Variable kernel density estimation

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 varieddepending 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. 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 varieddepending 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. Given a set of samples, { x → i } {displaystyle lbrace {vec {x}}_{i} brace } , we wish to estimate thedensity, P ( x → ) {displaystyle P({vec {x}})} , at a test point, x → {displaystyle {vec {x}}} : where n is the number of samples, K is the 'kernel', h is its width and D is the number of dimensions in x → {displaystyle {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 sampleswill fall in the tails of the filter with very low weighting, while regions of highdensity will find an excessive number of samples in the central region with weightingclose to unity. To fix this problem, we vary the width of the kernel in differentregions 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 dependingon the location of the sample. For multivariate estimators, the parameter, h, can be generalized tovary not just the size, but also the shape of the kernel. This more complicated approachwill not be covered here.