# how do you create a model for a sum of exponentials?

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carenar on 25 Jun 2015
Answered: Image Analyst on 8 Sep 2020
I am attempting to write a code with my input data and model it as a sum of exponentials. Depending on the curve, the algorithm needs to determine if the fit is robust enough (probably compare an error value) or if it is not to add another exponential term(ie a*exp(b*t)). Thus creating an equation of exponentials to properly fit the data the best it can. I have a formula to calculate the sum of squares error. I was going to use that value to determine if another term should be added.
Anyone know some sources that may be of use to help create the algorithm?
Thanks!

the cyclist on 2 Jul 2015
If you have the Statistics and Machine Learning Toolbox, you can fit this type of model straightforwardly using the nlinfit function.

Arturo Gonzalez on 8 Sep 2020
Per this answer, you can do it with the following matlab code
clear all;
clc;
% get data
dx = 0.001;
x = (dx:dx:1.5)';
y = -1 + 5*exp(0.5*x) + 4*exp(-3*x) + 2*exp(-2*x);
% calculate n integrals of y and n-1 powers of x
n = 3;
iy = zeros(length(x), n);
xp = zeros(length(x), n+1);
iy(:,1) = cumtrapz(x, y);
xp(:,1) = x;
for ii=2:1:n
iy(:, ii) = cumtrapz(x, iy(:, ii-1));
xp(:, ii) = xp(:, ii-1) .* x;
end
xp(:, n+1) = ones(size(x));
% get exponentials lambdas
Y = [iy, xp];
A = pinv(Y)*y;
Ahat = [A(1:n)'; [eye(n-1), zeros(n-1, 1)]];
lambdas = eig(Ahat);
lambdas
% get exponentials multipliers
X = [ones(size(x)), exp(lambdas'.*x)];
P = pinv(X)*y;
P
% show estimate
y_est = X*P;
figure();
plot(x, y); hold on;
plot(x, y_est, 'r--');

Image Analyst on 8 Sep 2020
The attached demo fits any number of Gaussians to a signal. The demo uses 6 as an example, but you can change it to however many Gaussians you want.