Argonauts:Parzen Windows
From Wasteland
Contents |
Sun 12 February 2006
Spatio-temporal coordinates
WMVL, 11am to 12.30pm exactly.
Attendees (in alphabetical order)
- Ramón Casero Cañas.
- Olivier Noterdaeme.
- Ingmar Posner.
Minutes
We are halfway through Hilary now. Are we any wiser than we were 4 weeks ago? Arguably not.
This weekend Ramon suggested doing Parzen Windows and the Mean Shift algorithm. Well, it turns out that 1) Parzen Windows are really easy to understand and 2) had Ramón not missed the WMVL Reading Group a couple of weeks ago, he would know that the rest already knew about Mean Shift, having read the paper
D. Comaniciu and P. Meer. Mean shift: A robust approach toward feature space analysis. IEEE Trans. on Pattern Analysis and Machine Intelligence, 24(5), May 2002.
Duh!
Anyways, we arrived at Parzen's classic paper (Parzen was at the University of Stanford). The paper is from 1962, but available online thanks to that wonderful JSTOR site
E. Parzen. On estimation of a probability density function and mode. The Annals of Mathematical Statistics, 33(3), Sep 1962, pp. 1065--1076.
Now, seriously, there are a lot of good oldies in JSTOR, all the great stuff Statisticians have been doing for the last century, simple, powerful and elegant ideas that we should be more aware of.
A quick glance at the formulae and the crew felt intimidated and preferred an easier approach. Shame. In fact, a good and concise explanation can be found in the first 3 pages of
G.C. van der Eijkel et al. A modulated Parzen-Windows approach for probability density estimation. Procs. of the Second International Symposium, IDA-97, London, UK, August 1997. Lecture Notes in Computer Science, vol. 1280, Springer, 1997.
We also had a look at sec. 4.3 of
R.O. Duda et al. Pattern classification. Wiley. 2000.
So putting everything together, given a sample of a random variable X for which we want to estimate its pdf, we put a kernel function (e.g. a Gaussian) on top of each point, and the estimated pdf is the linear combination of them, normalized so that the integral of the pdf is 1.
The catch is how to decide on the width of the kernel function.
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