Critiques on some combination rules forprobability theory based on optimization

techniquesMihai Cristian FloreaInformatique WGZ inc.

Quebec, CanadaEmail: mihai-cristian.florea@wgz.ca

Eloi BosseDRDC - Valcartier, DSS Section

Quebec, CanadaEmail: eloi.bosse@drdc-rddc.gc.ca

Abstract— A crucial point in the decision-level identityfusion is to combine information in an appropriate wayto generate an optimal decision, according to the individ-ual information coming from a set of different sensors.An interesting approach was developed for the decision-level identity fusion, which use optimization techniquesto minimize an objective function which measure thedissimilarities between the combination result and the setof initial sensor reports. Several objective functions werealready proposed for the Similar Sensor Fusion (SSF) andthe Dissimilar Sensor Fusion (DSF) models. In this paper,we present these fusion methods, we raise some questionsand make some improvements, and finally we study thebehaviour of these fusion rules on several examples.Keywords: Probability theory, combination rules,Similar Sensor Fusion, Dissimilar Sensor Fusion

I. INTRODUCTION

The aim of a fusion process is to provide a resultwhich best fit the information provided by a set ofsensors. A distinction should however be made betweentwo different situations : the fusion of information pro-vided by similar sensors and the fusion of informationprovided by dissimilar sensors. From this point of view,two different fusion models can be defined :

• the Similar Sensor Fusion (SSF) model : is usedto fuse information from similar sensors, provid-ing reports on common characteristics. This fusionmodel can also be applied to fuse distinct reportsfrom the same sensor, acquired at different mo-ments. It should not decrease the uncertainty in oneor more elements from the frame of discernment,but it should only help to eliminate inconsistenciesamong reports. The sensors can only confirm eachother’s reports and the goal of this fusion model isto find a result which is most consistent with all thesensor reports.

• the Dissimilar Sensor Fusion (DSF) model :is used to fuse information from dissimilar andindependent sensors, providing reports on differentcharacteristics. Reports from these sensors can rein-force each other to decrease the uncertainty in oneor more elements from the frame of discernment.The goal of this fusion model is to find a resultwhich best represents the increased certainty in theelements of the frame of discernment, according toall the sensor reports.

More details about these two fusion models can befound in [1], and the authors show through some ex-amples the goals and the mechanisms of each of thesefusion models.

Based on this classification, each combination rule de-veloped in the last years for probability theory, evidencetheory, possibility theory or fuzzy set theory, shouldclearly be categorized for the SSF model or for the DSFmodel.

In [1], [2] three combination rules for probabilitytheory are presented. These three combination ruleswere developed according to an optimization process.Moreover, the technique used to developed the threecombination rules can be extended to an entire classof combination rules based on optimization techniques.First two techniques were developed for the SSF model,and the third technique was developed for the DSFmodel.

In this paper we wish to make an analysis (throughsome examples) of these three fusion methods developedusing some optimization techniques.

This paper is structured as follows. Section II drawsa list of the possible sensor reports. In Section III wepresent the three combination rules developed in [1], [2].Some examples are studied in Section IV. Section V isthe conclusion.

II. SENSOR REPORT FORMS

Let Ω = a1, . . . , aN denote the set of N exhaustiveand mutually exclusive hypotheses or objects. Let’sconsider each sensor report as a collection of subsets ofΩ and the associated belief or likelihood values. Let alsoconsider a degree of reliability associated to each sensorreport, denoted as Degree of Confidence (DoC), which isa non-negative number. In [1], the author presents severalparticular representations of the sensor reports.

1) probability report :

D =

DoC = w,

P (ai) = ri ∈ [0, 1], ∀i = 1, . . . , N

(1)

whereN∑

i=1

ri = 1. Usually, DoC = w ∈ [0, 1], but it is

possible to consider DoCmax = ∞. Here, ri representsthe probability of the singleton ai.

