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.


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.

  • 1
    $\begingroup$ 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 ? $\endgroup$
    – denis
    Oct 10, 2012 at 14:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.