Local polynomial regression

Local polynomial regression fits a low-degree polynomial by weighted least squares near each target point.

The local polynomial Gram matrix is the weighted matrix of inner products of the local monomial basis. Its invertibility makes the local fit unique.

Effective kernel weights are the coefficients $W_i(x_0)$ for which a local polynomial estimate can be written $\hat{m}(x_0)={\sum }_i{W}_i(x_0)Y_i$.

Local polynomial weights of degree $p$ reproduce every multivariate polynomial of total degree at most $p$: applying the estimator to its design values returns its value at the target point.