Clarification of the rleationship between two concepts

Question

In my studies I had contact with two concepts. The first is polynomial regression. The second concept could be called reconstruction of a function, which also should be treated in school. For the reconstruction of a function you have mostly a set of points $S:=\left\{(x_1|y_1),...,(x_n|y_n)\right\}$ ($x_i \wedge y_i \in \mathbb{R}$) which are believed to be part of the graph of a function $f$. In my case a ploynomial function $f$. With the approach of solving the system

$$ \begin{aligned} f(x_1) &=& y_{1} \\ f(x_2)&=& y_{2} \\ & & & \vdots & \\ f(x_n) &=& y_{n} \end{aligned} $$

one might get the needed coefficients of the polynomial function.

Now my question is the following one:

Is the reconstruction of a function something like a special case of polynomial regression?

Answer 1

I think the solution of interpolation (your second concept) and the solution of regression are different. Interpolation and Regression are two different curve fitting method

Concept

1. Curve Fitting

   • Often have data, $y$, that is a function of some independent variable, $x$, but the underlying relationship is unknown

     • Know $y$’s and $x$’s (perhaps only approximately), but don’t know $f(x)$

   • Determine a function (i.e., a curve) that “best” describes relationship between $x$ and $y$

     • An approximation to (the unknown) $f(x)$

     • This is Curve Fitting

2. Regression vs. Interpolation

   We usually have two categories of curve fitting:

   • Least‐squares regression

     • Noisy data – uncertainty in $y$ value for a given $x$ value

     • Want “good” agreement between $f(x)$ and data points

       • Curve (i.e.$f(x)$) may not pass through any data points

   • Polynomial interpolation

     • Data points are known exactly – noiseless data

     • Resulting curve passes through all data points