I collected data about the response function, $f$, for each and every combination of three parameters $x$, $y$ and $\theta$, that represent horizontal position, vertical position and orientation, respectively. The data is shown in Figure 1, sliced at discrete values of $\theta=30,60,...,360$.
It can be clearly seen that higher response varies radially as $\theta$ changes. My first attempt to model this 3D response function, I considered the following linear model: $$f(x, y, \theta)=\alpha_0+\alpha_x \cdot x+\alpha_y \cdot y + \alpha_\theta \cdot \theta$$ The result (using standard least squares optimization methods in Matlab) is shown on Figure 2.
After looking at the result I realized that my model (which represents a hyperplane?) is wrong. However, I can't seem to come up with any other justified model. If a multivariate polynomial can be fitted to the data above, how should it look like?
Thanks for your time.
If your data are linear with respect to each parameter, then the equation of the generalized linear model should be $$f(x, y, \theta)=a_0+a_1 x+a_2 y + a_3 \theta+a_4 x y+ a_5 x \theta+a_6 y \theta+a_7 x y\theta$$