2) ratio type report :

D =

DoC = w = r1 + · · · + rL,

P (ω1) : · · · : P (ωL) = r1 : · · · : rL

(2)

where ω ⊆ Ω and r ≥ 0,∀ = 1, . . . , L. Thus,DoCmax = ∞. Here, r represents the likelihood valuebetween the subsets ω.

Note that a ratio type report can be seen as a partialknowledge of a probability type report. From a probabil-ity type report one can compute an infinity of equivalentratio type reports, which incorporate the whole knowl-edge of the probability type report. Inversely, from a ratiotype report, one can compute an infinity of probabilitytype reports, which are not necessarily equivalent.

3) partition type report

D =

DoC = w = r1 + · · · + rL,P (ω1) : · · · : P (ωL) = r1 : · · · : rL

(3)

where ω ⊆ Ω and r ≥ 0,∀ = 1, . . . , L with ωi∩ωj =∅ for i = j, and ω1 ∪ · · · ∪ ωL = Ω. Thus, DoCmax =∞. Here, r represents the likelihood value between thesubsets ω.

Note that the partition type report is a special caseof the ratio type report. From probabilistic point ofview, the comparisons expressed in the ratio-type reportand partition-type report can be expressed in variousforms using intersections and unions of the ω’s withoutchanging the probabilities of the original subsets. Sucha comparison can also be broken into multiple compar-isons. For example, P1(a1) : P1(a2) : P1(a3) = 1 : 1 : 1can be expressed as P1(a1) : P1(a2 ∨ a3) : P1(a3) =1 : 2 : 1. Although the probabilistic interpretationis the same, an optimal representation should be used

when computing the fusion result. In [1], the authorconsiders the optimal representation as the one havingno redundancy in it, but there is no constraint saying thatall the elements from the frame of discernment shouldbe part of the report. In this situation, the sensor reportsD1 and D2

D1 =

w = 1,P1(a1) : P1(a3) = 0.9 : 0.1

D2 =

w = 1,P2(a1) : P2(a2) : P2(a3) = 0.9 : 0 : 0.1

should not be considered as different sensor reports ifthe two are defined over the same frame of discernment.

We propose then to define the optimal representationfor the sensor reports to be the one with no redundancyin it, and moreover, all the elements from the frameof discernment have to be part of the report even ifsome null likelihood values have to be added to thereport representation. For example, let Ω = a1, a2, a3be a three element frame of discernment. The sensorreport D1 is not an optimal representation and shouldbe replaced by the sensor report D2.

III. COMBINATION RULES FOR PROBABILITY THEORY

BASED ON OPTIMIZATION PROCESSES

In this section we present three combination rules forprobability theory, developed using some optimizationtechniques which [1], [2]. Two combination rules weredesigned for the SSF model and one for the DSF model.

First, we introduce the general principle for the threecombination rules. Let Dk, k = 1, . . . , K be a set ofsensors reports to combine and let Df be the fusedreport. The fusion result is a probability distributiongiven by the set of probabilities Pf (ai) = pi,∀i =1, . . . , N such that

∑Ni=1 pi = 1, which best fits the

knowledge from the given sensor reports. A global costfunction CfK

is defined to measure the discrepancybetween the fusion result and the K sensor reports.

The three combination rules have to minimize theglobal cost function CfK

(P ), according to the con-straints imposed by a final probability function : pi ≥0,∀i = 1, . . . , N and

∑Ni=1 pi = 1. The DoC of the

fused information is also computed. These combinationrules are in fact constrained optimization problems whichcan be solved by existent interior point techniques givenin [3] and [4].

The fusion methods can operate in both sequentialfusion mode and batch fusion mode. However, the twofusion modes are providing different results. The com-plexity of the two fusion modes (batch and sequential) isidentical since the optimization problem is not dependentof the number of sensors to be fused, but only of the

2

cardinality of the frame of discernment. Thus, when theorder of the reports to be fused is not important, andthe same weight should be assign to each report, thebatch mode has to be considered. For the combinationof streaming data, the last report is more important thanthe first reports, and the sequential fusion mode shouldbe considered.

