Basis function regression for 3 parameters

I am currently working on finding the coefficients for a linear regression model, given its dependence on basis functions. The model is y(x1, x2, x3) = a_0 f_0(x1, x2, x3) + a_1 f_1(x1, x2, x3) + a_2 f_2(x1, x2, x3).... up to a_10 f_10(x1, x2, x3). The functions are, for example, f_0 = 1, f_1 = sin(x1*x2), f_2 = x1*x3^2 and so on.

Given this, I am aware that for a bivariate regression model y(x1, x2) , the matrices would look something like the attached picture. Then we can simply solve for X given the model is a linear system Ax = B

My question is, is there an analogous set of matrices for 3 predictors? I would imagine this would make A an (11x11x11) matrix, and B would still be (11x1) matrix. But when I construct these matrices in MATLAB and try to solve the system using the expression

X = A\B;

it informs me the matrices must be 2 dimensional.

Answer 1

[Torsten]

Put the equations

y(x1(i),x2(j),x3(k)) = a_0*f_0(x1(i), x2(j), x3(k)) + a_1*f_1(x1(i), x2(j), x3(k)) + a_2*f_2(x1(i), x2(j), x3(k)) + ... + a_10*f_10(x1(i), x2(j), x3(k))

in matrix form

Y = F*a

and solve for a as

a = F\Y

[Yash Bhatia]

I don't see how I can decompose y(x1, x2, x3) into matrices F and a. Can you go more into detail on that?

[Torsten]

Write

y(x1(i),x2(j),x3(k)) = a_0*f_0(x1(i), x2(j), x3(k)) + a_1*f_1(x1(i), x2(j), x3(k)) + a_2*f_2(x1(i), x2(j), x3(k)) + ... + a_10*f_10(x1(i), x2(j), x3(k))

as

y(x1(i),x2(j),x3(k)) = [f_0(x1(i), x2(j), x3(k)) f_1(x1(i), x2(j), x3(k)) f_2(x1(i), x2(j), x3(k)) ...f_10(x1(i), x2(j), x3(k))]*[a_0 ;a_1 ;a_2 ;...;a_10]   

Can you see what the (ijk)-th row of F and the (ijk)-th entry of Y is ?