Fitting exponential function, coefficients and errors

Dear experts,

I have experimental data for exponential distribution a*exp(b*x). I need to find coefficients a, b and their errors. I have used function fit(B,C, 'exp1') and I got some results. I have problem because some data points in my file have higher error rate because of the nature of experiment.

1.) Which algorithm or function in Matlab can give the smallest error ? 2.) How can I use/adopt some function in Matlab to put smaller weight (when I calculate coefficients) on data that drastically differ from exponential function ?

Thank you.

Answer 1

[Jonathan LeSage]

You have quite a few options that you might want to check out to improve your exponential fit! As an option within the fit function, you can exclude outlier data points from the your exponential fit. For example, here is how you would find an exponential fit with the 25th and 30th data points removed (assuming these are outlier points):

[expFit,goodness] = fit(xdata,ydata,'exp1','Exclude', [25 30]);

You can also provide a vector of weights to augment your exponential fit. In the following example, the weight vector is used in the fit:

[expFit,goodness] = fit(xdata,ydata,'exp1','Weights',weightVec);

As you mentioned, you could decrease the influence of outliers by weighting them accordingly. Another recommendation would be to explore the curve fitting tool that allows you to graphically fit your data. Use cftool to open the user interface.

Here are some links to additional documentation to help get you started:

• http://www.mathworks.com/help/curvefit/fit.html
• http://www.mathworks.com/help/curvefit/predint.html
• http://www.mathworks.com/help/curvefit/excludedata.html
• http://www.mathworks.com/help/curvefit/interactive-curve-and-surface-fitting-.html

Answer 2

[Amar Yimam]

%% COF_ExponentialFit_All.m

% Exponential least-squares fitting for all four lubricants

% Time range: 2400–3600 s, with extrapolation to 5000 s

clc; clear; close all;

%% --- 1. Load Excel Data ---

filename = 'Lubricants_COF.xlsx';

data = readtable(filename);

t_all = data.Time;

cof_data = {data.Luba, data.Lubb, data.Lubc, data.Lubd};

lubricants = {'Luba','Lubb','Lubc','Lubd'};

%% --- 2. Select Final Phase (2400–3600 s) ---

idx_final = (t_all >= 2400 & t_all <= 3600);

t_final = t_all(idx_final);

for i = 1:length(cof_data)

cof_data{i} = cof_data{i}(idx_final);

end

%% --- 3. Normalize COF (optional for better stability) ---

cof_norm = cell(size(cof_data));

for i = 1:length(cof_data)

cof_min = min(cof_data{i});

cof_max = max(cof_data{i});

cof_norm{i} = (cof_data{i} - cof_min) / (cof_max - cof_min);

end

%% --- 4. Define exponential model: y = a * exp(k * t)

expModel = @(p,t) p(1) * exp(p(2) * t);

results = struct();

%% --- 5. Fit exponential model for each lubricant ---

for i = 1:length(lubricants)

name = lubricants{i};

y = cof_norm{i};

t = t_final;

% Normalize time to reduce numerical range

t_scaled = t / 1000;

% Initial guesses (like Mathematica)

p0 = [0.01, 1e-6];

lb = [0, -1]; % allow slow decay or growth

ub = [10, 1];

opts = optimoptions('lsqcurvefit','Display','off');

[p_fit,~,resid] = lsqcurvefit(expModel,p0,t_scaled,y,lb,ub,opts);

% Fitted curve

t_smooth = linspace(min(t_scaled), 5, 500);

y_fit = expModel(p_fit,t_smooth);

% Goodness of fit (R²)

SSE = sum(resid.^2);

SST = sum((y - mean(y)).^2);

R2 = 1 - SSE/SST;

% Store

results.(name).params = p_fit;

results.(name).R2 = R2;

results.(name).t_smooth = t_smooth * 1000;

results.(name).y_fit = y_fit;

% Display

fprintf('\n=== %s ===\n', name);

fprintf('a = %.6f, k = %.6e\n', p_fit(1), p_fit(2));

fprintf('R² = %.4f\n', R2);

% Plot

figure('Color','w'); hold on;

scatter(t, y, 40, 'filled', 'MarkerFaceColor',[0.85 0.33 0.10], ...

'MarkerEdgeColor','k', 'DisplayName','Measured COF');

plot(t_smooth*1000, y_fit, 'LineWidth',2, 'Color',[0 0.45 0.74], ...

'DisplayName','Exponential Fit');

xlabel('Time (s)');

ylabel('Normalized COF');

title(sprintf('Exponential Fit – %s (R² = %.4f)', name, R2));

legend('Location','best'); grid on; hold off;

endtimes = linspace(xdata(1),xdata(end));

plot(xdata,ydata,'ko',times,fun(x,times),'b-')

legend('Data','Fitted exponential')

title('Data and Fitted Curve')