How to fit polynomial into some error bars data - matlab

Question

I need to fit data e.g. x, y, CI (where CI is confidence index of y) in Matlab.
Now, I use this code:
pf = polyfit(x, y, 2);
x1 = min(x):.1:max(x);
y1 = polyval(pf, x1);
figure
hold on
errorbar(x, y, CI, 'ko');
plot(x1, y1, 'k');
hold off
Of course, the fit comes out of some errors bars, and it's correct.
I would like obtain a fit curve closer to the points with a low confidence index, and discard the points with a high confidence index.
Thank you and bye,
Giacomo

Answer 1

What you are looking for are Weighted Least Squares. You can compute them with the function lscov. There is a nice example in its help page, but I'll try to make it clearer.
Let us construct a simple parabola, with a corrupted point
x = (0:0.1:1)';
y = 0.5*x.^2;
y(5) = 3*y(5);
and give some weights
w = ones(size(y));
w(5) = 0.1;
Next build the Vandermonde matrix (see here for the code) and solve the system
%// V = [x.^2 x ones(size(x))];
V = bsxfun(#power, x, 2:-1:0);
coeff = lscov(V, y, w);
The estimated coefficients, with and without the weights, are
x^2 x 1
with weights [0.4797 0.0186 -0.0004]
no weights [0.3322 0.1533 -0.0034]
Note that in your case w will have to be inverted.
If you don't like to build the Vandermonde matrix, and you have a license for the Curve Fitting Toolbox, you can use the following code
ft = fittype('poly2');
opts = fitoptions('Method', 'LinearLeastSquares');
opts.Weights = w;
fitresult = fit(x, y, ft, opts);
and you'll obtain the same result.