Kernel Smoother - Local Polynomial Regression

Local Polynomial Regression

Instead of fitting locally linear functions, one can fit polynomial functions.

For p=1, one should minimize:

with

In general case (p>1), one should minimize:

\begin{align} & \hat{\beta }(X_{0})=\underset{\beta (X_{0})}{\mathop{\arg \min }}\,\sum\limits_{i=1}^{N}{K_{h_{\lambda }}(X_{0},X_{i})\left( Y(X_{i})-b(X_{i})^{T}\beta (X_{0}) \right)}^{2} \\ & b(X)=\left( \begin{matrix} 1, & X_{1}, & X_{2},... & X_{1}^{2}, & X_{2}^{2},... & X_{1}X_{2}\,\,\,... \\ \end{matrix} \right) \\ & \hat{Y}(X_{0})=b(X_{0})^{T}\hat{\beta }(X_{0}) \\ \end{align}

Read more about this topic:  Kernel Smoother

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