What is the equation for multivariate kernel density estimation techniques?

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I was reading about non-parametric kernel density estimation. http://en.wikipedia.org/wiki/Kernel_density_estimation

For uni-variate where D = 1, we can write like

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For Multivariate Kernel density estimation (KDE), more preciously for d=3 and X = (x,y,z) can we write:

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Is this technically correct? Can any one help with this?

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Ami Tavory On

This is very difficult to do on your own, and you really should do this through some package. Nevertheless, the definition is:

fH(x)= 1 / n \sum{i=1}n KH (x - xi), where

  • x = (x1, x2, …, xd)T, xi = (xi1, xi2, …, xid)T, i = 1, 2, …, n are d-vectors;

  • H is the bandwidth (or smoothing) d×d matrix which is symmetric and positive definite;

  • K is the kernel function which is a symmetric multivariate density;

  • KH(x) = |H|−1/n K(H−1/2x).