A. Combination rules for the SSF model

The aim of the SSF model is to eliminate inconsisten-cies among reports. Thus, a set of partial cost functionsCk(P ) is first defined to measure the inconsistenciesbetween the ideal fusion result Df and each sensor reportDk. Then, a global cost function is defined as a weightedsum of the set of Ck(P ), with weighting coefficientswhich are functions of wk. In the SSF model, wf isdetermined by the DoC values of all the reports andhow strong each report confirms with Df , by the meanof the cost functions Ck(P ). Specifically, wf should bethe sum of contributions from each wk. The strongerDk agrees with Df , the more wk should contribute towf . In the extreme case that Dk fully agrees with Df ,all the confidence associated with Dk should contributeto Df . In contrast, if Dk completely disagree with Df ,the fusion of Dk does not affect the confidence of thefusion result at all. In other words, no part of wk shouldbe added to wf .

1) Combination rule based on a convex quadraticoptimization (reviewed): The convex quadratic (CQ)fusion method was introduced first in [2] and later wasreviewed in [1]. Consider K sensors, each exploringsome common characteristics of an object and providingratio type reports (Dk). Let Ck(P ) be a set of convexquadratic cost functions, given by :

Ck(P ) = ck(P )2Lk∑=1

(rk

wk− Pf (ω

k)ck(P )

)2

(4)

where ck(P ) = Pf (ω1k) + · · · + Pf (ωLk

k ) and let theglobal cost function be :

CfK(P ) =

K∑k=1

(wk)2Ck(P ) (5)

Thus, the fusion result using the CQ fusion methodis obtained by minimizing the global cost function fromEquation (5), using the constraint given by a probabilitydistribution

∑Ni=1 pi = 1, with pi ≥ 0, ∀i.

The DoC of the fusion result is given in [1] as follows:

wf =K∑

k=1

wk · Ckmax− Ck(Df )

Ckmax

(6)

where Ck(Df ) is the value of the cost function Ck(P )evaluated using the probability assignments of the fusionresult Df . Ckmax

is given by

Ckmax= max

i = 1, . . . , N :ri = wk, rj = 0

∀ j = i

ck(P )2Lk∑=1

(r

wk− Pf (ω

k)ck(P )

)2

When all the K sensors assign probabilities to all thesingletons of Ω, each sensor report Dk can be writtenas a probability sensor report and it is straightforward toshow that :

Pf (ai) =∑K

k=1(wk)2rik∑K

k=1(wk)2, i = 1, . . . , N (7)

Thus, the optimal fusion result is obtained by aweighted average of the sensor reports using (wk)2 asweighting coefficients.

One drawback of the CQ fusion method definedpreviously, is that the results are dependent of whichrepresentation for the sensor reports is used. The costfunction from Equation (4) is dependent of the repre-sentation of the sensor reports. We propose to use theCQ fusion method only with the optimal representationof the sensor reports described in Section II. Example 1shows the differences between the results obtained usingthe various representations of sensor reports.

2) Combination rule based on Kullback-Leibler’s dis-tance: The combination rule based on Kullback -Leibler’s distance was introduced first in [1]. We presenthere this fusion method which we denote as the KLfusion method. Consider K sensors each exploring somecommon characteristics of an object and providing par-tition type sensor reports Dk.

Let Ck(P ) be a set of cost functions, given by theKullback-Leibler’s measure of cross entropy between thefinal probability distribution and the probability distribu-tion of the sensor report Dk:

Ck(P ) =Lk∑=1

rk

wkln

rk

wk

Pf (ωk)

(8)

and let the global cost function be :

CfK(P ) =

K∑k=1

wkCk(P ) (9)

Thus, the KL fusion result is obtained minimizingthe global cost function from Equation (9), using theconstraint given by a probability distribution

∑Ni=1 pi =

1, with pi ≥ 0, ∀i.The DoC associated with the fusion result Df is

3

defined as follows:

wf =K∑

k=1

wke−Ck(Df ) (10)

It can be shown that the expression given by Equation(10) as defined in [1], cannot always be computed,particularly, in situations when a null likelihood value r

k

is associated to a subset ωk in the Dk sensor report, and

the fused result associates also a null probability to thesubset ω

k. Equation (10) should be reviewed so it can beused in any situation. In a sequential process, if the DoCvalue cannot be updated, the global cost function givenin Equation (8) cannot be computed for future steps,and thus, the KL fusion method becomes useless (seeExample 3 from Section IV).

