1
Electrical and Computer Engineering Department
Saurabh Prasad
Pattern Recognition
Chapter 4
Pattern Recognition
ECE
‐
8443
Chapter 4
Nonparametric Classification Techniques
Electrical and Computer Engineering Department,
Mississippi State University.

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Electrical and Computer Engineering Department
Saurabh Prasad
Pattern Recognition
Chapter 4
•
Introduction
•
Density Estimation
•
Parzen Windows
Outline

3
Electrical and Computer Engineering Department
Saurabh Prasad
Pattern Recognition
Chapter 4
Introduction
•
All Parametric densities are unimodal (have a single local
maximum), whereas many practical problems involve multi
‐
modal densities
–
Exception: GMMs
•
Nonparametric procedures can be used with arbitrary
distributions and without the assumption that the forms of
the underlying densities are known
•
There are two types of nonparametric methods:
–
Estimating P
(x |
ω
j
)
–
Bypass probability and go directly to a
‐
posteriori probability
estimation

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4
Electrical and Computer Engineering Department
Saurabh Prasad
Pattern Recognition
Chapter 4
4
Density Estimation
–
Basic idea:
–
Probability that a vector
x
will fall in region R is:
–
P is a smoothed (or averaged) version of the density function
p(
x
). If we have a sample of size n, the probability that k points
fall in
R
is then:
and the expected value for k is:
E(k) = nP
(3)
∫
ℜ
=
(1)
'
dx
)
'
x
(
p
P
(2)
)
P
1
(
P
k
n
P
k
n
k
k
−
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=