Suppose I have a discrete data set. Let the data set be points from $\mathbb{Z}_p \times \mathbb{Z}_p$ represented by $\{(\bar{x_1},\bar{y_1}),(\bar{x_2},\bar{y_2}),\cdots,(\bar{x_n},\bar{y_n})\}$ where p is a prime.
Is there a way to obtain a polynomial interpolation or a model fitting or any kind of a relationship to some randomly selected points from this discrete set. As an example suppose I have randomly selected 3 points ${(\bar{x_2},\bar{y_2}),(\bar{x_5},\bar{y_5}),(\bar{x_8},\bar{y_8})}$.
The polynomial interpolation or what ever the relationship obtained should satisfy the randomly selected points only and there should be no chance that it will get satisfied by any other point in the data set.
Thanks a lot. I highly appreciate any help.
You could change the $y$ values in the other (nonselected) points (randomly or any other way) and then fit a polynomial of degree $n$ to all the points.