Comparative Study on BBA Determination Using Different Distances of Interval Numbers

2017·1.2 MB·

by  Jiankun Ding

20th International Conference on Information Fusion
Xi'an, China - July 10-13, 2017
Comparative Study on BBA Determination Using
Different Distances of Interval Numbers
Jiankun Ding, Deqiang Han Jean Dezert Yi Yang
Institute of Integrated Automation ONERA SKLSVMS
School of Electronic and Information Engineering The French Aerospace Lab School of Aerospace
Xi’an Jiaotong University Chemin de la Hunie`re, Xi’an Jiaotong University
Xi’an, Shaanxi, China 710049 F-91761 Palaiseau, France Xi’an, Shaanxi, China 710049
Email: [email protected]; [email protected] Email: [email protected] Email: [email protected]
Abstract—Dempster-Shafer theory (DST) is an important the- throughiterativeestimation.Dengetal.[9]definedasimilarity
oryforinformationfusion.However,inDSThowtodeterminate measure based on radius of gravity, and then the similarity
the basic belief assignment (BBA) is still an open issue. The
measure is used for determinating the BBAs. Boudraa et al.
intervalnumberbasedBBAdeterminationmethodissimpleand
[10] and Florea et al. [11] determinates the BBAs based on
effective, where the features of different classes’ samples are
modeled using the interval numbers, i.e., an interval number the membership functions. Han et al. [12] proposed a method
model is constructed for each focal element. Then, the distances for the transformation of fuzzy membership function into
ofintervalnumbersareusedformeasuringthesimilaritydegrees BBAs by solving a constrained maximization or minimization
between the testing sample and each focal element, and the
optimization problem. Recently, Kang et al. [6] designed a
similarity degrees are used for determinating the BBA. The
BBA determination method using the interval numbers.
definitionofintervalnumbers’distanceiscrucialfortheeffective-
ness of the interval number based BBA determination methods. Kang’s interval number based BBA determination method
In this paper, we use different interval numbers’ distances for is simple and effective. Kang’s method first constructs the
determinating BBAs. By using the artificial data set and the Iris intervalnumber[14]modelsforeachfocalelement(including
date set of open UCI data base, respectively, we compare and
the singleton focal elements with single class and the com-
analyze the determination of BBAs with different distances.
poundfocalelementswithmultipleclasses)basedonthesetof
IndexTerms—Dempster-Shafertheory,basicbeliefassignment,
distance of interval numbers, information fusion, classification. training samples. In Kang’s method, the Tran and Duckstein’s
[14],[16]intervalnumberdistance(TD-IND)isusedformea-
I. INTRODUCTION suring the similarity degree of the testing samples compared
Dempster-Shafer theory (DST) [1] was proposed by Demp- with different focal elements’ interval number models. In the
ster in 1960s, and was developed by Shafer [2]. In DST, the final,thesimilaritiesarenormalizedtogetthevaluesofBBA.
basic beliefs are assigned to the power set of the frame of The definitions of the interval numbers’ distances (INDs) are
discernment (FOD), which is used to describe the uncertain- crucial for the performance of the interval number based
ty of sources of evidence. The evidences (i.e., basic belief BBAdeterminationmethod.Thereexistmanypossiblechoices
assignments, BBAs) originated from different sources can be for INDs, e.g., the Gowda and Ravi’s distance [15] (GR-
fused using the Dempster’s combination rule [1]. DST has IND), the Tran and Duckstein’s distance [16] (TD-IND), the
been widely used in the information fusion fields [3]–[5]. Hausdorffdistance[17](H-IND)andtheDeCarvalho’snorm-
UsingDST,thefirststepistodeterminatetheBBAs,which q distance [18] (Nq-IND). In this paper, we implement the
is still an open issue. The determination of BBAs can mainly Kang’s interval number based method using different INDs.
categorized into two branches [6]: (1) The experts give the We analyze the differences of the BBAs determinated using
BBAs directly according to their personal experiences; (2) differentINDsbasedonnumericalexamples.Furthermore,we
The BBAs are determinated based on the samples using some useMonte-Carloexperimentsforcomparingtheperformances
special determination rules. In the first branch, the determi- of interval number based methods with different INDs by
nation of BBAs relies on the experts’ subjective points of classifying an artificial set and the iris set1.
