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Network‐based Hard/Soft Information Fusion:  Soft Information and its Fusion

Ronald R. Yager , Iona CollegeTel. 212 249 2047, E‐Mail: yager@panix.com

Objectives:• Support development of hard/soft information fusion• Develop methods for the aggregation of uncertain information• Provide formalisms for the representation  and modeling of soft information DoD Benefit:• Better use of available information

Scientific/Technical Approach• Fuzzy Set Theory• Monotonic Set Measure• Dempster Shafer Theory• Mathematical theory of aggregation•Computing with Words

Accomplishments• Poss‐Prob Fusion via Conditioning• Querying Under Uncertainty• Modeling Imprecise Language• Set measure RepresentationChallenges• Mixed uncertainty mode fusion• Complexity of Soft information

Computing with WordsComputing with Words

Representation(Translation)

Fusion

Inference Reasoning

Retranslation

Soft Information

HardInformation

FusionInstructions

Focus of ResearchIona College

Our focus is on the development of new

knowledge and fundamental directions and

understandings in the process of hard/soft

information fusion. This includes the

modeling of various types of information as

well as the development of technologies for

the aggregation and fusion of information

Focus of ResearchIona College

Connection with Teammates

• Scoring functions for data association

• Graph matching technology

• Modeling human observer information

• Multi-modal information fusion

• Numeric and symbolic processing

• Expertise in fuzzy and possibilistic approach

Publication ListIona College

Journals•Yager, R. R., "A measure based approach to the fusion of possibilistic and probabilistic uncertainty," Fuzzy Optimization and Decision Making 10, 91-113, 2011.•Yager, R. R., "On the fusion of imprecise uncertainty measures using belief structures," Information Sciences 181, 3199-3209, 2011.•Yager, R. R., "Reasoning with doubly uncertain constraints," International Journal of Approximate Reasoning 52, 554-561, 2011•Yager, R. R., "Lexicographic ordinal OWA aggregation of multiple criteria," Information Fusion 11, 374-380, 2010•Yager, R. R., "Criteria satisfaction under measure based uncertainty," Fuzzy Optimization and Decision Making 9, 307-331, 2010.•Yager, R. R., "Cumulative distribution functions, p-boxes and decisions under risk," International J. of Knowledge Engineering and Soft Data Paradigms 2, 275-283, 2010.•Yager, R. R., "Validating criteria with imprecise data in the case of trapezoidal representations," Soft Computing Journal 15, 601-612, 2011•Yager, R. R. and Rybalov, A., "Bipolar aggregation using the uninorms," Fuzzy Optimization and Decision Making 10, 59-70, 2011•Yager, R. R., ”On possibilistic and probabilistic information fusion,”International Journal of Fuzzy Systems Applications 1 (3), 1-14. 2011

Publication List (2)Iona College

Conferences•Rickard, T., Aisbett, J., Yager, R. R. and Gibbon, G., "Fuzzy weighted power means in evaluation decisions," Proceedings of the World Conference on Soft Computing, San Francisco State University, California, Paper #100-461, 2011•Yager, R. R., "On the fusion of possibilistic and probabilistic information in biometric decision-making," IEEE Workshop on Computational Intelligence in Biometrics and Identity Management, at SSCI, 109-114, 2011Manuscripts• Yager, R. R., "Conditional approach to possibility-probability fusion," Technical Report #MII-3021 Machine Intelligence Institute, Iona, College, New Rochelle, NY•Yager, R. R. and Filev, D. P., "Using Dempster-Shafer structures to provide probabilistic outputs in fuzzy systems modeling," Technical Report #MII-3110 Machine Intelligence Institute, Iona, College, New Rochelle, NY, 2011.•Yager, R. R., "Dempster-Shafer structures with general measures," Technical Report #MII-3016 Machine Intelligence Institute, Iona, College, New Rochelle, NY, 2010

