## Weibull Distributions

### About Weibull Distribution Models

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.

### Fit Weibull Models Interactively

Open the Curve Fitter app by entering

`curveFitter`

at the MATLAB^{®}command line. Alternatively, on the**Apps**tab, in the**Math, Statistics and Optimization**group, click**Curve Fitter**.In the Curve Fitter app, select curve data. On the

**Curve Fitter**tab, in the**Data**section, click**Select Data**. In the**Select Fitting Data**dialog box, select**X data**and**Y data**, or just**Y data**against an index.Click the arrow in the

**Fit Type**section to open the gallery, and click**Weibull**in the**Regression Models**group.

There are no fit settings to configure in the **Fit
Options** pane.

Optionally, in the **Advanced Options** section, specify
coefficient starting values and constraint bounds, or change algorithm
settings. The app calculates random start points for
**Weibull** fits, defined on the interval [0 1]. You
can override the start points and specify your own values in the
**Fit Options** pane.

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

### Selecting a Weibull Fit at the Command Line

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)

## See Also

### Apps

### Functions

`fit`

|`fittype`

|`fitoptions`