Polynomial regression

In the modelling framework saying we are doing polynomial regression is saying that we are picking the modelling paradigm of functions of the form

$$f(x) = c_0 + \sum_{p \in \Delta_n^k}^n c_{p} x^p$$

where

• $x = (x_1, x_2, \ldots, x_n) \in A$,
• $\Delta_k^n = \{(p_1, \ldots, p_n) \in \mathbb{N}^n \vert \sum_{i=1}^n p_i \leq k\}$, and
• $x^p = \prod_{i=1}^n x_i^{p_i}$

we say $k$ is the degree of our polynomial.

If $A$ was 1-dimensional this would be:

$$f(x) = \sum_{i=0}^k c_i x^i.$$