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Concept
Variable kernel density estimation
Summary
In statistics, adaptive or "variable-bandwidth" kernel density estimation is a form of kernel density estimation in which the size of the kernels used in the estimate are varied
depending upon either the location of the samples or the location of the test point.
It is a particularly effective technique when the sample space is multi-dimensional.
Rationale
Given a set of samples, \lbrace \vec x_i \rbrace, we wish to estimate the
density, P(\vec x), at a test point, \vec x:
:
P(\vec x) \approx \frac{W}{n h^D}
:
W = \sum_{i=1}^n w_i
:
w_i = K \left ( \frac{\vec x - \vec x_i}{h} \right )
where n is the number of samples, K is the
"kernel", h is its width and D is the number of dimensions in \vec x.
The kernel can be thought of as a simple, linear filter.
Using a fixed filter width may mean that in regions of low density, all samples
will fall in the tails of the filter with v
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