Publication List (3)Iona College

Articles in Books• Yager, R. R., "Human focused summarizing statistics using OWA operators," In Scalable Fuzzy Algorithms for Data Manmagement and Analysis, A. Laurent and Lesot, M-J. (Eds.), Information Science Reference, Hershey, PA, 238-253, 2010•Yager, R. R., "Learning methods for evolving intelligent systems," in Evolving Intelligent Systems: Methodology and Applications, edited by Angelov, P., Filev, D. and Kasabov, N., Wiley: New York, 1-19, 2010.•Yager, R. R., "Information fusion with the power average operator," In Preferences and Decisions, Greco, S., Marques Pereira, R. A., Squillante, M., Yager, R. R. and Kacprzyk, J. (eds), Springer: Berlin, 397-414, 2010.•Yager, R. R., "Partition measures for data mining," In Advances in Machine Learning, Vol. I, Koronacki, J., Ras, Z. W., Wierzchon, S. T. and Kacprzyk, J. (Eds.), Springer: Berlin, 299-319, 2010•Edited Books•Bouchon-Meunier, B., Magdalena, L., Ojeda-Aciego, M., Verdegay, J.-L. and Yager, R. R., Foundations of Reasoning Under Uncertainty, Springer: Heidelberg, 2010.•Greco, S., Marques Pereira, R. A., Squillante, M., Yager, R. R. and Kacprzyk, J., Preferences and Decisions, Springer: Heidelberg, 2010•Yager, R. R., Kacprzyk, J. and Beliakov, G., Recent Developments in the Ordered Weighted Averaging Operators: Theory and Practice Springer: Berlin 2011

Project Statistics and SummaryIona College

Students supported:-# of undergraduate and graduate students 0-# of post-doc and faculty members 1-# of degrees awarded (MS, PhD) 0Publications:- Journal papers -9- Conference papers - 2- Manuscripts -3- Book and book chapters - 7Technology Transitions:- Patents (disclosures) - noneAwards:-International Fuzzy Systems Association Award for 2011-Naval Research Lab Publication Award

Hard Information ⇒ Probabilistic

Soft Information ⇒ Possibilistic

Requirement for technology that can fuse Probabilistic and Possibilistic Information

FUSING HARD AND SOFT INFORMATION

Focus of ResearchIona College

Focus of ResearchIona College

Conditioning Approach to

Possibility-Probability Fusion

Focus of ResearchIona College

• Assume V has domain X = {x1, …, xn}

• Two sources of informationProbability distribution P: P(xj) = pjPossibility distribution Π: Π(xj) = πj

• We shall obtain from the two pieces ofinformation a probability distribution Q basedon a conditioning of P by Π.

Focus of ResearchIona College

• Here we make use of Zadeh's ideasrelating fuzzy sets and possibilitydistributions

• We associate with the possibilitydistribution Π a fuzzy subset F of X such foreach xj ∈ X, F(xj) = Π(xj) = πj

Focus of ResearchIona College

Using the subset F we condition ourprobabilistic information P with ourpossibilistic information and we now obtainthe probability distribution Q such that forany xj we have

Q(x j) = P(x j / F) =

P({x j}Ê∩ÊFÊ)

P(F)

Focus of ResearchIona College

We now recall that the probability of a

fuzzy subset F is the expected value of its

membership function

P(F) = F(x j)Ê⋅Êpj

jÊ=Ê1

n∑

Focus of ResearchIona College

• We note that {xj} ∩ F is fuzzy subset of X

• Hence P({xj) ∩ F) = pj F(xj).

