Local regression works by choosing a target point ($x_0$) and its nearby training observations in order to find a fit. Points are then assigned weights (given by $K_{i0} = K(x_i, x_0)$) based on their distance from $x_0$; points closest to $x_0$ are assigned the highest weight, and as points get farther from $x_0$, they are assigned a lower weight. The point farthest from $x_0$ will be assigned a weight of 0. A weighted least squares regression is then performed by minimizing $\sum_{i=1}^{n} K_{i0} (y_i - \beta_0- \beta_1x_i)^{2}$. The new fitted value of $x_0$ is then found using the new regression model.