This demo creates a noisy gaussian with unknown random parameters and then fits the data using lsqnonlin.
1. Create noisy gaussian with unknown parameters. The gaussian function below includes terms for location of peak (mu), width (sig), amplitude (amp), and vertical offset (vo).
gaus = @(x,mu,sig,amp,vo)amp*exp(-(((x-mu).^2)/(2*sig.^2)))+vo;
mu = rand(1)*randi(8) + 1;
y = gaus(x,mu,sig,amp,vo);
y = y + rand(size(x)).*y;
2. Fit the data using lsqnonlin. This requires estimating the parameters. The location of the peak and amplitude are estimated using max(). The width is assumed to be 1 but you may need to use a more rigorous estimation of the width. Vertical offset is assumed to be 0. This demo does not set the upper and lower bounds but you may want to include them.
FUN = @(p)gaus(x,p(1),p(2),p(3),p(4)) - y;
[ampHat, maxIdx] = max(y);
pEst = lsqnonlin(FUN,[muHat, 1, ampHat, 0]);
Local minimum possible.
lsqnonlin stopped because the final change in the sum of squares relative to
its initial value is less than the value of the function tolerance.
3. Plot the raw data and the fit. Show the fit parameters in the figure title.
plot(x,y, 'o','DisplayName','raw data')
xFit = linspace(min(x),max(x),100);
yFit = gaus(xFit,pEst(1),pEst(2),pEst(3),pEst(4));
plot(xFit, yFit,'r-','DisplayName','fit')
title(sprintf('mean=%.2f, sigma=%.2f, amp=%.2f, voff=%.2f',pEst))
