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Encyclopedia > Kernel density estimation

In statistics, the Parzen window method (or kernel density estimation), named after Emanuel Parzen, is a way of estimating the probability density function of a random variable. As an illustration, given some data about a sample of a population, the Parzen window method makes it possible to extrapolate the data to the entire population. A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Emanuel Parzen (born April 21, 1929 in New York City) is an American statistician. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... A random variable is a term used in mathematics and statistics. ... In mathematics, extrapolation is a type of interpolation. ...


If x1, x2, ..., xN is a sample of a random variable, then the Parzen window approximation of its probability density function is A sample is that part of a population which is actually observed. ...

where W is some kernel, i.e., some probability density function. Quite often W is taken to be a Gaussian function with mean zero and variance σ2: Gaussian curves parameterised for statistics A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c. ... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ...

Six Gaussians (red) and their sum (blue). The Parzen window density estimate ρ(x) is obtained by dividing this sum by 6, the number of Gaussians. The variance of the Gaussians was set to 0.5. Note that where the points are denser the density estimate will have higher values.
Six Gaussians (red) and their sum (blue). The Parzen window density estimate ρ(x) is obtained by dividing this sum by 6, the number of Gaussians. The variance of the Gaussians was set to 0.5. Note that where the points are denser the density estimate will have higher values.

See also

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

References

  • Parzen E. (1962). On estimation of a probability density function and mode, Ann. Math. Stat. 33, pp. 1065-1076.
  • Duda, R. and Hart, P. (1973). Pattern Classification and Scene Analysis. John Wiley & Sons. ISBN 0471223611.

  Results from FactBites:
 
R: Kernel Density Estimation (326 words)
the number of equally spaced points at which the density is to be estimated.
the left and right-most points of the grid at which the density is to be estimated.
This allows the estimated density to drop to approximately zero at the extremes.
KERNEL (400 words)
Kernel estimation is a semi-parametric method for approximating a probability distribution.
When KERNEL is used with two arguments, a Gaussian kernel regression of the first variable on the second is computed; the smoothed values of the dependent variable are stored in @FIT.
Neither estimator is very sensitive to the choice of kernel function, but both are sensitive to the choice of band width h.
  More results at FactBites »

 
 

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