view.Inthispaper,wefocusonthesecondbranchapproaches,
i.e., the BBAs are determinated based on available samples. II. BASICOFDEMPSTER-SHAFERTHEORY
Researchers have proposed many approaches in this branch. Dempster-Shafertheory(DST)(alsoknownastheEvidence
Selzer et al. [3] determinated the BBAs based on the number Theory) is an appealing mathematical framework which can
of classes and the environmental weighting coefficient. Shafer effectively describe the uncertainty information for the state
[2]proposedaBBAdeterminationmethodbasedonstatistical ofnature.InDST,theframeofdiscernment(FOD)isdenoted
evidences.Bietal.[7]designedakindoftriplefocalelements by (cid:2) = {(cid:18) ;(cid:18) ;··· ;(cid:18) }. The elements in (cid:2) are mutually
1 2 n
BBA in dealing with the text classification problem. Szlzen-
stein et al. [8] used the Gaussian model getting the BBAs 1http://archive.ics.uci.edu/ml/datasets/Iris
978-0-9964527-5-5©2017 ISIF 979
and exhaustive. The basic belief assignment (BBA) function a degenerate interval (a precise number) with a zero length.
assignsbasicbeliefsonthepowersetof(cid:2),i.e.,2(cid:2).TheBBA Kang’ method measures the distances between the testing
is also called the mass function which satifies: sample and different interval number models of the focal
∑
elements. The testing sample should have a higher similarity
m(A)=1;m(∅)=0 (1)
degree with the focal element when the distance is small, and
A⊆(cid:2) the corresponding focal element is assigned a higher basic
If A⊆(cid:2);m(A)>0, A is called a focal element. belief. The steps of Kang’s method are described as follows:
The Belief (Bel) and Plausibility (Pl) of A are defined as:
1) The interval number models of the focal elements with
∑
single class are constructed by finding the minimum
Bel(A)= m(B) (2)
andthemaximumofthecorrespondingclasses’training
B⊆A
∑ ( ) samples. Then, the interval number models of the focal
Pl(A)= m(B)=1−Bel A (3) elements with mixture classes are obtained by finding
B∩A=∅ theoverlappingregionofthecorrespondingsingleclass-
es’intervalnumbermodels.Theintervalnumbermodels
Theinterval[Bel(A);Pl(A)]iscallthebeliefinterval,which
of different focal elements are denoted by ~b ;f ∈2(cid:2).
represents the uncertainty of the support degree of A. f
2) Calculate the distances between the testing sample (de-
Different information sources can provide different evi-
notedbya~)and(differe)ntfocalelements’intervalnumber
dences, i.e., the BBAs. In DST, two BBAs associated with
models, i.e., D a~;~b ; ∀f ∈2(cid:2). Note that the length
two distinct sources of evidence can be combined according f
to the Dempster’s rule, as in Eq. (4). of a~ is 0, i.e., a+ =a−.
∑
3) Calculate the similarity degree based on the distances
 m (B)m (C)
B∩C=A 1 2 A̸=∅ according to Eq. (6).
m(A)= 1−K (4)
 ( )
0 A=∅ 1
∑ S a~;~bf = ( ) (6)
where K = B∩C=Am(B)m(C) denotes the conflicting 1+(cid:11)D a~;~bf
coefficient. Dempster’s combination rule is both commutative
and associative. where (cid:11) > 0 is the support coefficient. Empirically, it
To make a probabilistic decision, the fused BBA can be is proper to set (cid:11)=5 [6].
transformedintotheprobabilityusingthePignisticprobability 4) The BBA is determinated by normalizing the similarity
transformation: degrees of all the focal elements.
∑
m(A) Kang’s method define the similarity degrees using interval
Betp((cid:18) )= ; ∀(cid:18) ∈(cid:2) (5)
i |A| i numbers’distance,andtheBBAsareobtainedbynormalizing
(cid:18)i∈A; A⊆(cid:2) the(simila)ritydegrees.Thus,thedefinitionoftheIND(i.e.,the
where |A| denotes the cardinality of A. D a~;~b ) is crucial for this method. The differences of the
f
BBAs determinated by Kang’s method using different INDs
III. KANG’SBBADETERMINATIONMETHODBASEDON
are compared in the next section.
