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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: [email protected]
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
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Focus of ResearchIona College
Conditioning Approach to
Possibility-Probability Fusion
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• 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.
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• 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
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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)
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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∑
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• 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}
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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
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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∑
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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.
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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
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Dempster-Shafer View of
Possibility-Probability Fusion
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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.
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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∅
∑
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Representation of Probability Distribution
in D-S
• Bayesian belief structure m1
• Focal elements are singletons Ei = {xi}
• m1 (Ei) = pi
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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
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Fused Belief Structure
• Bayesian belief structure m
• Focal elements are singletons Ei = {xi}
•
m(Ei) =piπi
piπiiÊ=Ê1
n∑
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Measure Based Approach to the
Representation of Uncertain
Information
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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
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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
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The Fuzzy Measure has the Capability
of modeling in a unified framework
many different types of knowledge
about the value of a variable
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•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))
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Fuzzy Measures Closed Under
Aggregation Operations Needed
for Multi-Source Information
Fusion
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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
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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)
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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
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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
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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 ??
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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)
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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
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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)
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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)
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The measures of assurance and
opportunity generalize some
fundamental concepts used in
uncertainty modeling
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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