Fitting multiple datasets with unique parameters using lsqcurvefit to a normalized function and constraints between datasets.

Thank you for taking the time to help. I will do my best to be clear and concise.

I have 3 sets of data (atttached), each with x and y data - not necesarilly the same length.

Each y(x) function can be normalized into p(t) using 4 parameters:

1. A - a constant that is extracted from each individual y(x), 'A' looks like [A1, A2, A3]
2. B - a fitting parameter unique to each y(x)
3. C - a fitting parameter applied to all y(x)
4. D - a constraint variable dependent on A and B applied to all y(x)

So the set looks something like:

I would like to use lsqcurvefit to fit my data to my function p(t) and output the unique (C and D) values.

I have a final constraint to increase the accuracy of lsqcurvefit: , where D is another global constant like C.

Where I'm at now, I want to adjust C and D to fit my function p(t) to y(x), but I don't know how to also include the unique A values for each y(x).

I've pasted my current code below where I am able to get an individual fit for each y(x) but without consistent C and D parameters.

%% Data analysis
Results = NaN(3,2); % Result [C,D] for each individual fitting
for t = 1:3
    %TL is summary matrix of all y(x) for the 3 experiments, NaN fills in
    %where some y(x) are shorter than others.
    x = TL(~isnan(TL(:,2*t-1)),2*t-1);  %extracts x values from TL
    y = TL(~isnan(TL(:,2*t)),2*t);      %extracts y values from TL
    
    %P = parameters: P(1) = C | P(2) = D
    %fun below is the normalization for p(t)
    fun = @(P,x)A(t)*(1-1/(2*(1-P(1))))*(1.304*exp(-(P(2)*x/(r/1000)^2).^0.5)-0.304*exp(-0.254*P(2)*x/(r/1000)^2))+1/(2*(1-P(1)));
    P0 = [0.4, 1*10^-9];                %Initial parameters 
    lb = [0.1, 1*10^-10];               %Lower bound
    ub = [0.5, 1*10^-8];                %Upper bound

    Results(t,:) = lsqcurvefit(fun,P0,x,y,lb,ub);   %Result consolidation for individual fitting

    subplot(3,1,t)
    plot(x,y,'ko',x,fun(Results(t,:),x),'b-')
end

From looking at other posts on using lsqcurvefit with multiple datasets, I think my obstacle has come down to whether I should (1) run lsqcurvefit on all y(x) and somehow input the unique A constants or (2) run lsqcurvefit on individual y(x) and somehow correlate C and D to match between each fitting.

I hope this is presented clearly, looking forward to your thoughts and discussion - Thanks.

Answer 1

[Torsten]

If your model functions share common parameters, you must run "lsqcurvefit" on all yi with all parameters present (in your case eight as far as I can see).

Since the computation of the simulated values is more complex than it can be done in a one-liner by a function handle, you should write a separate function for this.

[Nolan Miller]

To clarify, there are only 2 global fitting parameters to normalize all y(x).

I see where you get the 8 parameters, but A and B are dependent on D.

Is there a way to run lsqcurvefit varying two parameters and taking in a set of constants?

[Torsten]

I see where you get the 8 parameters, but A and B are dependent on D.

Why do you define D at all as a fitting parameter if it can be replaced by A1/B1 ?

Is there a way to run lsqcurvefit varying two parameters and taking in a set of constants?

You don't need to "take constants in", you can just define them in the function where you supply the simulated yi.

[Nolan Miller]

For the first part, it is because each y(x) corresponds to a specific A#, B#. There is a constraint equation that , so this goes back to running all datasets for C and D or individual datasets. Ultimately, I want to extract best fit parameters C and D because those must be the same across fitting all y(x) to p(t).

With my current code, I'm getting the output you suggest by ignoring the D constraint where I use specific A# and B# to fit y(x) to p(t), but then between different y(x) I get different C and D values (which should not be the case).

In regards to the constants, I suppose it would be more precise to say that they are not "constants" with respect to lsqcurvefit, but rather that each A value does not change with respect to fitting lsqcurvefit to its own y(x) and only changes with respect to fitting different y(x).

Thank you for your help