting can be described schematically as follows:

Least Squares Algorithm

If the distribution of a Weibull-distributed random variable is plotted in the form log(log(1/(1-F(x)))) = β*log(x) + log(α) in a double logarithmic diagram (Weibull network), the result is a straight line with slope β and y-intercept log(α). Exactly this principle is now used for least squares Weibull curve fitting. The empirical distribution function F_emp serves as an estimator for the distribution of the sample (with Hazen's formula as the calculation mode according to the EmpiricalDistribution function). The slope and y-intercept of log(log(1/(1-F_emp))) are then determined using linear regression analysis. This finally provides estimators for the parameters β and log(α), and thus also for α.

Details can be found in [1, Chapter 13, Section 3.2.1] or [2, Chapter 5, Section 5.1.2.2].

Maximum Likelihood Algorithm

This algorithm is based on the usual maximum likelihood principle, according to which (in simple terms) those parameters α and β are selected as estimates according to whose distribution the realization of the observed data appears most plausible. This can be formulated mathematically equivalent in the form of a maximization problem, according to which the parameters are to be selected in such a way that the so-called log-likelihood function is maximized. Extreme points of the log-likelihood function in turn correspond to zero points of the derivative of the log-likelihood function according to the parameters α and β. In the present case of the two-parametric Weibull distribution, the latter zero determination does not have an analytically closed solution, but can be determined iteratively using numerical equation solvers. A fixed point iteration is used as the numerical equation solver. The fixed-point iteration is terminated if the relative value change of the parameters is smaller than the specified delta (see argument MLETolerance). The maximum number of iterations is determined by the argument MLEMaximumNumberOfIterations. If the maximum number of iterations has been reached and the relative value change has not yet fallen below the specified tolerance value, the algorithm has not converged and a corresponding error message is displayed.

One disadvantage of maximum likelihood parameter estimation is its bias when the sample size is small. To correct this disadvantage, appropriate correction factors (depending on the sample size) can be taken into account (see argument MLECorrectionFactor).

All details can be found in [2, Chapter 5, Section 5.1.2.3] and [3].

Available in

Option Statistics

Examples

WeibullFit(Sample)

Performs a least squares curve fit using the two-parameter Weibull distribution of a sample and returns the α and β parameters found as a list. The sample contains only the failed objects with corresponding failure times in the Y component.

Dim p = WeibullFit(Sample)
Dim x = Series(0, 5*Maximum(Sample), 0.1)
Signal(Distribution(x, DISTRIBUTION_WEIBULL, p.alpha, p.beta), x)
 

Performs a Weibull curve fit of a sample and calculates the fitted Weibull distribution (lifetime distribution) in the interval [0, 5*Maximum(Sample)].

Dim p = WeibullFit(Sample)
Dim x = Series(0, 5*Maximum(Sample), 0.1)
Signal(1 - Distribution(x, DISTRIBUTION_WEIBULL, p.alpha, p.beta), x)
 

Performs a Weibull curve fit of a sample and calculates the resulting survival plot in the interval [0, 5*Maximum(Sample)].

WeibullFit(Sample, 7)

As an extension to the first example, the sample is now supplemented by 7 additional units that have not yet failed by the end of the measurement. The lifetime of these additional 7 test specimens is therefore greater than the maximum of all failure times in the Y component of the sample.

WeibullFit(Sample, , WEIBULLFIT_ALGORITHM_MLE)

Performs a maximum likelihood curve fit using the two-parameter Weibull distribution of a sample and returns the α and β parameters found as a list. The sample contains only the failed objects with corresponding failure times in the Y component.

See Also

Distribution Function

EmpiricalDistribution Function

NonLinCurveFit Function

Statistics Option

References

[1] Hartung, Joachim: Statistik, 9. Auflage. Oldenbourg Verlag GmbH, München, 1993. ISBN 3-486-22055-1.

[2] Wilker, Holger: Weibull-Statistik in der Praxis, 2. Auflage. Books on Demand GmbH, Norderstedt, 2010. ISBN 3-8391-6241-5.

[3] T. Kernane and Z. A. Raizah: Fixed point iteration for estimating the parameters of extreme value distributions. In: Communications in Statistics - Simulation and Computation, Taylor and Francis, Vol. 38, No. 10, Pages 2161-2170. Prentice Hall, 2009.