Definitions

Kernel (statistics)

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.

Definition

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

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

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−1u), where λ > 0. This can be used to select a scale that is appropriate for the data.

Kernel functions in common use

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

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

Uniform

$K\left(u\right) = frac\left\{1\right\}\left\{2\right\} 1_\left\{\left(|u|leq1\right)\right\}$

Triangle

$K\left(u\right) = \left(1-|u|\right) 1_\left\{\left(|u|leq1\right)\right\}$

Epanechnikov

$K\left(u\right) = frac\left\{3\right\}\left\{4\right\}\left(1-u^2\right) 1_\left\{\left(|u|leq1\right)\right\}$

Quartic

$K\left(u\right) = frac\left\{15\right\}\left\{16\right\}\left(1-u^2\right)^2 1_\left\{\left(|u|leq1\right)\right\}$

Triweight

$K\left(u\right) = frac\left\{35\right\}\left\{32\right\}\left(1-u^2\right)^3 1_\left\{\left(|u|leq1\right)\right\}$

Gaussian

$K\left(u\right) = frac\left\{1\right\}\left\{sqrt\left\{2pi\right\}\right\}e^\left\{-frac\left\{1\right\}\left\{2\right\}u^2\right\}$

Cosine

$K\left(u\right) = frac\left\{pi\right\}\left\{4\right\}cosleft\left(frac\left\{pi\right\}\left\{2\right\}uright\right)1_\left\{\left(|u|leq1\right)\right\}$