THEINTERVALNUMBERS’DISTANCES
Using the DST, the determination of the BBAs is the first
IV. COMPARISONSOFINTERVALNUMBERBASEDBBA
step, which is an still a challenging task. Interval number,
DETERMINATIONMETHODUSINGDIFFERENTINDS
whichcandescribetheuncertaintyorinsufficientinformation,
isusefulfordeterminatingtheBBAs.Thedefinitionofinterval
As aforementioned, the definition of the IND is crucial for
numbers is as follows: An interval number a~ in R is a
theintervalnumberbasedBBAdeterminationmethods.Many
set of real numbers that lie between two real numbers, i.e.,
INDs have been proposed. Here, we introduce four widely
a~ = [a−;a+] = {x|a− ≤x≤a+};a−;a+ ∈ R and a− ≤
used INDs.
a+. Kang et al. [6] proposed a BBA determination method
based on the interval number models, where the basic beliefs
A. Introduction of the interval number’s distances
assigned to different focal elements are determinated based
on the interval numbers’ distances between the testing sample Suppose a~ = [a−;a+] and ~b = [b−;b+] are two interval
and the interval number models of focal elements. Here, we numbers. Then [13], [14], c~ = a~ ⊕ ~b = [c−;c+], where
recall the Kang’s interval number based BBA determination c− = min(a−;b−) and c+ = max(a+;b+). The length (or
method first. width) of the interval number a~ is (cid:22)(a~) = a+ − a−. D
d
Kang’s method determinates BBAs on different single fea- is the length of the domain [14] of the interval numbers. To
tures respectively. In a single feature, Kang’s method models measure the difference between two interval numbers, many
different focal elements (including the focal elements with intervalnumbers’distances(INDs)havebeenproposed.Here,
single class and the focal elements with multiple classes) we introduce four widely used INDs, which are introduced as
using interval numbers, and the testing sample is treated as follows:
980
Gowda and Ravi (1995) [15]: In 1995 Gowda and Ravi
proposedametric(denotedbyGR-IND)combiningaposition
and a size component, as follows Class 3
( ) ( ) ( ) Overlapping region
of class 2 and class 3
DGR a~;~b =Dp a~;~b +Ds a~;~b (7) Class 2 Overlapping region
of class 1 and class 2
where the position component is defined as,
( ) [( ) ] Class 1
|a−−b−| (cid:25)
D a~;~b =cos 1− × (8)
p (cid:22)(D ) 2 1 2 3 4 5 6 7 8
d Feature values
and the size component is defined as
 ( )  Fig.1. Featurevalues’rangesofdifferentclasses
( ) (cid:22)(a~)+(cid:22) ~b
D a~;~b =cos ( ) × (cid:25) (9)
s 2×(cid:22) a~⊕~b 2 THEINTERVALNUMBERTSAMBOLDEELISOFFOCALELEMENTS.
Tran and Duckstein (2002) [16]: In the framework of Focalelements Intervalnumbermodel
fuzzy data analysis, Tran and Duckstein proposed the interval f(cid:18)1g [1;4]
numbers’ distance (TD-IND): f(cid:18)2g [3;7]
DT2D(a~;~b)=∫ 21 ∫ 12 {[12(a++a(cid:0))+x(a+(cid:0)a(cid:0))] ff(cid:18)(cid:18)31g;(cid:18)2g [[53;;84]]
[(cid:0)12 (cid:0)12 ]} f(cid:18)2;(cid:18)3g [5;7]
(cid:0) 1(b++b(cid:0))+y(b+(cid:0)b(cid:0)) 2dxdy f(cid:18)1;(cid:18)3g N/A
2 (10) f(cid:18)1;(cid:18)2;(cid:18)3g N/A
[( ) ( )]
= 1 a(cid:0)+a+ (cid:0) b(cid:0)+b+ 2
+41 [(a+(cid:0)a(cid:0))2+(b+(cid:0)b(cid:0))2] that in this example {(cid:18)1;(cid:18)3} and {(cid:18)1;(cid:18)2;(cid:18)3} do not have
12 intervalnumbermodels,becausethe{(cid:18) }’sand{(cid:18) }’sinterval
1 3
number models do not have overlapping region.
