Polynomial Regressors
Why
A simple example of an embedding.1
Definition
Fix $d \in \N $.
A polynomial feature map
of degree $d$ is a function $\phi : \R \to
\R ^d$ with
\[
\phi (x) = \pmat{1 & x^2 & \cdots & x^d}^\top .
\]
A polynomial regressor is a least squares linear predictor using a polynomial feature embedding (of any degree, but to be precise one must specify the degree). The task of consructing a linear predictor is often referred to as polynomial regression.
Given a dataset of paired records $(x^1, y^1), \dots , (x^n, y^n) \in \R ^2$, one can construct a predictor $g: \R \to \R $ for $y$ by embedding the dataset $(\phi (x^1), \dots , \phi (x^n))$ and finding the least squares linear regressor $f: \R ^d \to \R $ for $y$. One defines the predictor $g: \R \to \R $ by $g(\phi (x))$.
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