[FoRK] Kernel Density Technique (Parzen Window) help

Damien Morton dmorton at bitfurnace.com
Sun Apr 27 19:59:48 PDT 2014


Hi Stephen,

Thanks for the pointer to The Wand and Jones book.

It appears that the denominator is indeed an attempt at a adaptive kernel
width.

On page 43 they describe variable kernel estimators, and the denominator
from my paper corresponds to Wan and Jone's alpha(Xi) term.

"... suggests that each alpha(Xi) term should depend on the true density in
a roughly inverse way"

It appears that the |Xi - Xi+w| term is a crude attempt to estimate the
density around Xi

I had started to experiment with calculating the variance of the values in
the Xi....i+w window, and using that as the denominator, but I didnt
understand what I was doing. Now I have some basis to continue that.

Plus, the book also talks about clipping for extreme values, and then says
its out of the scope of the book. Harumph.




On Sun, Apr 27, 2014 at 2:58 PM, Stephen D. Williams <sdw at lig.net> wrote:

> The Wand and Jones (1995) must refer to this:
> https://encrypted.google.com/books?id=GTOOi5yE008C&q=
> parzen+window#v=onepage&q=parzen%20window&f=false
>
> 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!
>
> See also:
> 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
>>
>
>
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