Hausdorff distance [17]: Considering two sets A and B
Supposewehaveatestingsamplewhosefeaturevalueis2,
of points of Rn, and a distance d(x;y), where x ∈ A and
i.e., a~=[2;2], as the purple dot on X-axis of Figure 1. Then
y ∈B. The Hausdorff distanc (H-IND) is defined as follows:
( ) weusedifferentINDs,i.e.,theGR-INDasinEq.(7),theTD-
INDasinEq.(10),theH-INDasinEq.(12),andtheNq-IND
D (A;B)=max sup inf d(x;y);sup inf d(x;y)
H
x∈Ay∈B y∈Bx∈A as in Eq. (13) (with q = 2 in Nq-IND), for measuring the
(11) distance between the a~ and different focal elements’ interval
Ifd(x;y)istheManhattandistance(alsocalledtheCityblock number models, respectively. The distances are listed in Table
distance), i.e., d(x;y) = |x−y|, then Chavent et al. (2002) II.
proved that
( ) ((cid:12) (cid:12) (cid:12) (cid:12))
D a~;~b =max (cid:12)a−−b−(cid:12);(cid:12)a+−b+(cid:12) (12) TABLEII
H THEINDSBETWEENTHEa~ANDFOCALELEMENTS’INTERVALNUMBER
MODELS.
De Carvalho et al. (2006) [18]: A family of distances
between interval numbers has been proposed by De Carvalho Focalelements GR-IND TD-IND H-IND Nq-IND
et al. based on the bounds of interval numbers. The metric of f(cid:18)1g 0.9296 1.0000 2.0000 2.2361
norm-q (Nq-IND) is defined as: f(cid:18)2g 1.0315 3.2146 5.0000 5.0990
( ) ((cid:12) (cid:12) (cid:12) (cid:12) ) f(cid:18)3g 1.5474 4.5826 6.0000 6.7082
DNq a~;~b = (cid:12)a−−b−(cid:12)q+(cid:12)a+−b+(cid:12)q q1 (13) ff(cid:18)(cid:18)12;;(cid:18)(cid:18)23gg 11..15476445 14..50247155 25..00000000 25..28336110
B. Numerical example
Different INDs can be used for implementing the BBA Then, using the distances the similarity degrees are calculated
determinations. Here, we use a numerical example for com- according to Eq. (6), where the support coefficient is set to
paringtheintervalnumberbasedBBAdeterminationmethods (cid:11) = 5. By normalizing the similarity degrees the BBAs are
using different INDs. The BBA determination methods using obtained as listed in Table III.
different INDs are applied on a three-classes classification As the BBAs in Table III, the basic beliefs assigned to
problem. In this numerical example, we give the features’ different focal elements have small differences using GR-
rangesofdifferentclassesdirectly,asshowninFigure1,where IND compared with that using TD-IND, H-IND and Nq-
thefeature’srangeofclass1((cid:18) )is[1;4],class2((cid:18) )is[3;7] IND. For example, using GR-IND the basic beliefs assigned
1 2
and class 3 ((cid:18) ) is [5;8]. to {(cid:18) } and {(cid:18) } are 0.2552 and 0.2305, which have small
3 1 2
From the Figure 1, the interval numbers models of focal differences.UsingTD-IND,thebasicbeliefsof{(cid:18) }and{(cid:18) }
1 2
elements can be constructed, which is listed in Table I. Note are 0.4086 and 0.1289, whose difference is larger. The BBAs
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TABLEIII tively. In the Nq-IND, we have taken q = 2. The parameter
THEBBASDETERMINATEDBASEDONDIFFERENTINDS. (cid:11) in the generation of the similarity degrees in the interval
numberbasedBBAdeterminationmethod(asinEq.(6))isset
BBAs
Focalelements to 5. The Monte-Carlo classification experiments are repeated
GR-IND TD-IND H-IND Nq-IND
f(cid:18)1g 0.2552 0.4086 0.3184 0.3163 100 times with random testing samples. The effectiveness of
f(cid:18)2g 0.2305 0.1289 0.1281 0.1394 the interval number based BBA determination methods using
f(cid:18)3g 0.1546 0.0906 0.1069 0.1061 different INDs are compared using the average accuracy of
f(cid:18)1;(cid:18)2g 0.2078 0.2693 0.3184 0.3162 the 100 runs.
