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## Weibull Distributions

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 parameter

• A 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

1. Open the Curve Fitting app by entering cftool. Alternatively, click Curve Fitting on the Apps tab.

2. 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.

3. 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.

### 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)
This Weibull model is defined with three parameters: the first scales the curve along the horizontal axis, the second defines the shape of the curve, and the third scales the curve along the vertical axis. Notice that while this curve has almost the same form as the Weibull probability density function, it is not a density because it includes the parameter c, which is necessary to allow the curve's height to adjust to data.