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Let's say that I have a set of $N$ vectors $X_i$ of dimension $n$ :

$$ X_i = [x_1,x_2,...x_n] $$

For each vector $X_i$, I have a corresponding scalar $y_i$.

My aim is to find the function $y=f([x_1,x_2,...,x_n])$ that minimize the quantity :

$$ \chi^2 = \sum_{i=1}^N [y_i-f(X_i)]^2 $$

I have reasons to believe that $f$ is an N dimensional parabola (i.e. a polynomial of order 2 in $x_1,x_2,...,x_n$). Is there a straightforward, mathematical way to deduce this parabola from the $X_i$ and $y_i$ that I have at my disposal ?

Ideally, I would like to express this "best-fit" parabola as : $$ f(X_i) = (X_i - \mu) C^{-1} (X_i - \mu)^T + K $$ with $\mu$ a vector of dimension $n$, C a $n\times n$ symmetric matrix, and $K$ a constant.

Thanks in advance !

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  • $\begingroup$ What hypotheses you put on f. I think the result depends on the class of function you choose $\endgroup$ – Youem Aug 1 '17 at 18:15
  • $\begingroup$ Sorry if t wasn't clear enough : $f$ has to be a second order polynomial, i.e. here a $n$-dimensional parabola. $\endgroup$ – S.I. Aug 1 '17 at 19:38

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