Kernel (statistics)

The term kernel is used in statistical analysis to refer to a window function.

The term "kernel" has several distinct meanings in different branches of statistics.

[1] Note that such factors may well be functions of the parameters of the pdf or pmf.

In addition, in Bayesian analysis of conjugate prior distributions, the normalization factors are generally ignored during the calculations, and only the kernel considered.

At the end, the form of the kernel is examined, and if it matches a known distribution, the normalization factor can be reinstated.

For many distributions, the kernel can be written in closed form, but not the normalization constant.

Its probability density function is and the associated kernel is Note that the factor in front of the exponential has been omitted, even though it contains the parameter

In nonparametric statistics, a kernel is a weighting function used in non-parametric estimation techniques.

Kernels are also used in time series, in the use of the periodogram to estimate the spectral density where they are known as window functions.

An additional use is in the estimation of a time-varying intensity for a point process where window functions (kernels) are convolved with time-series data.

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

A kernel is a non-negative real-valued integrable function K. For most applications, it is desirable to define the function to satisfy two additional requirements: 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) = λK(λu), where λ > 0.

Several types of kernel functions are commonly used: uniform, triangle, Epanechnikov,[2] quartic (biweight), tricube,[3] triweight, Gaussian, quadratic[4] and cosine.