Kernel density estimation: Difference between revisions

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The '''Parzen window''' method is a way of estimating the [[probability density function]] of a [[random variable]].

If ''x''₁, ''x''₂, ..., ''x''_N 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 [[normal distribution#Probability densityGaussian function|Gaussian]] with mean zero. and [[variance]] &sigma;²:

:<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 &rho;(''x'') (is in blue); the Gaussians inwhich theadd sumup to &rho;(red''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==