证据理论应用举例

16
Clustering
problem formulation
x1 A11 , A12 , , A1T x A , A , , A 2T 2 21 22 x A , A , , A NT N N1 N 2
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 ))
Special Report
Applications of evidence theory
5
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
Special Report Applications of evidence theory
26
State Estimation
belief state estimation (BSE) [15] an extension of bounded error estimation (BEE) problem formulation
30/60/120/180 training samples 1000 test samples
Special Report
Applications of evidence theory
10
EKNN
Special Report
Applications of evidence theory
11
Special Report
Applications of evidence theory
12
EKNN
example for BKNN
Special Report
Applications of evidence theory
13
Classifier Fusion
original output of a classifier
Applications of evidence theory
23
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 ))
Applications of evidence theory
3
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
EKNN
an extension: BKNN [2] consider meta-class
mi (Cq ) , mi ( ) 1 mi ({C p , Cq }) , mi ( ) 1
lower error rate extra uncertain rate
x A1 , A2 , , AT y ?
training set
Special Report
Applications of evidence theory
6
EKNN
k-nearest neighbors (KNN)
Special Report
Applications of evidence theory
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)
Special Report
Special Report
10 250 250 250 10 Baidu Nhomakorabea50 250 250 10
Applications of evidence theory
25
Image Restoration
example
more references [13,14]
1
Content
Basics Pattern Recognition State Estimation Belief Rule Base Conclusion
Special Report
Applications of evidence theory
2
Basics
mass function
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
Special Report
(2)
Special Report
Applications of evidence theory
4
Pattern Recognition
Classification
EKNN classifier fusion
Clustering
ECM
Image restoration
PBMF
mi (Cq ) mi ( ) 1 (3)
0e
d / d cq
d-- distance to neighbor i dcq-- mean distance in class q
Special Report
Applications of evidence theory
( 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}
Special Report Applications of evidence theory
14
Classifier Fusion
13 original classifiers
AOD, NaiveBayes, SMO, IBk, IB1, KStar, DecisionStump, J48, RandomForset, DecisionTable, JRip, NNge, PART
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)
Special Report
Special Report
Applications of evidence theory
24
Image Restoration
evidence generation
difference in intensity 1
di | x(k ) xi (k ) |, c sort (d )
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 ))
Applications of evidence theory
19
ECM
evidential c-means [8]
an extension of c-means consider meta-cluster consider outliers
objective function
17
c-means
1 2
3
4
Special Report
Applications of evidence theory
18
Fuzzy c-means
cluster center
vk

u xi ik i 1
n i 1 ik
n
u

, k 1, , c
(4)
membership degree
21
Clustering
more references
RECM [9], belief c-means [10], credal c-means [11], etc.
Special Report
Applications of evidence theory
22
Image Restoration
n n
J ECM ( M , V )
i 1 j : A j

c j mij d 2 mi
2 ij i 1


(7)
Special Report
Applications of evidence theory
20
ECM
Special Report
Applications of evidence theory
Special Report
Applications of evidence theory
15
Classification
belief decision tree more references
refer to [4-7] for more details
Special Report
Applications of evidence theory
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
Special Report
Special Report on Applications of Evidence Theory
Lab of vibration control and vehicle control
Special Report
Applications of evidence theory
7
EKNN
KNN
Special Report
Applications of evidence theory
8
EKNN
evidential KNN (EKNN) [1]
each neighbor corresponds to an evidence degree of support depends on the distance neighbor i with label q will generate
y1 y 2 ? yN
Objective: Minimize inner-cluster distance Maximize outer-cluster distance
Special Report Applications of evidence theory