证据理论应用举例

partition belief mean filter (PBMF) [12] filter window
w(k ) {x1 (k ), , x9 (k )}
PBMF
y (k ) x (k ) (k )( x(k ) x (k )) (8)
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Image Restoration
evidence generation
difference in intensity 1
di | x(k ) xi (k ) |, c sort (d )
1
Content
Basics Pattern Recognition State Estimation Belief Rule Base Conclusion
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Basics
mass function
Applications of evidence theory
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Image Restoration
evidence generation distance to mean gray level
g (k ) | x(k ) wave (k ) | mg ( F ) (1 mg ( N )) mg ( NF ) 1 (1 mg ( N ))
ration of difference to total difference
i | xi (k ) x (k ) |, i 1, ,9
p( k )
5 i
m p ( F ) (1 m p ( N )) m p ( NF ) 1 (1 m p ( N ))
ui , j
2/ ( 1) dij

c k 1
d
2/ ( 1) ij
, i 1, n; j 1, , c
(5)
objective function
n c
2 J FCM (U , V ) uij d ij
i 1 j 1
(6)
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EKNN
example for BKNN
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Classifier Fusion
original output of a classifier
Applications of evidence theory
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ECM
evidential c-means [8]
an extension of c-means consider meta-cluster consider outliers
objective function
r (k )
4
i 1 i
c
4 mr ( F ) (1 mr ( N )) mr ( NF ) 1 (1 mr ( N ))
difference in intensity 2
l (k ) c1 c2 2 ml ( F ) (1 ml ( N )) ml ( NF ) 1 (1 ml ( N ))
9
EKNN
example
dataset: 3 Gaussian distributions
1 1 0 , 1 , 1 1 1 2 3 1 0 1 1 I , 2 I , 3 2I
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c-means
1 2
3
4
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Applications of evidence theory
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Fuzzy c-means
cluster center
vk

u xi ik i 1
n i 1 ik
n
u

, k 1, , c
(4)
membership degree
EKNN
an extension: BKNN [2] consider meta-class
mi (Cq ) , mi ( ) 1 mi ({C p , Cq }) , mi ( ) 1
lower error rate extra uncertain rate
definition
m() 0; m( A) 0, A ; m( A) 1. A (1)
example
{1 , 2 , 3 } m(1 ) 0.6, m( 2 , 3 ) 0.2, m() 0.2
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Classifier Fusion
13 original classifiers
AOD, NaiveBayes, SMO, IBk, IB1, KStar, DecisionStump, J48, RandomForset, DecisionTable, JRip, NNge, PART
(2)
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Applications of evidenition
Classification
EKNN classifier fusion
Clustering
ECM
Image restoration
PBMF
21
Clustering
more references
RECM [9], belief c-means [10], credal c-means [11], etc.
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Applications of evidence theory
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Image Restoration
( x) {s1 , s2 , , sn }
normalized mass function
m({Ci }) si
s
j
triplet mass function [3]
m ({u}) m ({v}) m (C ) 1 u arg max{m({C1}), , m({Cn }) v arg max{m({Ci }) | Ci u}
30/60/120/180 training samples 1000 test samples
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EKNN
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x A1 , A2 , , AT y ?
training set
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EKNN
k-nearest neighbors (KNN)
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Applications of evidence theory
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Classification
problem formulation
attributes label
x1 A11 , A12 , , A1T y1 x A , A , , A y 2T 2 2 21 22 xN AN 1 , AN 2 , , ANT yN
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Clustering
problem formulation
x1 A11 , A12 , , A1T x A , A , , A 2T 2 21 22 x A , A , , A NT N N1 N 2
Applications of evidence theory
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Basics
Dempster’s rule
A, B , A B C m1 ( A)m2 ( B ) , m(C ) 1 K 0, K A B m1 ( A)m2 ( B) X X
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Classification
belief decision tree more references
refer to [4-7] for more details
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Applications of evidence theory
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Image Restoration
example
more references [13,14]
Special Report Applications of evidence theory