Approximation: LOESS approximation

LOESS is a kind of linear weighted least square regression with tricube kernel.

That is,

where

All its offdiagonal terms are zeros while its diagonals are calculated by

for 0<=u<=1

otherwise

Here,

We calculate h form k-nearest neighbor distance. This k is usually set by

k=floor(*n)

given parameter alpha.

We modify the result of gird method in the following way:

where red points are the results from grid method.

As we increase the parameter of LOESS, the resulting curve tends to be dull.

As we decrease the parameter of LOESS, the resulting curve tends to fit to the data but

it may be too noisy when the parameter is smaller than or around 0.1.