quantspec (version 1.0-0)

kernels: Kernel function.

Description

Implementations of kernel functions

Usage

W0(x)

W1(x)

W2(x)

W3(x)

WDaniell(x, a = (pi/2))

Arguments

x

real-valued argument to the function; can be a vector

a

real number between 0 and $\pi$

Details

Daniell kernel function W0: $$\frac{1}{2\pi} I{|x| \leq \pi}.$$ Epanechnikov kernel W1 (i. e., variance minimizing kernel function of order 2): $$\frac{3}{4\pi} (1-\frac{x}{\pi})^2 I{|x| \leq \pi}.$$ Variance minimizing kernel function W2 of order 4: $$\frac{15}{32\pi} (7(x/\pi)^4 -10(x/\pi)^2+3) I{|x| \leq \pi}.$$ Variance minimizing kernel function W3 of order 6: $$\frac{35}{256\pi} (-99(x/\pi)^6 + 189(x/\pi)^4 - 105(x/\pi)^2+15) I{|x| \leq \pi}.$$ Kernel yield by convolution of two Daniell kernels: $$\frac{1}{\pi+a} \Big(1-\frac{|x|-a}{\pi-a} I{a \leq |x| \leq \pi}\Big).$$

Examples

plot(x=seq(-8,8,0.05), y=W0(seq(-8,8,0.05)), type="l")
plot(x=seq(-8,8,0.05), y=W1(seq(-8,8,0.05)), type="l")
plot(x=seq(-8,8,0.05), y=W2(seq(-8,8,0.05)), type="l")
plot(x=seq(-8,8,0.05), y=W3(seq(-8,8,0.05)), type="l")
plot(x=seq(-pi,pi,0.05), y=WDaniell(seq(-pi,pi,0.05),a=(pi/2)), type="l")