Question

I have a system with 4 sensors (say $s_1..s_4$) which I want to combine into a single signal.

I have logged the 4 outputs as well as a "control" sensor ($s_c$) which has the desired ouput signal. Since $s_1..s_4$ should contain enough information to recreate the control-signal, I now try to find a (non-linear) function to describe their relationship:$f(s_1,s_2,s_3,s_4)= s_c$ (4 independent variables, 1 dependent).

I know that there are tools available for $f(x)=y$ (2D) and $f(x,y)=z$ (3D) like curve- and surface-fitting of Matlab. Downside of those is that you already need a general idea of the function to fit to, but using (high order) fourier series often give good representations.

I thought regression analysis (least-squares) might be helpful, but then a column of additional data is required for every possible term in the function. This leads to a vast amount of data, a fourier series (sin and cos for each variable) effectively would mean $2^8$ possible combinations (terms) for just the 1st order.

Long story short: I can't seem to find a tool nor a mathematical method for "5D" fitting. Is there a way to achieve this?

Any help pointing me in the right direction is very much appreciated.

Answer 1

You might try radial basis function interpolation. RBFs have the advantage of having the same form in any dimension.

Let's say you have $ N $ sample points $ s^j = (s_1,s_2,s_3,s_4)$, and at each one you know from your measurements that $ f(s^j) = s_c^j $.

Assume \begin{equation} f(s) = \sum_{j=1}^N c_j \phi(|s-s^j|) \end{equation} where $ \phi $ is a radial basis function.

Then you can find the coefficients $ c_j $ by solving the matrix system $ Ac = s_c $ where $ A_{ij} = \phi(|s^i - s^j|) $, that is, you enforce that the approximation goes through all the sample points.

There are many choices for the RBF but standard ones are the Gaussian $ \phi(r) = e^{(-\epsilon r)^2}$ or the multiquadric $ \phi(r) = \sqrt{1 + (\epsilon r)^2} $, where $ \epsilon $ is a parameter you get to choose based (roughly) on how closely your points are spaced.

Answer 2

Also you could try Thin plate spline (TPS) approximation. Matlab/Octave code: http://www.cs.rtu.lv/jekabsons/Files/RBF.zip