{x j} ∩F = {

F(x j)

x j}

Focus of ResearchIona College

Combining these probabilities we get

Q(x j)Ê=Êpj F(x j)

F(xk )pkk=Ê1

n∑

=pj π j

πkpkkÊ=Ê1

n∑

Here then Q is the probability distribution that results from combining our two sources of information

Focus of ResearchIona College

Furthermore for any subset B, fuzzy orcrisp, we have

Q(B)Ê=Ê B(x j)jÊ=Ê1

n∑ ÊQjÊ=Ê

pj π j B(x j)jÊ=Ê1

n∑

pj π jjÊ=Ê1

n∑

Focus of ResearchIona College

Example

• X={x1, x2, x3, x4} p1 = 0.3, p2 = 0.2, p3 = 0.4, p4 = 0.1π� = 1, π2 = 0.6, π3 = 0.8, π4 = 0.2

• p1π1 = 0.3, p2π2 = 0.12, p3π3 = 0.32

p4π4 = 0.02

• Σ pjπj = 0.76.

Focus of ResearchIona College

Q(x j) =

pj π j0.76

We get:

Q(x1) = 0.39. Q(x2) = 0.16, Q(x3) = .0.42, Q(x4) = 0.03.

If B = {x1, x3} then Q(B) = 0.81

Here then with

Focus of ResearchIona College

Dempster-Shafer View of

Possibility-Probability Fusion

Focus of ResearchIona College

Uncertainty Representation in D-S Framework

• Use belief structure

• A collection Fj of subsets of X called focal elements

• Mapping m which associates with each focal element a value m(Fj) ∈ [0, 1] such that ∑jm(Fj) = 1.

Focus of ResearchIona College

Fusion of Information in D-S Framework

• Uses Dempster's rule• m1 and m2 are two belief structures with focal elements Ei and Fj respectively• Fusion is m = m1 ⊗ m2 its focal elements are all Ei ∩ Fj ≠ ∅ and

m(Ei ∩Fj) =mi(Ei)Ê⋅Êm2(Fj)

m(Ei)Ê⋅Êm2(Fj)EiÊ∩ÊFjÊ≠Ê∅

∑

Focus of ResearchIona College

Representation of Probability Distribution

in D-S

• Bayesian belief structure m1

• Focal elements are singletons Ei = {xi}

• m1 (Ei) = pi

Focus of ResearchIona College

Representation of Possibility ∏ Distribution in D-S

• Nested belief structure m2

• Assume elements indexed with πi ≥ πj if i < j

• Focal elements Fj = {x1, …, xj}, for j = 1 to n

(Nested: Fj ⊂ Fj+1) )

• m2(Fj) = πj - πj + 1

Focus of ResearchIona College

Fused Belief Structure

• Bayesian belief structure m

• Focal elements are singletons Ei = {xi}

•

m(Ei) =piπi

piπiiÊ=Ê1

n∑

Focus of ResearchIona College

Measure Based Approach to the

Representation of Uncertain

Information

Focus of ResearchIona College

Definition of a Fuzzy Measure

A fuzzy measure on X is a mapping μ: 2X → [0, 1] such that

1) μ(Ø) = 02) μ(X) = 1 (Normality Condition)3) μ(A) ≥ μ(B) if B⊆ A (Monotonicity)

It associates with subsets of X a number in the unit interval

Focus of ResearchIona College

Modeling Uncertain Information Using a Measure

• Assume V is variable with domain X

• Assume A is subset of X

•μ(A) indicates the anticipation of finding the value of V in A

Focus of ResearchIona College

The Fuzzy Measure has the Capability

of modeling in a unified framework

many different types of knowledge

about the value of a variable

Focus of ResearchIona College

•Certain Knowledge V = x*μ(B) = 1 if x* ∈ B μ(B) = 0 if x* ∉ B

• Probability Distribution

μ({xj}) = Pj Σ Pj = 1 μ(A ∪ B) = μ(A) + μ(B) if A ∩ B = ∅

• Possibility Distributionμ({xj}) = Πj Max[Πj]= 1μ(A ∪ B) = Max(μ(A), μ(B))

Focus of ResearchIona College

Fuzzy Measures Closed Under

Aggregation Operations Needed

for Multi-Source Information

Fusion

Focus of ResearchIona College

Definition: G is an aggregation function ofq arguments if G: [0, 1]q → [0, 1] and1. G(0, 0, …, 0) = 0, 2. G(1, 1, …, 1) = 1

