lsqcurvefit enables you to fit parameterized
nonlinear functions to data easily. You can use lsqnonlin as
well; lsqcurvefit is simply a convenient way
to call lsqnonlin for curve fitting.
In this example, the vector xdata represents
100 data points, and the vector ydata represents
the associated measurements. Generate the data using the following
script:
rng(5489,'twister') % reproducible
xdata = -2*log(rand(100,1));
ydata = (ones(100,1) + .1*randn(100,1)) + (3*ones(100,1)+...
0.5*randn(100,1)).*exp((-(2*ones(100,1)+...
.5*randn(100,1))).*xdata);The modeled relationship between xdata and ydata is
| (10-14) |
The script generates xdata from 100 independent
samples from an exponential distribution with mean 2. It generates ydata from Equation 10-14 using a = [1;3;2], perturbed by adding normal
deviates with standard deviations [0.1;0.5;0.5].
The goal is to find parameters , i = 1, 2, 3, for the model that best fit the data.
In order to fit the parameters to the data using lsqcurvefit,
you need to define a fitting function. Define the fitting function predicted as
an anonymous function:
predicted = @(a,xdata) a(1)*ones(100,1)+a(2)*exp(-a(3)*xdata);
To fit the model to the data, lsqcurvefit needs
an initial estimate a0 of the parameters. Enter
a0 = [2;2;2];
Run the solver lsqcurvefit as follows:
[ahat,resnorm,residual,exitflag,output,lambda,jacobian] =... lsqcurvefit(predicted,a0,xdata,ydata); Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the default value of the function tolerance.
To see the resulting least-squares estimate of , enter:
ahat
ahat =
1.0169
3.1444
2.1596The fitted values ahat are within 8% of a = [1;3;2].
If you have Statistics and Machine Learning Toolbox™ software, use the nlparci function
to generate confidence intervals for the ahat estimate.