Matlab find the best constants for a fitting model - matlab

Question

Please find the data in the link below, or if you can send me your private email, I can send you the data
https://dl.dropboxusercontent.com/u/5353938/test_matlab_lefou.xlsx
In the excel sheet, the first column is y, the second is x and the third is t, I hope this will make things much more clear, and many thanks for the help.
I need to use the following model because it is the one that fits best my data, but what I don't know is how to find the best values of a and b, that will allow me to get the best fit, (I can attach a file if you need the values), I already have the values of y, x and t:
y= a*sqrt(x).exp(b.t)
Thanks

Answer 1

Without the dependency on the curve fitting toolbox, this problem can also be solved by using fminsearch. I first generate some data, which you already have but didn't share with us. An initial guess on the parameters a and b must be made (p0). Then I do the optimiziation by minizmizing the squared errors between data and fit resulting in the vector p_fit, which contains the optimized parameters for a and b. In the end, the result is visualized.
% ----- Generating some data for x, y and t (which you already got)
N = 10; % num of data points
x = linspace(0,5,N);
t = linspace(0,10,N);
% random parameters
a = rand()*5; % a between 0 and 5
b = (rand()-1); % b between -1 and 0
y = a*sqrt(x).*exp(b*t) + rand(size(x))*0.1; % noisy data
% ----- YOU START HERE WITH YOUR PROBLEM -----
% put x and t into a 2 row matrix for simplicity
D(1,:) = x;
D(2,:) = t;
% create model function with parameters p(1) = a and p(2) = b
model = #(p, D) p(1)*sqrt(D(1,:)).*exp(p(2)*D(2,:));
e = #(p) sum((y - model(p,D)).^2); % minimize squared errors
p0 = [1,-1]; % an initial guess (positive a and probably negative b for a decay)
[p_fit, r1] = fminsearch(e, p0); % Optimize
% ----- VISUALIZATION ----
figure
plot(x,y,'ko')
hold on
X = linspace(min(x), max(x), 100);
T = linspace(min(t), max(t), 100);
plot(X, model(p_fit, [X; T]), 'r--')
legend('data', sprintf('fit: y(t,x) = %.2f*sqrt(x)*exp(%.2f*t)', p_fit))
The result can look like
UPDATE AFTER MANY MANY COMMENTS
Your data are column vectors, my solution used row vectors. The error occured when the errorfunction tryed to compute the difference of a column vector (y) and a row-vector (result of the model-function). Easy hack: make them all to row vectors and use my approach. The result is: a = 0.5296 and b = 0.0013.
However, the Optimization depends on the initial guess p0, you might want to play around with it a little bit.
clear variables
load matlab.mat
% put x and t into a 2 row matrix for simplicity
D(1,:) = x;
D(2,:) = t;
y = reshape(y, 1, length(y)); % <-- also y is a row vector, now
% create model function with parameters p(1) = a and p(2) = b
model = #(p, D) p(1)*sqrt(D(1,:)).*exp(p(2)*D(2,:));
e = #(p) sum((y - model(p,D)).^2); % minimize squared errors
p0 = [1,0]; % an initial guess (positive a and probably negative b for a decay)
[p_fit, r1] = fminsearch(e, p0); % Optimize
% p_fit = nlinfit(D, y, model, p0) % as a working alternative with dependency on the statistics toolbox
% ----- VISUALIZATION ----
figure
plot(x,y,'ko', 'markerfacecolor', 'black', 'markersize',5)
hold on
X = linspace(min(x), max(x), 100);
T = linspace(min(t), max(t), 100);
plot(X, model(p_fit, [X; T]), 'r-', 'linewidth', 2)
legend('data', sprintf('fit: y(t,x) = %.2f*sqrt(x)*exp(%.2f*t)', p_fit))
The result doesn't look too satisfying though. But that mainly is because of your data. Have a look here:

Answer 2

With the cftool-command (curve fitting toolbox) you can fit to your own functions, returning the variables that you need (a,b). Make sure your x-data and y-data are in separate variables. you can also specify weights for your measurements.