How can I use the LSQNONLIN function within the Optimization Toolbox to obtain the weighted least squares fit?

The LSQNONLIN and LSQCURVEFIT functions determine the least squares fit without weighting. I would like to introduce weights to the error used in the least squares fit.

Answer 1

[MathWorks Support Team]

To use LSQNONLIN to do a weighted least square fit, you need an equation to which you want to fit your data. Below is a short example demonstrating how to use LSQNONLIN to obtain a weighted fit. First it creates a data set using two different equations, adding in some noise. Then it calls LSQNONLIN, using a handle to the nested function, MYCURVE. Finally, it plots the data and the fitted line. To run this example copy all the MATLAB code below into a single file, save it as myFit.m, and then run the function MYFIT.

function myFit

%%create the first half of the data

xdata1 = 0:.01:1;

ydata1 = exp(-3*xdata1) + randn(size(xdata1))*.05;

weight1 = ones(size(xdata1))*1;

%%create the second half of the data 

% use a different function and with higher weights

xdata2 = 1:.01:2;

ydata2 = (xdata2-1).^2 + randn(size(xdata2))*.05;

weight2 = ones(size(xdata2))*10;

%%combine the two data sets

xdata = [ xdata1 xdata2 ];

ydata = [ ydata1 ydata2 ];

weight = [ weight1 weight2 ];

%%call |LSQNONLIN|

parameter_hat = lsqnonlin(@mycurve,[.10 -1],[],[],[],xdata,ydata)

%%plot the original data and fitted function in a figure

plot(xdata,ydata,'b.')

hold on

fitted =  exp(parameter_hat(1).*(parameter_hat(2) +xdata));

plot(xdata,fitted,'r')

xlabel('x'); ylabel('y')

legend('Data', 'Fit')

%%function that reports the error

      function err = mycurve(parameter,real_x, real_y)

          fit =  exp(parameter(1).*(real_x + parameter(2)));

          err = fit - real_y;

          % weight the error according to the |WEIGHT| vector

          err_weighted = err.*weight;

          err = err_weighted;

      end

end

Comparing the fitted line with the original data, you will find that the second half of the data is fitted well, while the first half is not. Also note that the functional form of the model being used in MYCURVE is not the same function used to create the original data. However, an exponential can approximate a second order polynomial, over a small range, well enough.

Weighting the first half of the data set more than the second half would get LSQNONLIN to only fit the decaying exponential part (first part) of the data and not the polynomial. Getting LSQNONLIN to converge to the first half of the data might also require a change in the initial solution estimate as well.

For more information on how to use LSQNONLIN syntax, refer to the Optimization Toolbox User's Guide.