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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.

Buddhini Angelika
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  • Please delete this earlier version of your queston. https://math.stackexchange.com/questions/2583244/will-polynomial-interpolation-give-a-polynomial-satisfying-those-points-only – Ethan Bolker Dec 31 '17 at 18:26

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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.

Ethan Bolker
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  • Thanks. But that doesn't meet my requirement. Any other way? – Buddhini Angelika Jan 01 '18 at 07:07
  • No way that occurs to me. This does do what you asked for - if it doesn't work you should edit the question to clarify the requirements. A small example with a satisfactory hand built model/interpolation might help. So might a discussion of why you need this. – Ethan Bolker Jan 01 '18 at 12:58