Main Content

The Weibull distribution is widely used in reliability and life (failure rate) data analysis. The toolbox provides the two-parameter Weibull distribution

$$y=ab{x}^{b-1}{e}^{-a{x}^{b}}$$

where *a* is the scale parameter and *b* is
the shape parameter.

Note that there are other Weibull distributions but you must create a custom equation to use these distributions:

A three-parameter Weibull distribution with

*x*replaced by*x – c*where*c*is the location parameterA one-parameter Weibull distribution where the shape parameter is fixed and only the scale parameter is fitted.

Curve Fitting Toolbox™ does not fit Weibull probability distributions to a sample of data. Instead, it fits curves to response and predictor data such that the curve has the same shape as a Weibull distribution.

Open the Curve Fitting app by entering

`cftool`

. Alternatively, click Curve Fitting on the Apps tab.In the Curve Fitting app, select curve data (

**X data**and**Y data**, or just**Y data**against index).Curve Fitting app creates the default curve fit,

`Polynomial`

.Change the model type from

`Polynomial`

to`Weibull`

.

There are no fit settings to configure.

(Optional) Click **Fit Options** to specify
coefficient starting values and constraint bounds, or change algorithm
settings.

The toolbox calculates random start points for Weibull models, defined on the interval [0,1]. You can override the start points and specify your own values in the Fit Options dialog box.

For more information on the settings, see Specifying Fit Options and Optimized Starting Points.

Specify the model type `weibull`

.

For example, to load some example data measuring blood concentration of a compound against time, and fit and plot a Weibull model specifying a start point:

time = [ 0.1; 0.1; 0.3; 0.3; 1.3; 1.7; 2.1;... 2.6; 3.9; 3.9; ... 5.1; 5.6; 6.2; 6.4; 7.7; 8.1; 8.2;... 8.9; 9.0; 9.5; ... 9.6; 10.2; 10.3; 10.8; 11.2; 11.2; 11.2;... 11.7; 12.1; 12.3; ... 12.3; 13.1; 13.2; 13.4; 13.7; 14.0; 14.3;... 15.4; 16.1; 16.1; ... 16.4; 16.4; 16.7; 16.7; 17.5; 17.6; 18.1;... 18.5; 19.3; 19.7;]; conc = [0.01; 0.08; 0.13; 0.16; 0.55; 0.90; 1.11;... 1.62; 1.79; 1.59; ... 1.83; 1.68; 2.09; 2.17; 2.66; 2.08; 2.26;... 1.65; 1.70; 2.39; ... 2.08; 2.02; 1.65; 1.96; 1.91; 1.30; 1.62;... 1.57; 1.32; 1.56; ... 1.36; 1.05; 1.29; 1.32; 1.20; 1.10; 0.88;... 0.63; 0.69; 0.69; ... 0.49; 0.53; 0.42; 0.48; 0.41; 0.27; 0.36;... 0.33; 0.17; 0.20;]; f=fit(time, conc/25, 'Weibull', ... 'StartPoint', [0.01, 2] ) plot(f,time,conc/25, 'o');

If you want to modify fit options such as coefficient starting
values and constraint bounds appropriate for your data, or change
algorithm settings, see the table of additional properties with `NonlinearLeastSquares`

on
the `fitoptions`

reference page.

Appropriate start point values and scaling `conc/25`

for the two-parameter
Weibull model were calculated by fitting a 3 parameter Weibull model using
this custom
equation:

f=fit(time, conc, ' c*a*b*x^(b-1)*exp(-a*x^b)', 'StartPoint', [0.01, 2, 5] ) f = General model: f(x) = c*a*b*x^(b-1)*exp(-a*x^b) Coefficients (with 95% confidence bounds): a = 0.009854 (0.007465, 0.01224) b = 2.003 (1.895, 2.11) c = 25.65 (24.42, 26.89)

`fit`

| `fitoptions`

| `fittype`