Variable kernel density estimation#Use for statistical classification

Given a set of samples, $\backslash lbrace\; \backslash vec\; x\_i\; \backslash rbrace$, we wish to estimate the

density, $P(\backslash vec\; x)$, at a test point, $\backslash vec\; x$:

: $$

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 $\backslash vec\; x$.

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 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.

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.

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

: $$

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

where k is a constant.

If we back-substitute the estimated PDF, and assuming a Gaussian kernel function,

we can show that W is a constant:

: $$

W = k^D (2 \pi)^{D/2}

A similar derivation holds for any kernel whose normalising function is of the order {{math|h^D}}, although with a different constant factor in place of the {{math|(2 π)^D/2}} term. This produces a generalization of the k-nearest neighbour algorithm.

That is, a uniform kernel function will return the

KNN technique.{{Cite journal

| last = Mills

| first = Peter

| title = Efficient statistical classification of satellite measurements

| year = 2011

| volume = 32

| issue = 21

| pages = 6109–6132

| journal = International Journal of Remote Sensing

| doi = 10.1080/01431161.2010.507795

| arxiv = 1202.2194| bibcode = 2011IJRS...32.6109M

| s2cid = 88518570

}}

There are two components to the error: a variance term and a bias term. The variance term is given as:

: $$

e_1 = \frac{P \int K^2}{n h^D}

.

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:

: $$

e_2 = \frac{h^2}{n} \nabla^2 P

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

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.

{{Cite journal

| last = Taylor

| first = Charles

| title = Classification and kernel density estimation

| journal = Vistas in Astronomy

| volume = 41

| issue = 3

| pages = 411–417

| year = 1997

| doi=10.1016/s0083-6656(97)00046-9| bibcode = 1997VA.....41..411T

}}

Alternatively, we can divide up the sum based on the class of each sample:

: $$

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

where c_i is the class of the ith sample.

The class of the test point may be estimated through maximum likelihood.

• [http://www.mathworks.com/matlabcentral/fileexchange/58309-adaptive-kernel-density-estimation-in-one-dimension akde1d.m] - Matlab m-file for one-dimensional adaptive kernel density estimation.
• [http://libagf.sf.net libAGF] - A C++ library for multivariate adaptive kernel density estimation.
• [http://www.mathworks.com/matlabcentral/fileexchange/58312-kernel-density-estimator-for-high-dimensions akde.m] - Matlab function for multivariate (high-dimensional) variable kernel density estimation.

*

Category:Statistical classification

Category:Estimation of densities

Category:Nonparametric statistics