Section 2.4 Higher Degree Regression
Continuing in a similar fashion to the previous section, consider now an approximation using a quadratic function \(f(x) = ax^2 + bx + c\text{.}\) In this case, the total squared error would be of the form
\begin{equation*}
TSE(a,b,c) = \sum_{k=0}^n (a x_k^2 + b x_k + c - y_k)^2.
\end{equation*}
Taking all three partials gives
\begin{equation*}
TSE_a = \sum_{k=1}^n 2(a x_k^2 + b x_k + c - y_k) \cdot x_k^2
\end{equation*}
\begin{equation*}
TSE_b = \sum_{k=1}^n 2(a x_k^2 + b x_k + c - y_k) \cdot x_k
\end{equation*}
\begin{equation*}
TSE_c = \sum_{k=1}^n 2(a x_k^2 + b x_k + c - y_k) \cdot 1 .
\end{equation*}
Once again, setting equal to zero and solving gives the normal equations for the best-fit quadratic
\begin{equation*}
a \sum_{k=1}^n x_k^4 + b \sum_{k=1}^n x_k^3 + c \sum_{k=1}^n x_k^2 = \sum_{k=1}^n x_k^2 y_k
\end{equation*}
\begin{equation*}
a \sum_{k=1}^n x_k^3 + b \sum_{k=1}^n x_k^2 + c \sum_{k=1}^n x_k = \sum_{k=1}^n x_k y_k
\end{equation*}
\begin{equation*}
a \sum_{k=1}^n x_k^2 + b \sum_{k=1}^n x_k + c \sum_{k=1}^n 1 = \sum_{k=1}^n y_k.
\end{equation*}
One can easily use software to perform these calculations of course. Further, you can extend the ideas from above and derivate formulas for a best-fit cubic. The interactive cell below determines the optimal quadratic polynomial for a given set of data points and by commenting and uncommenting as suggested will also determine the best fit cubic.