Suppose the following dataset $\mathcal{D}=\{z_1,z_2,...,z_{80}\}$, was obtained by taking measurements on a natural phenomenon that follows the law: $z=f(x,y)=ax^2+by+c$, where $a,b,c \in \mathbb{R}$.
$\mathcal{D}=\{-19,28,49,34,28,26,19,4,-11,64,29,-9,13,38,14,33,3,54,-6,48,-14,-24,91,24,34,101,51,91,-16,53,24,13,-12,28,81,-7,61,8,-16,24,3,71,54,43,3,21,8,64,18,18,101,16,71,96,51,61,1,-6,-7,-4,8,-11,38,3,49,24,33,19,-2,13,-1,96,76,44,9,74,53,86,18,6 \} $
Edit: The values of x and y can only be integers.
In advance I know what the model that governs the phenomenon is, however I do not know the constants (weights) involved. We can assume that the error obtained when taking these data is practically zero.
I would like to find the values of the constants (weights) from these data.
Clearly I can't use techniques like linear regression, because my data is not in the form $\mathcal{D}=\{\{x_1,y_1,z_1\},...,\{x_{80},y_{80},z_{80}\}\}$, which is necessary to fit my weights.
My questions are:
- What is this type of problem called?
- What kind of techniques can be used to solve a problem like this?
I have tried to generate data artificially so that later I can use linear regression, but I don't get the results I want.
P.D. the true model is governed by the equation: $z=f(x,y)=3x^2-5y+1$
Thanks everyone for your help