Curve fitting a function that is dependent on a difference of 2d functions

I have collected and plotted some data. The function I have plotted is the following:

    for i=1:15
        for j=i+1:15
             DDTV_r0(i,j)=((D^(1/3)*((DTV(i)-DTV(j))))/(2*((cX(i)-cX(j))+(cY(i)-cY(j)))*.1816*Lambda^2))^(-3/5);
        end
    end

where DDTV_r0 is the value Im after. DTV on the right-side of the equation was calculated based on the data collected, and the rest of the variables are constants. DDTV_r0 has a theoretical value of (0.0025) for all valid combinations of (i,j) where j>i as shown in the for-loop. I would like to do some sort of curve fitting process that allows me to find what the values of DTV should be to get the theoretical DDTV_r0 values and compare those values to the DTV from my real data. What's confusing me is how to do this when DTV appears two times, as a difference, and the term (DTV(i)-DTV(j)) is constrained such that j>i. Im not sure how to handle this is the curvefitting toolbox. Any suggestions?

Here is an image of the plot for DDTV_r0(i,j) shown below from the data I collected. How do I use the curvefitting toolbox to create a similar plot for but for the theoritical values of DDTV_r0, and somehow give me the error between the fitted DTV values and the DTV from my data?

Thank you for any advice you can give!

[Sam Chak]

@Scott Kaiser, I tried, but it doesn't plot. Do you have any idea?

for i=1:15
    for j=i+1:15
         DDTV_r0(i,j)=((D^(1/3)*((DTV(i)-DTV(j))))/(2*((cX(i)-cX(j))+(cY(i)-cY(j)))*.1816*Lambda^2))^(-3/5);
    end
end

Unrecognized function or variable 'D'.

[Scott Kaiser]

You may have trouble plotting since (cX(i)-cX(j))+(cY(i)-cY(j) are all each from a look-up table. Is this the issue you are having?

[Sam Chak]

@Scott Kaiser, Thanks for your clarification. Can you attach the look-up table so that we can test out @Matt J's suggestion?

Answer 1

[Matt J]

The Curve Fitting Toolbox is meant for fitting a small number of parameters. Here, you have 15 parameters, so you should use lsqcurvefit. lsqcurvefit doesn't care about the implementation details of the prediction function. You can put it inside a function that does whatever you need it to do.

upperTri=triu( true(15) ,1);

xdata={cX(:),cY(:),D,lambda,upperTri};

ydata=0.0025*ones(nnz(upperTri),1);

DTV = lsqcurvefit(@F, DTVguess, xdata,ydata);

DDTV0=F(DTV,xdata);


function pred=F(x,xdata)  

    DTV=x(:);
    
    [cX,cY,D,lambda,upperTri]=deal(xdata{:});

    
    Numerator=D^(1/3)*(DTV-DTV');
    Denominator=2*(  (cX-cX') + (cY-cY')   )*.1816*Lambda^2;
    
    DDTV_r0=(Numerator./Denominator).^(-3/5);

    
    pred=DDTV_r0(upperTri);
    
end

Answer 2

[Catalytic]

There appears to be a simple analytical solution -

constant=(0.0025)^(-5/3)*(2*Lambda^2*0.1816)/D^(1/3);

DTV=(cX+cY)*constant