Useful link: Kernel function

Wednesday, July 10, 2013

Kernel function

Source from wiki:

http://en.wikipedia.org/wiki/Kernel_(statistics)

Kernel functions in common use[edit]

Several types of kernel functions are commonly used: uniform, triangle, Epanechnikov, quartic (biweight), tricube, triweight, Gaussian, and cosine.

In the table below, 1_{…} is the indicator function.

Kernel Functions, K(u)			$\textstyle \int u^2K(u)du$	$\textstyle \int K^2(u)du$
Uniform	$K(u) = \frac12 \,\mathbf{1}_{\{\|u\|\leq1\}}$		$\frac13$	$\frac12$
Triangular	$K(u) = (1-\|u\|) \,\mathbf{1}_{\{\|u\|\leq1\}}$		$\frac{1}{6}$	$\frac{2}{3}$
Epanechnikov	$K(u) = \frac{3}{4}(1-u^2) \,\mathbf{1}_{\{\|u\|\leq1\}}$		$\frac{1}{5}$	$\frac{3}{5}$
Quartic (biweight)	$K(u) = \frac{15}{16}(1-u^2)^2 \,\mathbf{1}_{\{\|u\|\leq1\}}$		$\frac{1}{7}$	$\frac{5}{7}$
Triweight	$K(u) = \frac{35}{32}(1-u^2)^3 \,\mathbf{1}_{\{\|u\|\leq1\}}$		$\frac{1}{9}$	$\frac{350}{429}$
Tricube	$K(u) = \frac{70}{81}(1- {\left\| u \right\|}^3)^3 \,\mathbf{1}_{\{\|u\|\leq1\}}$		$\frac{35}{243}$	$\frac{175}{247}$
Gaussian	$K(u) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}u^2}$		$1\,$	$\frac{1}{2\sqrt\pi}$
Cosine	$K(u) = \frac{\pi}{4}\cos\left(\frac{\pi}{2}u\right) \mathbf{1}_{\{\|u\|\leq1\}}$		$1-\frac{8}{\pi^2}$	$\frac{\pi^2}{16}$

All of the above Kernels in a Common Coordinate System[edit]

All of the above kernels in a common coordinate system

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