solve multidimensional equation using least square method in matlab

Question

How do I get the coefficient a and b from this equation using least square method? What is the best way to solve this?

Let's say θ(k1,k2) is a matrix of 60x60 (constant/values), that is theta=rand(60,60) , but

How do I solve for a and b in matlab? Any easy function to do it?

Thanks in advance!

Reference paper: Here (section III)

Answer 1

You can use the regress function to do this. Here is an example:

% Generate an example
n = 60;
theta = rand(n);

% Create regressors
[M,N] = meshgrid(1:n,1:n);
X = [M(:), N(:)];

% Regress
B=regress(theta(:), X);

% Compare the results
theta_hat = reshape(X*B,n,n);
plot3(M,N,theta,'o');
hold on;
surf(M,N,theta_hat);

Notice that the regression is done on theta(:) which is a (3600,1) vector containing the values of theta(k1,k2) uses the corresponding coordinates in X which is (3600,2). The first column of X is k1, the second is k2.

The result of calling regress gives you B=[a;b] the coefficients that best fit the data in theta.

One final note is that the least squares could be solved directly using

B=inv(X'*X)*X'*theta(:)

which should give the same result, but regress is the preferred MATLAB method.