In the special case where all the K sensors assignprobabilities to all the singletons of Ω, each sensor reportDk can be written as a probability sensor report as inEquation (1) and it is straightforward to show that :

Pf (ai) =∑K

k=1 wkrik∑K

k=1 wk

, i = 1, . . . , N (11)

Thus, the optimal fusion result is obtained by aweighted average of the sensor reports using wk asweighting coefficients.

Note that the solution provided by the CQ and KLfusion methods is not always unique. In this situation,the fusion result can be represented as a partition typereport.

B. Combination rule based on the analytic center forthe DSF model (reviewed)

The aim of the DSF model is to best represent theincreased certainty in the elements of the frame ofdiscernment, according to all the sensor reports. ConsiderK independent sensors observing an object. Each sensorexplores some different characteristics of the object andsends its own identity declaration in a sensor report tothe fusion center. A global cost function, defined in [1],involves only the probability distributions of each sensorand the final probability distribution.

CfK(P ) =

K∑k=1

wk

N∑i=1

1rik

pi −N∑

i=1

ln pi (12)

One drawback of the formulation proposed in [1] isthat it cannot be used if the probability distributions tobe combine have null probabilities associated to somesingletons (ri

k = 0 for some k and i). We review thisformulation for the analytic center fusion method and

we propose the global cost function :

CfK(P ) =

K∑k=1

wk

N∑i=1

rik =0

1rik

pi −N∑

i=1

ln pi (13)

The fusion solution is obtained solving the convexoptimization problem which is to minimize Equation(13), using the constraint given by a probability distri-bution

∑Ni=1 pi = 1, with pi ≥ 0, ∀i. This problem is

a special case of the analytic center problem in linearprogramming presented in [4]. We denote this fusionmethod as AC. The DoC wfK

associated with fusionresult DfK

is defined as :

wfK= 1 −

K∏k=1

(1 − wk) (14)

The analytic center fusion method can operate in boththe sequential and batch fusion modes.

IV. EXAMPLES

In this section we consider some simple examples totest the fusion methods (CQ, KL and AC) presented inSection III and we analyze the results. A comparisonwith the basic average (AV) is made for the SSF fusionmethods. Ratio type reports and partition type reportsare first transformed into probability type reports, usingthe pignistic transformation from [5], before using theAV fusion method.

The first example shows the differences between theresults obtained using the optimal representation andnon-optimal representations for the sensor reports.

Example 1 Let Ω = a1, a2, a3 be a frame of discern-ment with three elements and let D1 and D2 be twoconflictual but similar and independent sensor reports(example from [6]).

D1 =

w = 1,P1(a1) : P1(a3) = 0.9 : 0.1

D2 =

w = 1,P2(a2) : P2(a3) = 0.9 : 0.1

Let denote by D∗

1 and D∗2 the optimal representations

of D1 and D2 respectively in the frame of discernmentΩ. Table I shows the results when the CQ combinationrule is performed between the sensor reports D1 (orD∗

1) and D2 (or D∗2). We can remark that the CQ fusion

method is performing differently, according to the typeof representation for each sensor report. A comparisonwith the simple average is also presented.

We remark that not only the results are differentaccording to the type of representation used in the

4

combination process, but the conclusions can changeradically.

