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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.
<ref name="Terrell_Scott1992">{{Cite journal
| author1 = D. G. Terrell
| author2 = D. W. Scott
| title = Variable kernel density estimation
| journal = [[Annals of Statistics]]
| volume = 20
| pages = 1236–1265
| year = 1992
| doi=10.1214/aos/1176348768
}}
</ref>

==Rationale==

Given a set of samples, <math>\lbrace \vec x_i \rbrace</math>, we wish to estimate the
density, <math>P(\vec x)</math>, at a test point, <math>\vec x</math>:

: <math>
P(\vec x) \approx \frac{W}{n h^D}
</math>

: <math>
W = \sum_{i=1}^n w_i
</math>

: <math>
w_i = K \left ( \frac{\vec x - \vec x_i}{h} \right )
</math>

where ''n'' is the number of samples, ''K'' is the
[[Kernel (statistics)|"kernel"]], ''h'' is its width and ''D'' is the number of dimensions in <math>\vec x</math>.
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 very low weighting, while regions of high
density will find an excessive number of samples in the central region with weighting
close to unity. To fix this problem, we vary the width of the kernel in different
regions of the sample space.
There are two methods of doing this: balloon and pointwise estimation.
In a balloon estimator, the kernel width is varied depending on the location
of the test point. In a pointwise estimator, the kernel width is varied depending
on the location of the sample.<ref name="Terrell_Scott1992"/>

For multivariate estimators, the parameter, ''h'', can be generalized to
vary not just the size, but also the shape of the kernel. This more complicated approach
will not be covered here.

==Balloon estimators==

A common method of varying the kernel width is to make it proportional to the density at the test point:

: <math>
h = \frac{k}{\left [ n P(\vec x) \right ]^{1/D}}
</math>

where ''k'' is a constant.
If we back-substitute the estimated PDF,
we can show that ''W'' is a constant:<ref name="Mills2010"/>

: <math>
W = k^D (2 \pi)^{D/2}
</math>

This produces a generalization of the [[k-nearest neighbour algorithm]].
That is, a uniform [[Kernel (statistics)|kernel function]] will return the
KNN technique.<ref name="Mills2010">{{Cite journal
| last = Mills
| first = Peter
| title = Efficient statistical classification of satellite measurements
| year = 2011
| volume = 32
| issue = 21
| journal = International Journal of Remote Sensing
| doi = 10.1080/01431161.2010.507795
| url = http://peteysoft.users.sourceforge.net/TRES_A_507795.pdf
}}</ref>

There are two components to the error: a variance term and a bias term. The variance term is given as:<ref name="Terrell_Scott1992"/>

: <math>
e_1 = \frac{P \int K^2}{n h^D}
</math>.

The bias term is found by evaluating the approximated function in the limit as the kernel
width becomes much larger than the sample spacing. By using a Taylor expansion for the real function, the bias term drops out:

: <math>
e_2 = \frac{h^2}{n} \nabla^2 P
</math>

An optimal kernel width that minimizes the error of each estimate can thus be derived.

==Use for statistical classification==

The method is particularly effective when applied to [[statistical classification]].
There are two ways we can proceed: the first is to compute the PDFs of
each class separately, using different bandwidth parameters,
and then compare them as in Taylor.<ref name="Taylor1997">
{{Cite journal
| last = Taylor
| first = Charles
| title = Classification and kernel density estimation
| journal = Vistas in Astronomy
| volume = 41
| issue = 3
| pages = 441–417
| year = 1997}}</ref>
Alternatively, we can divide up the sum based on the class of each sample:

: <math>
P(j, \vec{x}) \approx \frac{1}{n}\sum_{i=1, c_i=j}^n w_i
</math>

where ''c<sub>i</sub>'' is the class of the ''i''th sample.
The class of the test point may be estimated through [[maximum likelihood]].

Many kernels, Gaussian for instance, are smooth. Consequently, estimates
of joint or conditional probabilities are both continuous and differentiable.
This makes it easy to search for a border between two classes by zeroing
the difference between the conditional probabilities:

: <math>
R(\vec x) = P(2 | \vec x) - P(1 | \vec x)
= \frac{P(2, \vec x) - P(1, \vec x)}{P(1, \vec x) + P(2, \vec x)}
</math>

For example, we can use a one-dimensional [[root-finding algorithm]] to zero
''R'' along a line between two samples that straddle the class border.
The border can be thus sampled as many times as necessary.
The border samples along with estimates of the gradients of ''R''
determine the class of a test point through a dot-product:

: <math>
j = \arg \underset{i}{\min} | \vec{b_i} - \vec x | \,
</math>

: <math>
p = (\vec x - \vec{b_j}) \cdot \nabla_{\vec x} R |_{\vec x=\vec{b_j}} \,
</math>

: <math>
c = ( 3 + p/|p| ) / 2 \,
</math>

where <math>\lbrace \vec{b_i} \rbrace</math> sample the
class border and ''c'' is the estimated class.
The value of ''R'', which determines the conditional probabilities,
may be extrapolated to the test point:

: <math>
R(\vec x) \approx \tanh p \,
</math>
<ref name="Mills2010"/>

Two-class classifications are easy to generalize to multiple classes.

==External links==
* [http://libagf.sf.net libAGF] - A library for multivariate, adaptive kernel density estimation.

==References==

<references/>

[[Category:Classification algorithms|*]]
[[Category:Statistical classification]]
[[Category:Estimation of densities]]
[[Category:Non-parametric statistics]]

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en>Headbomb: Various citation cleanup, replaced: | url=http://dx.doi.org/ → | doi= (2), {{MathSciNet | id = 0445498}} → {{MR|0445498}} (4), | id={{MR|0445498}} → | mr=0445498 (4), added orphan tag using AWB