Local linear regression ($\hat{f}(x)={\hat{a}}_x$)

Local linear regression fits $a+b(X_i-x)$ to responses near each target $x$ by weighted least squares, then returns the fitted intercept. With weights $w_i=K((X_i-x)/h)$, put $S_k={\sum }_i{w}_i(X_i-x{)}^k$ and $T_k={\sum }_i{w}_i(X_i-x{)}^k{Y}_i$. If $S_0{S}_2-S_1^2>0$, then $\hat{f}(x)=(S_2{T}_0-S_1{T}_1)/(S_0{S}_2-S_1^2)$. Its polynomial reproduction property of local polynomial regression reproduces affine functions, reducing boundary bias relative to a local constant fit.