Global Fit larger dataset to nonlinear equation with one global variable , and N number of variables, better objective function needed

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Velencia
Velencia 2024-9-3
编辑： Velencia 2024-9-15
I have a very larger data set (m x N) decays
i need to fit the larger data set to a nonlinear equation y = b_N + e(-i*c*d_N) where c is a global parameter and B_N and d_N are unique to each line in the matrix.
I am looking for a good example to follow or format. The exists examples are for one or two data set but I have over 20 decays.
I have the following code but the fit is very bad. I think the objective function need to change but I am not sure how to improve it.
% Define the number of datasets
N = length(Y_N)
tau = tau_N; % Cell array to store tau_N values for each dataset
Y = Y_N; % Cell array to store Y_N values for each dataset
% function is
% exp((-8*(G^2)*(gamma^2)*tau*(D_zz(omega)*tau)/T_2*(pi^2)*(omega^2))
%exp( (-8*(G^2)*(gamma^2)/(pi^2))*(tau^2*D__zz(i))/(T_2*(omega^2))
%42.577e9 -> (gamma/2pi)
%G is 35 T/M for the PM2
factor_PM2 = 2*(35^2)*((42.577e9)^2); % is equal to values for 8*(G^2)*(gamma^2)/(pi^2))
% where D_zz is param(N+i), param(i) is an Amplitude, and param(end) is the Global T2
%where omega is (1/tau_N_1) which is the first x values of each data set
model = @(params, tau_N, tau_N_1, i) ...
params(i) * exp(-factor_PM2 * params(N+i) * tau_N.^2 / (params(end) * ((1 / tau_N_1)^2)));
% Initial guesses for A_N, B_N, C_N for each dataset
A_guess = 5 * ones(N, 1); % Initial guess for A_N
B_guess = .00000001* ones(N, 1); % Initial guess for B_N
T2_guess = 10; % Initial guess for the global parameter T2
% Combine initial guesses for all datasets into a single parameter vector
initial_params = [A_guess; B_guess; T2_guess];
% Initial guesses for A_N, B_N, C_N for each dataset
A_guess = .1 * ones(N, 1); % Initial guess for A_N
B_guess = 5 * ones(N, 1); % Initial guess for B_N
C_guess = 10e-9 * ones(N, 1); % Initial guess for C_N
T2_guess = 1; % Initial guess for the global parameter D
% Combine initial guesses for all datasets into a single parameter vector
initial_params = [A_guess; B_guess; T2_guess];
% Define the objective function for fitting all datasets
objective_function = @(params) ...
sum(arrayfun(@(i) sum((Y{i} - model(params, tau{i}, tau{i}(1), i)).^2), 1:N));
% Perform the fitting using lsqnonlin (requires Optimization Toolbox)
options = optimoptions('lsqnonlin', 'Display', 'iter'); % Option to display the fitting progress
fitted_params = lsqnonlin(objective_function, initial_params, [], [], options);
% Extract fitted parameters
A_fit = fitted_params(1:N);
B_fit = fitted_params(N+1:2*N);
T2_fit = fitted_params(end);
% Display results
fprintf('Fitted parameters:\n');
for i = 1:N
fprintf('Dataset %d: A = %.4f, B = %.4f\n', i, A_fit(i), B_fit(i));
end
fprintf('Global parameter T2 = %.4f\n', T2_fit);
% Plotting the data and fits
figure;
hold on;
for i = 1:N
% Plot the original data points
plot(tau{i}, Y{i}, 'o', 'DisplayName', sprintf('Data %d', i));
% Calculate the fitted curve using the fitted parameters
fitted_curve = A_fit(i) * exp(-factor_PM2 * B_fit(i) * tau{i}.^2 / (T2_fit * (1 / tau{i}(1)^2)));
% Plot the fitted curve
plot(tau{i}, fitted_curve, '-', 'DisplayName', sprintf('Fit %d', i));
end
xlabel('\tau');
ylabel('Y');
legend show;
PLease advise.
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Velencia
Velencia 2024-9-10
编辑：Velencia 2024-9-15
no the data set is too large to provide and is data athat i cannot share. I can provide assimulated data set would that be useful @Aastha

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回答（1 个）

Subhajyoti
Subhajyoti 2024-9-9
Hi @Velencia,
To fit a large dataset to a non-linear equation, you can use the ‘lsqcurvefit’ function. It solves nonlinear curve-fitting (data-fitting) problems in least-squares sense.
You can refer to the following link for more information on ‘lsqcurvefit’ function:
https://www.mathworks.com/help/optim/ug/lsqcurvefit.html
You can follow these steps for the given curve-fitting problem:
  1. Prepare the Data: Ensure your data matrix is organized such that each row corresponds to a different decay, and your independent variable (e.g., time) is appropriately defined.
  2. Initial Guess: Provide an initial guess for the parameters. This can significantly affect the convergence of the optimization.
  3. Optimization: Use the optimization function - ‘lsqcurvefit’ to fit the model to your data.
Additionally, you can refer to these resources to learn more about curve-fitting in MATLAB:

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