[FoRK] Kernel Density Technique (Parzen Window) help

Stephen D. Williams sdw at lig.net
Sun Apr 27 11:58:13 PDT 2014

```The Wand and Jones (1995) must refer to this:

Page 44 of that book, in the free Google view of it, mentions the use of the mean value theorem although that appears to be just one
step.
https://en.wikipedia.org/wiki/Mean_value_theorem

It seems like the most likely meaning of  p sub(w) (a sub(i) ) ...   |a sub(i) - a sub(i+w)|, is that w=window.  It isn't clear what
was meant by the denominator.  Doesn't seem workable as is.

Not only does the following explain kernel density estimation well, but it points to a numpy function that computes with a Gaussian
kernel and determines the bandwidth (i.e. w) automatically:
https://en.wikipedia.org/wiki/Kernel_density_estimation
And, rug plot!

http://research.cs.tamu.edu/prism/lectures/pr/pr_l7.pdf
https://www.cs.utah.edu/~suyash/Dissertation_html/node11.html

See section 4 here.  Also good illustration of different kernels.
http://arxiv.org/pdf/1212.2812.pdf

Looks interesting for both machine learning and image cleanup / compression / understanding.  Although most of those methods already
have much more efficient ways of solving an equivalent problem.  You might want to look at both machine learning and image
processing algorithms to do "peak detection in time-series".

Really interesting:
https://en.wikipedia.org/wiki/Manifold_learning#Manifold_learning_algorithms

http://iihm.imag.fr/daassi/papierInfoVisJournal/papierExemple/9500051a.pdf
http://people.csail.mit.edu/rosman/tcie_ijcv.pdf
http://tx.technion.ac.il/~rc/diffusion_maps.pdf

http://shogun-toolbox.org/

General purpose machine learning toolkit:
http://waffles.sourceforge.net/

sdw

On 4/26/14, 12:04 PM, Damien Morton wrote:
> Hi everyone,
>
> I am trying to implement a technique described in the following paper,
>
> http://www.tcs-trddc.com/trddc_website/pdf/SRL/Palshikar_SAPDTS_2009.pdf
>
> I have struck a problem and am looking for some help.
>
> on page 7, function S4 is defined
>
> Part of that is function pw(ai)
>
> The right hand term inside the summation has a denominator |a[i] - a[i+w]|
>
>   The kernel density technique (also called Parzen window)  is being used
>
> The problem is that the denominator is 0 when a[i]==a[i+w] - I want to know
> how best to handle this event
>
>   now, I have searched all kinds of material on the kernel density technique
> and nowhere do i see the denominator defined this way
>
>   from what I can see, the denominator is usually expressed as a constant
> proportional to the standard deviation of the samples
>
>   I _think_ the term  |a[i] - a[i+w]| is being used as some kind of adaptive
> kernel width. Its really not clear to me why this term is being used and
> what it means
>
> I tried contacting the author, but got no response
>
> If anyone can point me in the right direction or give me some advice, t
> would be very much appreciated

```