Parzen window

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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 extrapolate the data to the entire population.

If x1, x2, ..., xN ∈ R is a sample of a random variable, then the Parzen window approximation of its probability density function is

\rho(x)=\frac{1}{N}\sum_{i=1}^N W(x-x_i)

where W is some stochastic kernel, i.e., some probability density function. Quite often W is taken to be a Gaussian function with mean zero and variance σ2:

W(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-{x^2 / 2\sigma^2}}.
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.

[edit] See also

[edit] 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 0-471-22361-1.

[edit] External links

  • Free Online Software (Calculator) computes the Kernel Density Estimation for any data series according to the following Kernels: Gaussian, Epanechnikov, Rectangular, Triangular, Biweight, Cosine, and Optcosine.
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