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

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:

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

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)