f(cid:18)2;(cid:18)3g 0.1519 0.1026 0.1282 0.1220
A. Experiment on artificial set
Theartificialsetgeneratedcontains3classes.Eachclasshas
determinatedbasedonH-INDandNq-INDaresimilartoeach
50 samples, and each sample has 3 features. The features of
other.
different clas(ses ar)e generated according to Gaussian distribu-
Here, we use the Pignistic probability transformation (as in tion, i.e., G (cid:22);(cid:27)2 . The standard deviations ((cid:27)) of different
Eq.(5))fortransformingtheBBAstoprobabilitiesfordecision
classes’ different features are all set as (cid:27) = 1. The mean
making. The probabilities of the testing sample belonging to
((cid:22)) settings of different classes’ different features are listed in
different classes are listed in Table IV.
Table V.
TABLEIV
TABLEV
THEPIGNISTICPROBABILITIESOBTAINEDBASEDONDIFFERENTINDS. THEMEAN((cid:22))SETTINGSOFDIFFERENTCLASSES’DIFFERENTFEATURES.
Pignisticprobabilities
Classes Mean((cid:22))
GR-IND TD-IND H-IND Nq-IND Classes
Feature1 Feature2 Feature3
CCClllaaassssss123((((cid:18)(cid:18)(cid:18)123))) 000...342513900136 000...531414341389 000...431757710749 000...431756487451 CCClllaaassssss123((((cid:18)(cid:18)(cid:18)123))) 8150 5911 1690
Intuitively, the testing sample belongs more likely to class
The features of different classes in the artificial set we
1, as shown in Fig. 1. According to Table IV, the methods
generated are shown in Figures 2–4.
using the TD-IND, H-IND and Nq-IND all can make right
classifications. According to the probabilities originated from
Feature 1
the GR-IND, the testing sample should be classified to class
2. Revisiting the BBA determinated based on GR-IND, the
basic beliefs assigned to the focal elements with single class Class 3
has the right tend, i.e., m({(cid:18) }) > m({(cid:18) }) > m({(cid:18) }).
1 2 3 Class 2
However, the Pignistic probabilities originated from the GR-
INDiscounter-intuitive,wherethebeliefsassignedtothefocal Class 1
elementswithmultipleclassesarecountedtogether.Fromthis

perspective, the BBA determinated based on GR-IND is not 2 4 6 8 10 12 14
Feature values
sogood.Inthisnumericalexample,theintervalnumberbased
methodsusingtheTD-IND,H-INDandNq-INDperformmore
Fig.2. Artificialsamples’feature1ofdifferentclasses.
properfortheBBAdeterminationthanthatusingtheGR-IND
if the decision-making is based on max of BetP.
V. EXPERIMENT Feature 2
To compare the interval number based BBA determination
Class 3
methodusingdifferentINDs,weuseMonte-Carloexperiments
on the classification of the artificial set and the iris set. The Class 2
information fusion based classification is implemented as fol-
lows.Ineachclassification,theintervalnumberbasedmethod Class 1
is used for determinating the BBA in each single feature.

2 4 6 8 10 12 14
Then these multiple BBAs are combined using Dempster’s
Feature values
combination rule as in Eq. (4). Then the combined BBA
is transformed into probabilities using Pignistic probability
Fig.3. Artificialsamples’feature2ofdifferentclasses.
transformation as in Eq. (5). The testing sample is classified
as the class which has the largest Pignistic probability. As shown in Figures 2–4, the class 3 is linearly separable
Intheexperiment,theintervalnumberbasedmethodsusing fromclass1andclass2,andclass1andclass2arenotlinearly
different INDs are used for determinating the BBAs respec- separable from each other in feature 1. Similarly, class 2 and
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class3arenotlinearlyseparablefromeachotherinfeature2, classes are the same), and all the samples are used as the
and class 1 and class 3 are not linearly separable from each testingsamples.TheresultsoftheintervalnumberbasedBBA
other in feature 3. determination methods based on different INDs are shown in
Figure 5.
Feature 3

1
0.9
Class 3
0.8
CCllaassss 12 Classification accuracy00000.....34567
0.2 GD−IND
2 4 6 8 10 12 14 0.1 THND−q−I−NIINNDDD
Feature values 0 691215 30 45 60 75 90 105 120 135
Number of train samples
Fig.4. Artificialsamples’feature3ofdifferentclasses. Fig. 5. Performances of the interval number based methods using different
INDswithdifferentscalesoftrainingsamplesonirisdataset.
