Hi everyone!
I am new to MATLAB and I have the following issue. I have the following data set:
x11 = [0.091, 0.068, 0.086, 0.091, 0.097, 0.064, 0.052, 0.066, 0.074];
x12 = [0.707, 0.920, 0.491, 0.527, 0.656, 0.474, 0.49, 0.267, 0.552];
y = [0.07, 0.029, 0.063, 0.077, 0.155, 0.043, 0.023, 0.057, 0.085];
where x11 and x12 are the measured inputs, and y is the known output.
I need to solve for the unknown coefficients a1, a2 and a3 in the following original equation:
y = a1*(x11^a2)*(x12^a3)
This is a non-linear problem of course, but I do linearize it by taking log on both sides, so that I have:
y2 = log10(y);
x1 = log10(x1);
x2 = log10(x2);
I then perform a linear regression analysis using the pinv function which makes use of a pseudoinverse matrix to find the coefficients.
My current code compiled based on the information I found on the Internet is following:
clear all close all
x11 = [0.091, 0.068, 0.086, 0.091, 0.097, 0.064, 0.052, 0.066, 0.074];
x12 = [0.707, 0.920, 0.491, 0.527, 0.656, 0.474, 0.49, 0.267, 0.552];
y = [0.07, 0.029, 0.063, 0.077, 0.155, 0.043, 0.023, 0.057, 0.085];
x1=log10(x11);
x2=log10(x12);
y2=log10(y);
n=length(x1);
a=[ones(n,1), x1', x2'];
c=pinv(a)*y2';
a1=c(1,:);
a2=c(2,:);
a3=c(3,:);
%here I take antilog to get back the original power relations:
y_pred = 10^(log10(a1)).*(x11.^a2).*(x12.^a3);
Although I do get a solution for all of the coefficients I am having two issues:
- I am not really satisfied with the output (poor R^2 coefficient)
- I want to make some constraints so that all of the coefficient stay positive, for example: using lsqlin didn't help much as the solution the is highly biased by my initial suggestions and the coefficient values either take the upper or lower bound value, which seems not to be correct.
I obtained a better solution using fminsearch approach without pre-linearizing the equation, but was curious if I can get comparable results taking the linear way.
I would be grateful for your help!