Combination Pf (a1) Pf (a2) Pf (a3) DoC

CQ(D1,D2) 0.4737 0.4737 0.0526 2CQ(D∗

1 ,D2) 0.9217 0.0298 0.0485 1.29CQ(D1,D∗

2) 0.0298 0.9217 0.0485 1.29CQ(D∗

1 ,D∗2) 0.4500 0.4500 0.1000 1.33

KL 0.4500 0.4500 0.1000 NaNAV 0.4500 0.4500 0.1000 N/A

TABLE I

EXAMPLE 1. COMBINATION USING DIFFERENT REPRESENTATIONS

OF SENSOR REPORTS

Take for instance that when combining the sensorreport D1 under his optimal representation D∗

1 with thesensor report D2, the result is providing singleton a1

with higher probability than provided in D1 even if a1 isabsent in the sensor report D2. This is a counter-intuitivebehaviour for the SSF fusion model, since there shouldbe no reinforcement in the singleton beliefs. We canremark that when combining the optimal representations,a simple average is performed (as stated in Section III-A.1). So, to avoid any counter-intuitive behaviours, thesolution is to use the optimal representation proposed inSection II. We notice that the KL and AV fusion methodsprovide the same results as the CQ fusion method appliedto the optimal representation of the sensor reports.However, the KL fusion rule cannot compute the DoCvalue (see discussion in Section III-A.2).

In the next example we show the capability for the CQand KL combination rules to extract the real informationfrom partial probability distributions.

Example 2 Let Ω = a1, a2, a3. Suppose D1, D2 andD3 are three similar and independent partition typesensor reports :

D1 =

w1 = 1,P1(a1 ∨ a2) : P1(a3) = 0.8 : 0.2

D2 =

w2 = 1,P2(a1) : P2(a2 ∨ a3) = 0.2 : 0.8

D3 =

w3 = 1,P3(a1 ∨ a3) : P3(a2) = 0.4 : 0.6

Note that D1, D2 and D3 are partial knowledges of thesame probability distribution P0(a1) = 0.2, P0(a2) =0.6 and P0(a3) = 0.2. Since the three sensor reportsare provided by similar sensors, we combine them usingthe rules developed for the SSF model. A combinationrule for the SSF model should be able to fuse the partial

knowledge from the three sensor reports and find the realprobability distribution. Table II compares the resultsobtained using the combination rules CQ, KL and AV.

Pf (a1) Pf (a2) Pf (a3) DoCCQ 0.2000 0.6000 0.2000 3KL 0.2000 0.6000 0.2000 3AV 0.2667 0.4666 0.2667 N/A

TABLE II

EXAMPLE 2 : COMPARISON FOR THE SSF MODEL

We can remark that the result provided by the CQand KL combination rules recovers the real probabilitydistribution, which is not the case for the AV fusion rule.

We should notice that the CQ and KL combinationrules find the probability distribution which minimizethe distance between itself and the initial probabilitydistributions. In this example, the techniques based onthe optimization process (CQ and KL) perform well, andthe results are satisfactory.

In the following example we show how the CQ andKL combination techniques can have a counter-intuitivebehaviour in a sequential fusion mode.

Example 3 Let Ω = a1, a2, a3, a4, a5. Suppose D1,D2, . . . , D5 are five similar and independent ratio typereports :

D1 =

w1 = 1,P1(a1 ∨ a2) : P1(a3 ∨ a4 ∨ a5) = 0.6 : 0.4

D2 =

w2 = 1,P2(a1 ∨ a4) : P2(a2) : P2(a3 ∨ a5) == 0.4 : 0.2 : 0.4

D3 =

w3 = 1,P3(a1 ∨ a2 ∨ a5) : P3(a3 ∨ a4) = 0.6 : 0.4

D4 =

w4 = 1,P4(a1 ∨ a4) : P4(a2 ∨ a3) = 0.7 : 0.3

D5 =

w5 = 1,P5(a1 ∨ a2 ∨ a3) : P5(a5) = 0.5 : 0.5

Note that D1, D2 and D3 are partition type reports,

which is not the case for D4 and D5. Moreover, the sen-sor reports D1, D2 and D3 are presented by an optimalrepresentation as defined in Section II, while the sensorreports D4 and D5 are not in the optimal representation.Let D∗

