Kernel (statistics)

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A kernel is a weighting function used in non-parametric estimation techniques. Kernels are used in kernel density estimation to estimate random variables' density functions, or in kernel regression to estimate the conditional expectation of a random variable. Kernels are also used in time-series, in the use of the periodogram to estimate the spectral density. An additional use is in the estimation of a time-varying intensity for a point process.

Commonly, kernel widths must also be specified when running a non-parametric estimation.

[edit] Definition

A kernel is a non-negative real-valued integrable function K satisfying the following two requirements:

$\int_{-\infty}^{+\infty}K(u)\,du = 1\,;$
$K(-u) = K(u) \mbox{ for all values of } u\,.$

The first requirement ensures that the method of kernel density estimation results in a probability density function. The second requirement ensures that the average of the corresponding distribution is equal to that of the sample used.

If K is a kernel, then so is the function K* defined by K*(u) = λ⁻¹K(λu), where λ > 0. This can be used to select a scale that is appropriate for the data.

[edit] Kernel functions in common use

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

Below, the notation $1_{(p)}\,\!$ denotes the value 1 when p holds, and 0 when p is false.

Kernel Functions
Uniform	$K(u) = \frac{1}{2}\ 1_{(\|u\|\leq1)}$
Triangle	$K(u) = (1-\|u\|)\ 1_{(\|u\|\leq1)}$
Epanechnikov	$K(u) = \frac{3}{4}(1-u^2)\ 1_{(\|u\|\leq1)}$
Quartic	$K(u) = \frac{15}{16}(1-u^2)^2\ 1_{(\|u\|\leq1)}$
Triweight	$K(u) = \frac{35}{32}(1-u^2)^3\ 1_{(\|u\|\leq1)}$
Gaussian	$K(u) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}u^2}$
Cosine	$K(u) = \frac{\pi}{4}\cos\left(\frac{\pi}{2}u\right)1_{(\|u\|\leq1)}$