Approximation: Nadaraya Watson kernel regression

We calculate

where

for

Here, u is adjusted by variance of x. That is,

u=u/sqrt(Var(x)) or u=u*inv(chol(vcx(x))) in multivariate case.

where we use various types of kernels.

Uniform:

K(u)=0.5*|u| for –1<=u<=1

K(u)=0 otherwise

Gaussian:

K(u)= (1/sqrt(2*pi))*exp(-0.5*u^2)

Triangular:

K(u)=1-|u| for –1<=u<=1

K(u)=0 otherwise

Biweight:

K(u)=(15/16)*((1-u^2)^2) for –1<=u<=1

K(u)=0 otherwise

Triweight:

K(u)=(35/32)*((1-u^2)^3) for –1<=u<=1

K(u)=0 otherwise

Epanechnikov:

K(u)=0.75*(1-u^2) for –1<=u<=1

K(u)=0 otherwise

Cosine:

K(u)= pi/4*cos(pi/2*u) for –1<=u<=1

K(u)=0 otherwise