Scientific/Technical Approach

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1 Network based Hard/Soft Information Fusion: Soft Information and its Fusion Ronald R. Yager, Tel , 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 Hard Information Computing with Words Soft Information Representation (Translation) Fusion Inference Reasoning Retranslation Fusion Instructions 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 Representation Challenges Mixed uncertainty mode fusion Complexity of Soft information 1

2 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

3 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

4 Publication List Journals Yager, R. R., "A measure based approach to the fusion of possibilistic and probabilistic uncertainty," Fuzzy Optimization and Decision Making 10, , Yager, R. R., "On the fusion of imprecise uncertainty measures using belief structures," Information Sciences 181, , Yager, R. R., "Reasoning with doubly uncertain constraints," International Journal of Approximate Reasoning 52, , 2011 Yager, R. R., "Lexicographic ordinal OWA aggregation of multiple criteria," Information Fusion 11, , 2010 Yager, R. R., "Criteria satisfaction under measure based uncertainty," Fuzzy Optimization and Decision Making 9, , Yager, R. R., "Cumulative distribution functions, p-boxes and decisions under risk," International J. of Knowledge Engineering and Soft Data Paradigms 2, , Yager, R. R., "Validating criteria with imprecise data in the case of trapezoidal representations," Soft Computing Journal 15, , 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),

5 Publication List (2) 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 # , 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, , 2011 Manuscripts 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, Yager, R. R., "Dempster-Shafer structures with general measures," Technical Report #MII-3016 Machine Intelligence Institute, Iona, College, New Rochelle, NY, 2010

6 Publication List (3) 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, , 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, 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, , 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, , 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, 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

7 Project Statistics and Summary Students supported: -# of undergraduate and graduate students 0 -# of post-doc and faculty members 1 -# of degrees awarded (MS, PhD) 0 Publications: - Journal papers -9 - Conference papers Manuscripts -3 - Book and book chapters - 7 Technology Transitions: - Patents (disclosures) - none Awards: -International Fuzzy Systems Association Award for Naval Research Lab Publication Award

8 FUSING HARD AND SOFT INFORMATION Hard Information Probabilistic Soft Information Possibilistic Requirement for technology that can fuse Probabilistic and Possibilistic Information

9 Conditioning Approach to Possibility-Probability Fusion

10 Assume V has domain X = {x 1,,x n } Two sources of information Probability distribution P: P(x j )=p j Possibility distribution Π: Π(x j )=π j We shall obtain from the two pieces of information a probability distribution Q based on a conditioning of P by Π.

11 Here we make use of Zadeh's ideas relating fuzzy sets and possibility distributions We associate with the possibility distribution Π a fuzzy subset F of X such for each x j X, F(x j )=Π(x j )=π j

12 Using the subset F we condition our probabilistic information P with our possibilistic information and we now obtain the probability distribution Q such that for any x j we have Q(x j ) = P(x j /F)= P({x j }Ê ÊFÊ) P(F)

13 We now recall that the probability of a fuzzy subset F is the expected value of its membership function P(F) = n F(x j )Ê Êp j jê=ê1

14 We note that {x j } F is fuzzy subset of X {x j } F = { F(x j ) x j } Hence P({x j ) F) = p j F(x j ).

15 Combining these probabilities we get Q(x j )Ê=Ê p j F(x j ) n F(x k )p k k=ê1 = p j π j n π k p k kê=ê1 Here then Q is the probability distribution that results from combining our two sources of information

16 Furthermore for any subset B, fuzzy or crisp, we have n jê=ê1 Q(B)Ê=Ê B(x j ) ÊQ j Ê=Ê n jê=ê1 p j π j B(x j ) n p j π j jê=ê1

17 Example X={x 1, x 2, x 3, x 4 } p 1 = 0.3, p 2 = 0.2, p 3 = 0.4, p 4 = 0.1 π = 1, π 2 = 0.6, π 3 = 0.8, π 4 = 0.2 p 1 π 1 = 0.3, p 2 π 2 = 0.12, p 3 π 3 = 0.32 p 4 π 4 = 0.02 Σ p j π j = 0.76.