In each Monte-Carlo run, we randomly select 25 samples
According to Figure 5, the methods using TD-IND, H-IND
from each class (75 samples in total) as the set of training
andNq-INDperformwellinboththecaseswithsmallnumber
samples, and the remaining samples are used as the testing
oftrainingsamplesandlargenumberoftrainingsamples.The
samples. We first classify the testing sample according to the
method using TD-IND performs the best compared with the
BBA determinated based on each single feature, respectively.
methods using other three INDs. The results of the method
Then, we combine the BBAs determinated based on the 3
using GD-IND have a counter-intuitive behavior, since its
features,andusethecombinedBBAforclassifyingthetesting
accuracy decreases with the increasing of the number of the
sample. The results of the methods based on different INDs
training samples. When the number of training samples is
are listed in Table VI.
large, the interval numbers generated can better model the
features of corresponding classes, especially, for the mixture
TABLEVI
THERESULTSOFTHEMETHODSBASEDONDIFFERENTINDS. classes’ focal elements (i.e., the overlapping range of corre-
sponding classes’ interval number models). However, as dis-
INDs Classificationcorrectrate(%) cussed in the numerical example in section IV-B, the interval
Feature1 Feature2 Feature3 Combined
numberbasedmethodusingGD-INDisnotrecommendedfor
GD-IND 44.70 64.86 42.62 80.95
determinating the BBA, especially, counting the mixture class
TD-IND 67.71 84.13 61.66 94.84
focal elements together. That is why the method using GD-
H-IND 64.66 80.24 56.01 89.66
IND performs bad when the number of training samples is
Nq-IND 65.86 81.68 55.84 91.97
large.
InTableVI,thecolumns“Feature1”,“Feature2”and“Feature
VI. CONCLUSION
3”aretheresultsofthemethodsusingdifferentINDsbasedon
In this paper, we have tested different INDs for implement-
each single features. The column “Combined” are the results
ingtheintervalnumberbasedBBAdeterminationmethod.The
obtained by combining the BBAs determinated on different
effectiveness of the BBAs are compared based on the infor-
features with Demspter’s rule of combination. According to
mation fusion based classification problems. The experiments
Table VI, the classifications of the methods using different
validate that combining the BBAs determinated using interval
INDs based on each single feature does not perform well.
number based methods with different INDs performs well
However, the BBAs determinated based on different features
for the classification problems. The methods using the TD-
reflectdifferentaspects’informationofthesamples.Byfusing
IND,H-INDandNq-INDprovidequasisimilarperformances,
the BBAs based on different features, better classification
where the one using TD-IND is the best one. Using the
performances are obtained. Comparing the results of the
GD-IND, the basic beliefs construction is not very effective.
methods based on different INDs, the method based on GD-
With GD-IND, the differences of the basic beliefs assigned to
IND performs the worst. The performances of the methods
different focal elements are small, which is not discriminant
based on TD-IND, H-IND and Nq-IND are similar, where the
enoughformakingdecisions,especially,countingthemixture
one based on TD-IND is the best. The BBA built using the
classes’ focal elements. Therefore, the method using the GD-
GD-IND is not recommended for the BBA determination.
IND is not recommended.
B. Experiment on iris set
Up to now, the interval number based BBA determination
The iris set contains 3 classes. Each class has 50 samples, methodsareimplementedonthesinglefeature.Infuturework,
and each sample has 4 features. In this experiment, we we will try to use the interval numbers for determinating
randomly select different numbers of samples as the training the BBAs on the multiple features spaces, and compare the
samples (the number of the samples selected from different effectivenessoftheonesusingdifferentINDs.Wewillexplore
983
also different decision-making strategies (i.e. DSmP, min of
d BI, etc.), and test other rules of combination as well to see
if we can improve classification performances.
ACKNOWLEDGMENT
This work was supported by the National Natural Sci-
ence Foundation (Nos. 61573275, 61671370), Grant for S-
tate Key Program for Basic Research of China (973) (No.
2013CB329405), Science and Technology Project of Shaanxi
Province (No. 2013KJXX-46), Postdoctoral Science Founda-
tion of China (No. 2016M592790), and Fundamental Re-
search Funds for the Central Universities (No. xjj2014122,
xjj2016066).
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