Scaling Objective Function for Widely Varying Parameters: Data Fitting with lsqcurvefit

I have am struggling to fit a dataset to a function which has fit parameters may orders of magnitude different from each other. From what I can tell, I need to scale the parameters inside the function so that there are meaningful changes to the output value when the fitting routine tweaks the parameters looking for optimal values. I have tried scaling the p(1) parameter inside the function to fit, "PAnumberSpectra", but have not yet found a method that works. Could someone provide direction?

My code/data are attached below. Sorry it's long, but it should be self-contained, and suffient to run by itself. I am using Matlab R2019B on Windows 10.

Answer 1

[Thiago Henrique Gomes Lobato]

The main problem actually isn't the scaling but rather the optimization algorithm. Your function seems to be extremely non-smooth with plently of local minima, and thus a gradient based approach isn't able to find a good solution. Using a non-gradient based algorithm can somewhat solve this problem. Here is my result when I use fminsearch that optimizes it with the Nelder-Mead algorithm:

The changes that I made in your code were:

lb = NftBnds(1,:);         ub = NftBnds(2,:);
%[betaParams,~,residual]       = lsqcurvefit(@PAnumberSpectra,guessVec,indVar,binAvgNum,lb,ub,options); % Fit the function (defined at bottom).
[betaParams,residual]       = fminsearch(@PAnumberSpectra2,guessVec,options); % Fit the function (defined at bottom).

...
    
   function Nft      = PAnumberSpectra2(p,indVar)
        % Parameter Labels
        % p(1)        = A, the amplitude param
        % p(2)        = sigma, the width param
        % p(3)        = f0, the center frequency
        % p(4)        = A0, the constant offset
        % p(5)        = Gamma, the one-body loss rate
        % indVar(1)   = Frequency of the PA beam
        % indVar(2)   = Exposure time
        [binAvgFreq,tExp,binAvgNum] = loadData();
        indVar = [binAvgFreq,tExp];

        beta = p(1)*Gamma_mol/(sqrt(2*pi)*p(2))*exp((-(indVar(:,1)-p(3)).^2./(2*p(2)^2)))+p(4);
        numerator     = meanN0.*exp(-p(5).*indVar(:,2));
        denomFactor   = (meann0*beta*xi)./(2^(3/2)*p(5));
        denominator   = 1 + denomFactor.*(1 - exp(-p(5).*indVar(:,2)));
        Nft           = numerator./denominator;
        
        Nft = rms(binAvgNum-Nft);
        
    end

And you had a little error by the plotting:

plotNumFit          = PAnumberSpectra(betaParams,indVarPlot);
initGuessAtomNum    = PAnumberSpectra(guessVec,indVarPlot);%PAnumberSpectra(betaParams,indVarPlot);

[Joshua Hill]

Thank you for your help. That does indeed seem to have been the issue.