How to sum two data sets to get a best match of another data set?

I have three spectrum, which essentially shares the same wavelength (x-axis). I would like to add the first two spectrum (i.e. column y1 & y2) is such a way as describes below to match the 3rd spectrum:

Ideally: y3 = a*y1 + b*y2

But in reality, I would like to find coefficients a & b with minimised y3 - (a*y1 + b*y2)

All x-axis, y1, y2, and y3 are columns of 608X1

The problem I am facing is that I don't know how to optimise the coefficients a & b to achive what I want.

Thank you !

Answer 1

[Mathieu NOE]

hello

see demo below :

I added some noise on y3 to see the robustness of fminsearch

got the results :

a = 3.0379

b = 5.0149

which is pretty close to the input equation coefficients

    y1 = rand(1,20);
    y2 = 3*rand(1,20);
    noise = 0.1*rand(1,20);
    y3 = 3*y1+5*y2+noise;
    
% equation to minimize : y3-(a*y1+b*y2) %a = x(1), b=x(2)

% fminsearch optimization loop 
fun = @(x)norm(y3-(x(1)*y1+x(2)*y2));  %a = x(1), b=x(2)
x0 = [0, 0];
X = fminsearch(fun,x0);

a = X(1)
b = X(2) 

[Xiu Wen Kang]

Thank you Mathieu ! I tried your script and it works just fine, I manage to get the coefficients which generate a good spectrum. Another quick question, is it possible to add a upper bound and a lower bound for the coefficients optimisation?

[Mathieu NOE]

hello

I believe there is a variant of fminsearch with bounds available on the FEX , but I just dsicover it , never usd it yet :

fminsearchbnd, fminsearchcon - File Exchange - MATLAB Central (mathworks.com)

[Xiu Wen Kang]

ok, I will check it out :)