Fitting a complicated PDF

Hello All,

I have a data set that is a sum of a gaussian and exponential. I am attaching the x and y of it. I have the curve fitting toolbox but count not find a proper fit this this type of data. Could you please help?

I tried fitting this function:

a1*exp(-((x-b1)/c1)^2)+ a*exp(-b*x)

But it wont fit the tail. See picture.

Thanks!

load x.mat;
load y.mat;
stairs(y,x);

[Alex Sha]

Hi, the result is not bad:

Sum Squared Error (SSE): 0.000346048772160713
Root of Mean Square Error (RMSE): 0.00131538734249785
Correlation Coef. (R): 0.99638338591949
R-Square: 0.992779851736387

Parameter	Best Estimate
----------	-------------
a1	0.0805692205334533
b1	28.393379454596
c1	-3.12814917178287
a	0.000431608180208735
b	-0.00125973014058258

[Sanchit Sharma]

I want to fit the tail of the plot as well.

[Alex Sha]

and then you have to change the fitting model to meet your requirements

Answer 1

[Shivansh]

Hi Sanchit,

I understand that you are trying to fit a data set that is a sum of a gaussian and an exponential distribution using the curve fitting toolbox. You are facing issues in fitting the tail.

The below code can be used to fit the data:

% Load your data
load x.mat;  
load y.mat;

% Ensure x and y are column vectors
x = x(:);
y = y(:);

% Define the custom model with a stretched exponential component
ft = fittype('a1*exp(-((x-b1)/c1)^2) + a2*exp(-(b2*x)^c2)', ...
    'independent', 'x', ...
    'dependent', 'y', ... 
    'coefficients', {'a1', 'b1', 'c1', 'a2', 'b2', 'c2'});

% Provide initial guesses for the parameters
% These should be based on your data and may require adjustment
initialGuesses = [max(y), mean(x), std(x), min(y), 1/mean(x), 0.5];

% Set lower and upper bounds for the coefficients
% Adjust these bounds based on the physical constraints of your problem
lb = [0, min(x), 0, 0, 0, 0];   % Lower bounds  
ub = [Inf, max(x), Inf, Inf, Inf, 1]; % Upper bounds

% Set options for fitting, including bounds and initial guesses
options = fitoptions(ft);
options.StartPoint = initialGuesses;
options.Lower = lb;  
options.Upper = ub;

% Perform the fitting with bound  
[fitresult, gof] = fit(x, y, ft, options);

% Plot the original data and the fit
figure;
stairs(x, y, 'b'); hold on;
plot(x, fitresult(x), 'r', 'LineWidth', 2);
legend('Data', 'Fit');
xlabel('x');
ylabel('y'); 
title('Gaussian + Stretched Exponential Fit');

% Display fitting results
disp(fitresult);
disp(gof);

Output of the aboce code:

General model:

fitresult(x) = a1*exp(-((x-b1)/c1)^2) + a2*exp(-(b2*x)^c2)

Coefficients (with 95% confidence bounds):

a1 = 0.08091 (0.07971, 0.08211)

b1 = 28.39 (28.36, 28.43)

c1 = 3.151 (3.097, 3.205)

a2 = 0.001456 (-6.031, 6.034)

b2 = 2.345 (-6959, 6964)

c2 = 0.8275 (-838.4, 840.1)

sse: 3.8381e-04

rsquare: 0.9920

dfe: 194

adjrsquare: 0.9918

rmse: 0.0014

If the tail is not fitting well, you can try following workarounds to improve the fit:

1. Removing the outliers from the data.
2. If the exponential decay is not captured correctly, try using a different parameterization for the exponential, such as “a2*exp(-b2*(x-b3))”.
3. If the tail is long, you can try fitting with a stretched exponential distribution instead of the exponential distribution.

You can also work with the initial guess for the parameters or the expected behaviour of tail if you have domain knowledge of the problem.

You can refer to the following link for more information on the "fit" function of Curve fitting toolbox: https://in.mathworks.com/help/curvefit/fit.html.

Hope it helps!