3. G(a1, …aq) ≥ G(b1, …bq) if all aj ≥ bj

Theorem: Assume μj are q fuzzy measureson X. Then μ defined such that

μ(A) = G(μ1(A),…., μq(A))is a fuzzy measure

Focus of ResearchIona College

Conjunctive Aggregation of Poss and Prob•Source 1: V is μ1 (Hard Probabilistic)

• Source 2: V is μ2 (Soft Possibilistic)

• Agg Instruction: Satisfy Sources 1 and 2

• Fused Value: V is μ = V is μ1 and μ2

• Use product for “and”

• μ(A) = μ1(A) μ2(A) = Prob(A) Poss(A)

Focus of ResearchIona College

μ(A) = ( p jx j∈A∑ ) Max[π j]

x j∈A

μ(A) = p jx j∈A∑ πA

* (πA* = Max[π j]

x j∈A)

μ(x j) = p jπ j

μ(A) = ( (μ({x jx j∈A∑ })+ p j Δ j) Δ j = πA

* − π j

Quasi-Additive Measure

Focus of ResearchIona College

An important use of hard-soft information is

the determination of the validity of situation

based on known intelligence information

Difficulties arise when the intelligence

information contains uncertainty

Focus of ResearchIona College

Our plan of attack will work if the enemy

has less then 5000 defenders

Intelligence tells us they have between

3000 and 6000 defenders

Will our plan of attack work ??

Focus of ResearchIona College

DUAL OF MEASURE

• Can associate with any measure a dual.

• If μ is a measure we define its dual as

Negation of the anticipation of not A

• If μ is a measure its dual is also measure

•The dual of the dual is the original measure

öμ(A) = 1− μ(A)

Focus of ResearchIona College

Measures of Assurance and OpportunityMotivation

• μ(A) indicates our anticipation of A occurring

• Does μ(A) = 1 assure us that A will occur ??

• Consider the measure μ*(A) = 1 for all A ≠ ∅

• Here μ*(A) = 1, however also have

• Here we have just as strong an anticipation

that A will not occur

μ * (A) = 1

Focus of ResearchIona College

Measure of Assurance

• To be assured that A will occur we have to anticipate A will occur and also anticipate that will not occur. • Our anticipation that will not occur can be measured by 1 - μ( ). • This is the dual of μ, • We introduce measure λ called the assuranceof A defined as

A

AA

öμ(A)

λ(A) = μ(A) ∧ öμ(A)

Focus of ResearchIona College

Measure of Opportunity ψ

•

• ψ is a measure

• ψ(A) ≥ μ(A)

•. For measures that are duals have

. • ψ(A) is opportunity that value of V lies in A

ψ(A)=μ(A)∨ öμ(A)

ψμ(A) = ψ öμ(A)

Focus of ResearchIona College

The measures of assurance and

opportunity generalize some

fundamental concepts used in

uncertainty modeling

Focus of ResearchIona College

Assurance-Opportunity for Special Cases

• Probability Measure: ψ(A) = λ(A) = μ(A) Very Special !!!

• Possibility MeasuresAlways have ψ(A) = μ(A) and

ψ is possibility and λ is necessity

• Dempster-Shafer

ψ is plausibility and λ is belief

λ(A) = öμ(A)

2011‐2012 & Option PlansIona College

• Capability Goal: Advise team on appropriate algorithms for fusion

and uncertainty normalization

• Research Goals:

• Modeling Instructions for Fusing Information

• Providing representation of linguistically expressed Soft Information

• Continue working on measure based framework for fusion of

Information in different uncertain modalities

• Decisions with Hard-Soft Information

• Temporal alignment under imprecision

• Using copulas to join different type variables

• Adjudicating conflicting information

• Imprecise Matching

END !!!!!!!

Focus of ResearchIona College

Focus of ResearchIona College