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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 [[extrapolation|extrapolate]] the data to the entire population.
If ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''N''</sub> is a [[statistical sample|sample]] of a random variable, then the Parzen window approximation of its probability density function is
:<math>\rho(x)=\frac{1}{N}\sum_{i=1}^N W(x-x_i)</math>
where ''W'' is some kernel. Quite often ''W'' is taken to be a [[Gaussian function]] with mean zero and [[variance]] σ<sup>2</sup>:
:<math>W(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-{x^2 / 2\sigma^2}}.</math>
[[Image: Parzen_window_illustration.png|frame|center|The Parzen window density estimate ρ(''x'') is in blue; the Gaussians which add up to ρ(''x'') are in red. Six sample points were considered. The variance of the Gaussians was set to 0.5. Note that where the points are denser, the density estimate has higher values.</sup>]]
==See also==
*[[Density estimation]]
==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.
==External links==
*[http://mathworld.wolfram.com/ParzenWindow.html Parzen Window -- from MathWorld]
[[Category:Probability and statistics]]
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