Docstoc

towers_pde

Document Sample
towers_pde Powered By Docstoc
					    Overview of
  Non-Parametric
Probability Density
Estimation Methods

      Sherry Towers
  State University of New York
         at Stony Brook
All kernal PDF
estimation
methods
(PDE’s) are
developed
from a simple
idea…

 If a data point
lies in a region
where
clustering of
signal MC is
tight, and
bkgnd MC is
loose, the
point is likely
to be signal
                   S.Towers
 Toestimate a PDF, PDE’s
 use the idea that any
 continuous function can
 be modelled by sum of
 some “kernal” function

 Gaussiankernals are a
 good choice for particle
 physics

 So,a PDF can be
 estimated by sum of
 multi-dimensional
 Gaussians centred about
 MC generated points

                       S.Towers
 Best form of Gaussian kernal
 is a matter of debate:

                PDE method
   Static-kernal
   uses a kernal with covariance
   matrix obtained from entire
   sample

   TheGaussian Expansion
   Method (GEM), uses an
   adaptive kernal; the
   covariance matrix used for the
   Gaussian at each MC point
   comes from “local” covariance
   matrix.

                          S.Towers
S.Towers
GEM vs Static-Kernal PDE

 GEM  gives unbiased
 estimate of PDF, but
 slower to use because
 local covariance must be
 calculated for each MC
 point

 Static-kernalPDE
 methods have smaller
 variance, and are faster
 to use, but yield biased
 estimates of the PDF
                      S.Towers
Comparison of GEM and
static-kernal PDE:




                   S.Towers
PDE vs Neural Networks

 Both  PDE’s and Neural
  Networks can take into
  account non-linear
  correlations in parameter
  space
 Both methods are, in
  principle, equally
  powerful
 For most part they
  perform similarly in an
  “average” analysis
                       S.Towers
PDE vs Neural Networks

 But, PDE’s have far
 fewer parameters, and
 algorithm is more
 intuitive in nature (easier
 to understand)




                        S.Towers
Plus, PDE estimate of
PDF can be
visually examined:




                  S.Towers
PDE’s vs Neural Nets…
   There are some problems
    that are particularly well
    suited to PDE’s:




                          S.Towers
PDE’s vs Neural Nets…




                   S.Towers
PDE’s vs Neural Nets…




                   S.Towers
PDE’s vs Neural Nets…




                   S.Towers
        Summary

 PDE   methods are as
  powerful as neural networks,
  and offer an interesting
  alternative
 Very few parameters, easy to
  use, easy to understand, and
  yield unbinned estimate of
  PDF that user can examine
  in the multidimensional
  parameter space!


                        S.Towers

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:4
posted:10/17/2012
language:Unknown
pages:16