https://towardsdatascience.com/loess-373d43b03564

University of Michigan - Ann Arbor

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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 at $$x_0$$ is then found using the new regression model.

Local Regression

1. Gather the fraction s = $$\frac{k}{n}$$ of training points whose $$X_i$$ are closest to $$X_0$$.
2. Assign a weight $$K_{i0}=K(x_i,x_0)$$ to each point in this neighborhood, so that the point furthest from $$X_0$$ has weight zero, and the closest has highest weight. All but these k nearest neighbors get weight zero.
3. Fit a weighted least squares regression of the $$y_i$$ on the $$x_i$$ using the aforementioned weights, by finding $$\hat{\beta _0}$$ and $$\hat{\beta _1}$$ that minimize $$\sum^n _{i=1} K_{i0}(y_i-\beta_0-\beta_1 x_i)^2$$.
4. The fitted value at $$x_0$$ is given by $$\hat{f(x_0)}=\hat{\beta_0}+\hat{\beta_1}x_0$$.

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