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:
*$$int_{-infty}^{+infty}K(u)du = 1,;
*$$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⁻¹"K"(λ⁻¹"u"), 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{2pie^{-frac{1}{2}u^2}

Cosine

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

ee also

*Kernel density estimation
*Kernel smoother
*Stochastic kernel
*Density estimation

External links

* [http://people.revoledu.com/kardi/tutorial/Regression/KernelRegression/Kernel.htm Kernel Basis function] (with graphs).