4 and D∗5 be the optimal representations of the

sensor reports D4 and D5 respectively. D∗4 and D∗

5 arepartition type reports and will be used with the KL fusionmethod, instead of D4 and D5 respectively. We considerthat these reports do not arrive at the fusion center in the

5

same time : D1 arrives first and D5 arrives last. In thisexample, we study a sequential process, but at each stepwe combine all available sensor reports in a batch mode,which is equivalent to the fusion of streaming data.

First, we study the CQ combination rule in a sequen-tial process using the batch mode, when the non optimalrepresentations D4 and D5 are used. The results arepresented in Table III.

The probability distribution Pf2 is the best approxi-mation of the two partial knowledges given by the sensorreports D1 and D2. We notice that the fusion resultgives a null probability for singletons a4 and a5 evenif the probability of these two singletons were never nullin the initial sensor reports. This result is contrary toour intuition. When the first three sensor reports arecombined, the fusion result keeps a null probability forsingletons a4 and a5 even if the probability of these twosingletons were never null in the initial sensor reports.This result is also contrary to our intuition. Next, werealize the combination of the first four sensor reports.We remark that the probability assign to singletons a1

and a4 in D4 is greater than their probabilities fromDf3. Their probabilities in Df4 should be grater thanthose in Df3, but it is not the case for singleton a1.

Moreover, even if the singleton a5 has no probabilityin Df3, and there is no probability assign to it in D4,its probability is updated in Df4 and it increases evenmore than the probability of singleton a4, to which thesensor report D4 assign a non null probability. Finally,we realize the combination of all five sensor reports andthe results (Pf5). The probability of the singleton a4 isincreasing from Pf4 to Pf5, even if in the sensor reportD5 the singleton a4 has a null probability. Moreover,singleton a4 becomes more probable than singleton a1

which was the favourite until now, and which had a non-null probability in the D5 sensor report.

We can conclude to a completely non-intuitive be-haviour of the CQ fusion method when non optimalrepresentations for the sensor reports are used in asequential fusion process.

Next, let consider the combination of the five sensorreports with optimal representations (D1, D2, D3, D∗

4

and D∗5), in a sequential process, using the batch fusion

mode. The CQ, KL and AV fusion methods are analyzedand Table IV shows the result for this example. Somecounter-intuitive behaviours observed when fusing theD1 to D5 sensor reports with the CQ fusion methodvanished or decreased. However, we notice that between

Information to be combined Pf (a1) Pf (a2) Pf (a3) Pf (a4) Pf (a5) DoC

D1 ⊕ D2 0.4000 0.2000 0.4000 0.0000 0.0000 2D1 ⊕ D2 ⊕ D3 0.4000 0.2000 0.4000 0.0000 0.0000 3

D1 ⊕ D2 ⊕ D3 ⊕ D4 0.3970 0.1396 0.2129 0.1237 0.1268 3.93D1 ⊕ D2 ⊕ D3 ⊕ D4 ⊕ D5 0.2246 0.2112 0.0353 0.2481 0.2808 4.83

TABLE III

EXAMPLE 3. SSF FUSION MODEL : SEQUENTIAL PROCESS USING THE BATCH MODE AND THE CQ COMBINATION RULE.

Information to be combined Fusion method Pf (a1) Pf (a2) Pf (a3) Pf (a4) Pf (a5) DoC

D1 ⊕ D2

batch CQ 0.4000 0.2000 0.4000 0.0000 0.0000 2batch KL 0.4000 0.2000 0.4000 0.0000 0.0000 2

AV 0.2500 0.2500 0.1667 0.1667 0.1667 N/A

D1 ⊕ D2 ⊕ D3

batch CQ 0.4000 0.2000 0.4000 0.0000 0.0000 3batch KL 0.4000 0.2000 0.4000 0.0000 0.0000 3

