# towers_pde

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```					    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
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
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

```
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 views: 4 posted: 10/17/2012 language: Unknown pages: 16