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Kernel (statistics)
 
 
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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 statistics way of Density estimation the probability density function of a random variable....
 to estimate random variable
Random variable

In mathematics, random variables are used in the study of Randomness and probability. They were developed to assist in the analysis of Game of chance, stochastic events, and the results of experiment by capturing only the mathematical properties necessary to answer probability questions....
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 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 Hz, or energy per Hz....
. An additional use is in the estimation of a time-varying intensity for a point process
Point process

In mathematics, a point process is a random element whose values are "point patterns" on a Set S. While in the exact mathematical definition a point pattern is specified as a locally finite counting Measure , it is sufficient for more applied purposes to think of a point pattern as a countable set subset of S that has no limit points...
.

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

rnel 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 mathematics, a probability density function is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density ƒ, if ƒ is a non-negative Lebesgue integration function such that the probability of the interval [a, b] is given by...
.






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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 statistics way of Density estimation the probability density function of a random variable....
 to estimate random variable
Random variable

In mathematics, random variables are used in the study of Randomness and probability. They were developed to assist in the analysis of Game of chance, stochastic events, and the results of experiment by capturing only the mathematical properties necessary to answer probability questions....
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 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 Hz, or energy per Hz....
. An additional use is in the estimation of a time-varying intensity for a point process
Point process

In mathematics, a point process is a random element whose values are "point patterns" on a Set S. While in the exact mathematical definition a point pattern is specified as a locally finite counting Measure , it is sufficient for more applied purposes to think of a point pattern as a countable set subset of S that has no limit points...
.

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 mathematics, a probability density function is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density ƒ, if ƒ is a non-negative Lebesgue integration function such that the probability of the interval [a, b] is given by...
. 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) = ?-1K(?-1u), where ? > 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.

Below, the notation denotes the value 1 when p holds, and 0 when p is false.

Kernel Functions
Uniform
Triangle
Epanechnikov
Quartic
Triweight
Gaussian
Normal distribution

The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields....
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 statistics way of Density estimation the probability density function of a random variable....
  • Kernel smoother
    Kernel smoother

    A kernel smoother is a statistics technique for estimating a real valued function by using its noisy observations, when non-parametric statistics 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

    In probability and statistics,density estimation is the construction of an estimate, based on observed data, of an unobservable underlying probability density function....


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