Instead of fitting a high-degree polynomial over the entire range of $X$, piecewise polynomial regression involves fitting separate low-degree polynomials over different regions of $X$. For example, a piecewise cubic polynomial works by fitting a cubic regression model of the form
$$y_{i}=\beta_{0}+\beta_{1} x_{i}+\beta_{2} x_{i}^{2}+\beta_{3} x_{i}^{3}+\epsilon_{i}$$

where the coefficients $\beta_{0}, \beta_{1}, \beta_{2}$, and $\beta_{3}$ differ in different parts of the range of $X$. The points where the coefficients change are called knots.

For example, a piecewise cubic with no knots is just a standard cubic polynomial, as in $(7.1)$ with $d=3$. A piecewise cubic polynomial with a single knot at a point $c$ takes the form
$$y_{i}= \begin{cases}\beta_{01}+\beta_{11} x_{i}+\beta_{21} x_{i}^{2}+\beta_{31} x_{i}^{3}+\epsilon_{i} & \text { if } x_{i}<c \ \beta_{02}+\beta_{12} x_{i}+\beta_{22} x_{i}^{2}+\beta_{32} x_{i}^{3}+\epsilon_{i} & \text { if } x_{i} \geq c\end{cases}$$

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