Algorithm 1.

Loess Regression

Let X = {x1, x2,…, xp} be a set of p points (or observations), and let Y = {y1, y2, …, yp} be the corresponding data values. for i=1 to p do Let xi be the point to be fitted on the loess curve. Let D(·) be an appropriate distance metric. For each point xj ∈ X, i≠j, define the distance dj = D(xi,xj). Let d(1}, d(2), …, d(p) be the ordered set of distances. Let k be the number of neighbors of xi to be used in the local regression. The value of k is given as a percentage of the number of observations and is called the span of the regression. Define N(xi), the neighborhood of xi, as the set of points closest to xi using the distance metric D: xj ∈ N(xi) if d(j)<k Define Δ, the maximum distance in the span, as $$\mathrm{\Delta }=\underset{x_j\in N(x_i)}{max}\mid x_i-x_j\mid$$ Assign weights wj to each of the points xj $$w_j=W\left(\frac{\mid x_i-x_j\mid }{\mathrm{\Delta }}\right)$$ where W(·) is the tri-cube function $$W(u)=\{\begin{array}{ll}{\left(1-u^3\right)}^3,\hfill & 0\le u<1,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$$ Calculate the weighted least squares fit of Y on the neighborhood of X using the weights w. Let s(·), the smooth function, be given by s(xi) = ŷj at the point xi end for

Source: Adapted from Cleveland and Devlin (1988)