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:

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

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

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

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

Cosine

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

See also

• Kernel density estimation
• Kernel smoother
• Stochastic kernel
• Density estimation

External links

• Kernel Basis function (with graphs).