A kernel is a weighting function used in non-parametric estimation techniques. Kernels are used in kernel density estimation

Kernel density estimation

In statistics, kernel density estimation is a non-parametric way of estimating the probability density function of a random variable. Kernel density estimation is a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample...

 to estimate random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s' density functions, or in kernel regression

Kernel regression

The kernel regression is a non-parametric technique in statistics to estimate the conditional expectation of a random variable. The objective is to find a non-linear relation between a pair of random variables X and Y....

 to estimate the conditional expectation

Conditional expectation

In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution....

 of a random variable. Kernels are also used in time-series, in the use of the periodogram

Periodogram

The periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898 as in the following quote:...

 to estimate the spectral density

Spectral density

In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...

. An additional use is in the estimation of a time-varying intensity for a point process

Point process

In statistics and probability theory, a point process is a type of random process for which any one realisation consists of a set of isolated points either in time or geographical space, or in even more general spaces...

.

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:

• 
• 

The first requirement ensures that the method of kernel density estimation results in a probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

. 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 kernel functions are commonly used: uniform, triangle, Epanechnikov, quartic (biweight), tricube, triweight, Gaussian, and cosine.

In the table below, 1_{…} is the indicator function.
{| class="wikitable" style="background-color:white;text-align:left"
!colspan=3| Kernel Functions, K(u)
!
!
|-
! Uniform
|
|
|  
|  
|-
! Triangular
|
|
|  
|  
|-
! Epanechnikov
|
|
|  
|  
|-
! Quartic
(biweight)
|
|
|  
|  
|-
! Triweight
|
|
|  
|  
|-
! Tricube
|
|
|  
|  
|-
! Gaussian
|
|
|  
|  
|-
! Cosine
|
|
|  
|  
|}

All of the above Kernels in a Common Coordinate System

See also

• Kernel density estimation
  Kernel density estimation
  In statistics, kernel density estimation is a non-parametric way of estimating the probability density function of a random variable. Kernel density estimation is a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample...
  
  
• Kernel smoother
  Kernel smoother
  A kernel smoother is a statistical technique for estimating a real valued function f\,\,\left by using its noisy observations, when no parametric model for this function is known...
  
  
• Stochastic kernel
  Stochastic kernel
  In statistics, a stochastic kernel estimate is an estimate of the transition function of a stochastic process. Often, this is an estimate of the conditional density function obtained using kernel density estimation...
  
  
• Density estimation
  Density estimation
  In probability and statistics,density estimation is the construction of an estimate, based on observed data, of an unobservable underlying probability density function...