18 Here then with We get: Q(x j ) = p j π j 0.76 Q(x 1 ) = Q(x 2 ) = 0.16, Q(x 3 ) =.0.42, Q(x 4 ) = If B = {x 1, x 3 } then Q(B) = 0.81

19 Dempster-Shafer View of Possibility-Probability Fusion

20 Uncertainty Representation in D-S Use belief structure Framework A collection F j of subsets of X called focal elements Mapping m which associates with each focal element a value m(f j ) [0, 1] such that j m(f j )=1.

21 Fusion of Information in D-S Framework Uses Dempster's rule m 1 and m 2 are two belief structures with focal elements E i and F j respectively Fusion is m = m 1 m 2 its focal elements are all E i F j and m i (E i )Ê Êm 2 (F j ) m(e i F j ) = m(e i )Ê Êm 2 (F j ) E iê ÊF j Ê Ê

22 Representation of Probability Distribution in D-S Bayesian belief structure m 1 Focal elements are singletons E i = {x i } m 1 (E i ) = p i

23 Representation of Possibility Distribution in D-S Nested belief structure m 2 Assume elements indexed with π i π j if i < j Focal elements F j = {x 1,, x j }, for j = 1 to n (Nested: F j F j+1) ) m 2 (F j ) = π j - π j + 1

24 Fused Belief Structure Bayesian belief structure m Focal elements are singletons E i = {x i } m(ei ) = p i π i n p i π i iê=ê1

25 Measure Based Approach to the Representation of Uncertain Information

26 Definition of a Fuzzy Measure A fuzzy measure on X is a mapping μ: 2 X [0, 1] such that 1) μ(ø) = 0 2) 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

27 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

28 The Fuzzy Measure has the Capability of modeling in a unified framework many different types of knowledge about the value of a variable

29 Certain Knowledge V = x* μ(b) = 1 if x* B μ(b) = 0 if x* B Probability Distribution μ({x j }) = P j Σ P j = 1 μ(a B) = μ(a) + μ(b) if A B = Possibility Distribution μ({x j }) = Π j Max[Π j ]= 1 μ(a B) = Max(μ(A), μ(b))

30 Fuzzy Measures Closed Under Aggregation Operations Needed for Multi-Source Information Fusion

31 Definition: Gisanaggregation function of q arguments if G: [0, 1] q [0,1]and 1. G(0, 0,, 0) = 0, 2. G(1, 1,, 1) = 1 3. G(a 1, a q ) G(b 1, b q )ifalla j b j Theorem: Assume μ j are q fuzzy measures on X. Then μ defined such that μ(a) = G(μ 1 (A),., μ q (A)) is a fuzzy measure

32 Conjunctive Aggregation of Poss and Prob Source 1: V is μ 1 (Hard Probabilistic) Source 2: V is μ 2 (Soft Possibilistic) Agg Instruction: Satisfy Sources 1 and 2 Fused Value: V is μ = V is μ 1 and μ 2 Use product for and μ(a) = μ 1 (A) μ 2 (A) = Prob(A) Poss(A)

33 μ(a) = ( )Max[π j ] p j x j A x j A μ(a) = π A * p j x j A (π A * = Max[π j ] x j A ) μ(x j ) = p j π j μ(a) = ( * (μ({x j }) + p j Δ j ) Δ j =π A πj x j A Quasi-Additive Measure

34 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

35 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??

36 DUAL OF MEASURE Can associate with any measure a dual. If μ is a measure we define its dual as öμ(a) = 1 μ(a) 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

37 Measures of Assurance and Opportunity Motivation 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 μ *(A) = 1 Here we have just as strong an anticipation that A will not occur

38 Measure of Assurance To be assured that A will occur we have to anticipate A will occur and also anticipate that A will not occur. Our anticipation that A will not occur can be measured by 1 - μ( A). This is the dual of μ, μ(a) ö We introduce measure λ called the assurance of A defined as λ(a) = μ(a) öμ(a)

39 Measure of Opportunity ψ ψ(a) =μ(a) öμ(a) ψ is a measure ψ(a) μ(a). For measures that are duals have. ψ μ (A) = ψμ ö (A) ψ(a) is opportunity that value of V lies in A

40 The measures of assurance and opportunity generalize some fundamental concepts used in uncertainty modeling

41 Assurance-Opportunity for Special Cases Probability Measure: ψ(a) = λ(a) = μ(a) Very Special!!! Possibility Measures Always 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)

42 & Option Plans 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

43 END!!!!!!!

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