AV 0.2333 0.2333 0.1778 0.1778 0.1778 N/A

D1 ⊕ D2 ⊕ D3 ⊕ D∗4

batch CQ 0.4553 0.1211 0.2737 0.1026 0.0474 3.94batch KL 0.4500 0.1500 0.3000 0.1000 0.0000 NaN

AV 0.2625 0.2125 0.1708 0.2209 0.1333 N/A

D1 ⊕ D2 ⊕ D3 ⊕ D4 ⊕ D∗5

batch CQ 0.3595 0.1397 0.1428 0.1516 0.2065 4.73batch KL 0.4225 0.1259 0.2178 0.0243 0.2095 NaN

AV 0.2433 0.2033 0.1700 0.1767 0.2067 N/A

TABLE IV

EXAMPLE 3. SSF FUSION MODEL : COMBINATION OF INFORMATION IN A SEQUENTIAL PROCESS USING THE BATCH MODE

6

Information to be combined Pf (a1) Pf (a2) Pf (a3) Pf (a4) Pf (a5) DoC

(D1 ⊕ D2) ⊕ D3 0.4000 0.2000 0.4000 0.0000 0.0000 3((D1 ⊕ D2) ⊕ D3) ⊕ D∗

4 0.4273 0.1727 0.3727 0.0273 0.0000 3.91(((D1 ⊕ D2) ⊕ D3) ⊕ D∗

4) ⊕ D∗5 0.4107 0.1561 0.3561 0.0360 0.0410 4.69

TABLE V

EXAMPLE 3. SSF FUSION MODEL : COMBINATION OF INFORMATION IN A SEQUENTIAL PROCESS USING THE SEQUENTIAL MODE AND THE

CQ FUSION METHOD

the steps 2 and 3, there is an increasing value of theprobability of a5 even if a5 has a null probability inthe sensor report D∗

4 . The DoC value for the KL fusionmethod cannot be computed after the 3rd fusion step.

When the sequential fusion mode is used to fuse theD1 to D5 sensor reports in the optimal representation,the KL fusion method cannot be used. During the combi-nation of (D1⊕D2)⊕D3 the DoC value wf3 cannot becomputed, which blocks the sequential fusion process.The results obtained using the CQ fusion method arepresented in Table V.

In conclusion of this example, the CQ fusion methodhas a counter-intuitive behaviour in such a sequentialprocess, when the combination is realized by the batchfusion mode at each fusion step. We notice that theuse of the optimal representation for the senor reportsreduces the counter-intuitive behaviour, but does noteliminate it. The KL fusion method has a more intuitivebehaviour than the CQ fusion method. However, whenfusing only the first three sensor reports, the probabilitydistribution obtained using the KL combination rule isidentical to the one obtained using the CQ combinationrule, and both provide some counter-intuitive results(Pf (a4) = Pf (a5) = 0 even if the initial sensor reportswere not against a4 and a5). We do not recommend theuse of these fusion methods for the characterization ofa sequential fusion process.

In the following example we show how the AC fusionmethod acts in a high conflicting situation.

Example 4 Let Ω = a1, a2, a3 be a frame of discern-ment with three elements and let D1 and D2 be twodissimilar and independent sensor reports, in conflict,which are given in Example 1. The AC combinationrule for the DSF model is supposed to work withprobability sensor reports, as stated in Section III-B.The AC combination rule, as defined first in [1], cannotbe used to combine the D1 and D2 sensor reports,because they are not defined by complete probabilitydistributions. It cannot be used neither to combine theD∗

1 and D∗2 representations of the sensor reports D1

and D2 respectively because some null probabilities

are assigned. However, with the new formulation andrestriction given in Equation (13), the result obtainedusing the AC combination rule is given in Table VI.Note that this result is very close to the one given bythe CQ combination rule developed for the SSF model,which is a normal behaviour for the combination of highconflicting information.

