Nonlinear curve fitting with summation function

I have the following data (Rt vs Sm) that I would like to fit with the following equation:

where ri=r0+i*(Ss+Sm), r0=30. In the fitting, Rs, Rm and LT are fitting parameters with initial guesses [100, 0.2, 0.2].

Please excuse the poor coding as I am new to MATLAB and have been using ChatGPT to help. I have the following code but it returned multiple errors and a few fixes doesn't seem to help. I am attaching the data file. Thank you in advance!

% Load data from Excel file

data = readtable('results.xlsx');

% Designate first column as Sm and second column as Rt

Sm = data(:,1).Variables;

Rt = data(:,2).Variables;

% Define fitting function

ri = 30+(0:9).*(10+Sm);

rj = (1:9).*(10+Sm);

fun = @(x,Sm) ((x(1))/(2*pi))*sum((log((ri+Sm)./ri))+((x(2))*((1./ri)+(1./(ri+Sm)))))+(x(3))/(2*pi)*(sum(log((rj-(x(2)))./(ri-10+(x(2))))));

% Set initial guess for fitting parameters

x0 = [100, 0.2, 0.2];

% Perform nonlinear curve fitting

x = lsqcurvefit(fun,x0,Sm,Rt);

% Calculate fitted values

Rt_fit = fun(x,Sm);

% Calculate R-square value

Rsq = 1 - sum((Rt - Rt_fit).^2)/sum((Rt - mean(Rt)).^2);

% Plot data and fitted curve

figure;

plot(Sm,Rt,'o',Sm,Rt_fit);

legend('Data','Fitted Curve');

xlabel('Sm');

ylabel('Rt');

Answer 1

[Torsten]

Check again whether fun is correctly defined.

I changed ri to rj in the last log expression.

% Load data from Excel file
data = readtable('results.xlsx');

% Designate first column as Sm and second column as Rt
Sm = data(:,1).Variables;
Rt = data(:,2).Variables;

% Define fitting function
ri = 30+(1:9).*(10+Sm);
rj = (1:9).*(10+Sm);
fun = @(x,Sm) x(1)/(2*pi)*sum(log((ri+Sm)./ri)+x(2)*(1./ri+1./(ri+Sm)),2)+x(3)/(2*pi)*sum(log((rj-x(2))./(rj-10+x(2))),2);
% Set initial guess for fitting parameters
x0 = [100, 0.2, 0.2];

% Perform nonlinear curve fitting
x = lsqcurvefit(fun,x0,Sm,Rt);

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

% Calculate fitted values
Rt_fit = fun(x,Sm);

% Calculate R-square value
Rsq = 1 - sum((Rt - Rt_fit).^2)/sum((Rt - mean(Rt)).^2);

% Plot data and fitted curve
figure;
plot(Sm,Rt,'o',Sm,Rt_fit);
legend('Data','Fitted Curve');
xlabel('Sm');
ylabel('Rt');

[Chengyang]

This is absolutely correct! Thank you so much!

[Chengyang]

Could you please explain the ,2 part that you added to make this work? Thank you so much!

[Torsten]

While I tested your code, I changed

ri = 30+(0:9).*(10+Sm);

to

ri = 30+(1:9).*(10+Sm);

You should correct this in the above code since you want to sum from 0 to 9 in the first sum.

Concerning your question:

The expressions

log((ri+Sm)./ri)+x(2)*(1./ri+1./(ri+Sm))

and

log((rj-x(2))./(rj-10+x(2)))

are matrices of size (number of elements of Sm) x (number of elements of ri resp. rj).

To compute the sums over the ri resp rj, it's necessary to sum the columns of this matrix. That's what the "2" stands for.