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Given data points $x_i$ in $\mathbb{R}^d$ with function values $f_i$, one can estimate the function at a given $x$ by $\ \ \ \ \text{f}_{est}( x ) = \frac {\sum { w_i f_i }} {\sum { w_i }}$ with $w_i = k( |x - x_i| )$.
Here $k$() is a kernel function, often Gaussian. See also Radial basis function.

Consider this variant:
a) make f$_{est}$() scale-free, the same for $x_i$ at distances 1, 2, 3 $\dots$ as at 10, 20 30 $\dots$, by taking some number Nnear of the points near $x$ and scaling these Nnear distances by their average $Dav$: $\ \ \ \ wscaled_i = k( \frac {|x - x_i|} { Dav } )$

b) use the Catmull-Rom a.k.a. C-R spline kernel:

enter image description here

The C-R spline is widely used in signal and image processing because it approximates the sinc kernel, which reconstructs band-limited signals on uniform grids perfectly. (For scattered / non-uniform points, I have no idea.) As you see, C-R is 0 at 1, negative from 1 to 2, then 0 beyond 2. Now if the scaled distances $\frac {|x - x_i|} {Dav}$ are all 1, $\sum {wscaled_i}$ will be 0; hmm. Even if they're all near 1, e.g. $\sim \mathcal{N}(1, \text{small}\ \sigma)$, that's not so hot either.

Do scaled sinc-like kernels make sense for density estimation ?
Has anyone used them in practice ?

(Added): Inverse distance weighting is also scale-free, property a) above.
Fwiw, changing the $\tfrac{1}{distance}$ kernel in IDW with python to the M-N spline kernel makes not much of a difference there.

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1 Answer 1

There has been lots of work regarding the use of sinus cardinal function as a kernel for density estimation. Most of the work involving sinc kernels is actually for density deconvolution in error-in-variable models because the Fourier transform has a compact support and because it enjoys nice computational properties. In terms of implementations there is an implementation in R called 'deamer' and you may also consider a Matlab package EstimHidden by yves rozenholc.

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interesting, +1. Can you suggest an introduction t deconvolution methods ? Would you have any comments on fitting smooth f to not much data in 5d or so ? –  denis Oct 10 '12 at 14:33

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