باشگاه دانشجویی کارشناسی ارشد دانشگاه امیرکبیر

Example: Robust Fitting

1. Create a baseline sinusoidal signal:

   xdata = (0:0.1:2*pi)'; 
   y0 = sin(xdata);

2. Add noise to the signal with non-constant variance:

   % Response-dependent Gaussian noise
   gnoise = y0.*randn(size(y0));
   
   % Salt-and-pepper noise
   spnoise = zeros(size(y0)); 
   p = randperm(length(y0));
   sppoints = p(1:round(length(p)/5));
   spnoise(sppoints) = 5*sign(y0(sppoints));
   
   ydata = y0 + gnoise + spnoise;

3. Fit the noisy data with a baseline sinusoidal model:

   f = fittype('a*sin(b*x)'); 
   fit1 = fit(xdata,ydata,f,'StartPoint',[1 1]);

4. Identify “outliers” as points at a distance greater than 1.5 standard deviations from the baseline model, and refit the data with the outliers excluded:

   fdata = feval(fit1,xdata); 
   I = abs(fdata - ydata) > 1.5*std(ydata); 
   outliers = excludedata(xdata,ydata,'indices',I);
   
   fit2 = fit(xdata,ydata,f,'StartPoint',[1 1],'Exclude',outliers);

5. Compare the effect of excluding the outliers with the effect of giving them lower bisquare weight in a robust fit:

   fit3 = fit(xdata,ydata,f,'StartPoint',[1 1],'Robust','on');

6. Plot the data, the outliers, and the results of the fits:

   plot(fit1,'r-',xdata,ydata,'k.',outliers,'m*') 
   hold on
   plot(fit2,'c--')
   plot(fit3,'b:')
   xlim([0 2*pi])

   

7. Plot the residuals for the two fits considering outliers:

   figure 
   plot(fit2,xdata,ydata,'co','residuals') 
   hold on
   plot(fit3,xdata,ydata,'bx','residuals')