Combination Pf (a1) Pf (a2) Pf (a3) DoC

AC(D1,D2) 0.4762 0.4762 0.0476 1

TABLE VI

EXAMPLE 4. REVIEWED AC COMBINATION RULE FOR THE DSF

FUSION MODEL

Example 5 Consider the following independent and dis-similar sensor reports :

D1 =

w1 = 1P1(a1) : P1(a2) : P1(a3) = 0.8 : 0.1 : 0.1

D2 =

w2 = 1P2(a1) : P2(a2) : P2(a3) = 0.09 : 0.51 : 0.4

D′2 =

w3 = 1P3(a1) : P3(a2) : P3(a3) = 0.11 : 0.49 : 0.4

Note that D2 and D′2 sensor reports are very close. Ac-

cording to the continuity principle, when D1 is combinedwith D2 and D′

2 respectively, the results should be closetoo. Table VII shows the fusion results when the ACfusion method is used to combine these sensor reports.

Combination Pf (a1) Pf (a2) Pf (a3) DoC

AC(D1,D2) 0.3220 0.3697 0.3083 1AC(D1,D′

2) 0.4930 0.2682 0.2388 1

TABLE VII

EXAMPLE 5. AC COMBINATION RULE FOR THE DSF FUSION

MODEL

We notice that the two results are very different. Thus,we conclude that the continuity property is not satisfiedand that the AC fusion method is very sensitive tochanges, especially when the sensor reports to be fusedare in conflict. Moreover, when the Bayesian inference is

7

used, Singleton a1 is considered as the best probable inboth situation, which is not the case when combining D1

with D2 using the AC fusion method. We showed here acounter-intuitive behaviour of the AC fusion method.

V. CONCLUSION

In this paper we presented an interesting approachto combine ratio type and partition type reports and ina more general way probability type reports. A cleardistinction is made between the Similar Sensors Fusionmodel and Dissimilar Sensors Fusion model and thegoals for each model is presented. We analyze threecombination rules for probability theory which use op-timization techniques. Two combination rules are basedon a convex quadratic optimization and on the Kullback-Lieber’s distance respectively, which were developedfor the SSF model, and a combination rule is basedon the analytic center problem, which was developedfor the DSF model. One of the strengths of the twofusion methods for the SSF model is that they canwork with partial probability distributions. These fusionmethods are capable to recover a probability distributionfrom multiple partial knowledges of this distribution.However, as shown through some examples, the CQ andKL combination rules fail to provide intuitive behavioursin a sequential fusion process. We also proved thatthe AC combination rule is very sensitive to small

changes, and thus do not respect the continuity prin-ciple. We showed that these three fusion methods arenot well-characterized in some particular situations andwe tried to improve the fusion methods in some ofthese situations. Future works should try to improve thefusion methods presented here, try other cost functionswhich could be more appropriate according to particularapplications, and analyze more complex scenario-tests toprove the utility or the uselessness of the fusion methodsbased on optimization techniques in the identificationproblems.

REFERENCES

[1] L. Li, “Data fusion and filtering for target tracking and identifica-tion,” Ph.D. dissertation, McMaster University, 2003.

[2] L. Li, Z.-Q. Luo, K. M. Wong, and E. Bosse, “Convex optimizationapproach to identity fusion for multisensor target tracking,” IEEETrans. System, Man, and Cybernetics, Part. A, vol. 31, no. 3, pp.172 – 178, 2001.

[3] C. Roos, T. Terlaky, and J.-P. Vial, Theory and Algorithms forLinear Optimization: An Interior Point Approach. John Wiley &Sons, 1997.

[4] Y. Ye, Interior Point Algorithms: Theory and Analysis. Wiley,New York, 1997.

[5] P. Smets, “Constructing the pignistic probability function in a con-text of uncertainty,” Uncertainty in Artificial Intelligence, vol. 5,pp. 29–39, 1990.

[6] L. A. Zadeh, “Review of Shafer’s Mathematical Theory of Evi-dence,” AI Magazine, vol. 5, pp. 81–83, 1984.

8