Previous Accomplishments. Focus of Research Iona College. Focus of Research Iona College. Publication List Iona College. Journals

Transcription

1 Network-based Hard/Soft Information Fusion: Soft Information and its Fusion Ronald R. Yager, Tel , 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 Hard Information Computing with Words Soft Information Representation (Translation) Fusion Inference Reasoning Retranslation Fusion Instructions Accomplishments Poss-Prob Fusion Methods Querying Under Uncertainty Forminng Joint Variables Set measure Representation Challenges Mixed uncertainty mode fusion Complexity of Soft information 1 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 Previous Accomplishments Measure Theoretic Paradigm for Uncertainty Modeling Fusion and Aggregation of Uncertainty Measures Conditioning Approach to Poss-Prob Fusion Linguistic Expression of Fusion Rules Prioritized Aggregation Operation Modeling Doubly Uncertain Constaints Decision Making with Uncertain Information Quantification of Uncertainty Journals Publication List Yager, R. R., "On prioritized multiple criteria aggregation," IEEE Transactions on Systems, Man and Cybernetics: Part B 42, , Yager, R. R., "Participatory learning of propositional knowledge," IEEE Transactions on Fuzzy Systems 20, , Yager, R. R. and Alajlan, N., "Measure based representation of uncertain information," Fuzzy Optimization and Decision Making 11, , Yager, R. R., "Membership modification and level sets," Soft Computing 17, , Yager, R. R. and Abbasov, A. M., "Pythagorean membership grades, complex numbers and decision-making," International Journal of Intelligent Systems 28, , 2013.

2 Publication List (2) Conferences Yager, R. R., "On a view of Zadeh's Z-numbers," Advances in Computational Intelligence- Procceedings of the 14th International Conference on Information Processing and of Uncertanity in Knowledge-Based Systems (IPMU) Part 3, Catania, Italy, Springer:Berlin, , Yager, R. R. and Yager, R. L., "Social networks: querying and sharing mined information," Proceedings of the 46th Hawaii International Conference on System Science HICSS-46, IEEE Computer Society, , Yager, R. R. and Petry, F. E., "Intuitive decision-making using hyper similarity matching," Proceedings of the Joint IFSA Congress and NAFIPS Meeting, Edmonton, Canada, , Articles in Books Yager, R. R., "Intelligent aggregation and time series smoothing," In Time Series Analysis, Modeling and Applications, Pedrycz, W. and Chen, S. M. (Eds), Springer, Heidelberg, 53-75, 2013 Manuscripts Publication List (3) Yager, R. R., "Joint cumulative distribution functions for Dempster Shafer belief structures," Fuzzy Optimization and Decision Making, (To Appear). Yager, R. R., "On forming joint variables in computing with words," International Journal of General Systems, (To Appear). Yager, R. R. and Alajlan, N., "Probabilistically weighted OWA aggregation," IEEE Transactions on Fuzzy Systems, (To Appear). 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 -5 - Conference papers Manuscripts -3 - Book and book chapters - 1 Technology Transitions: - Patents (disclosures) - none Awards: None - Joint Variables with Hard and Soft Uncertainties

3 Joint Probability Distributions Assume U and V are random variables on X and Y with probability distributions P(x) and Q(y) Joint Random Variable (U, V) has probability distribution R(x, y). If there is some correlation between U and V the joint variable contains some additional information reducing marginal uncertainties Given uncertain information about the speed of movement of enemy unit Given uncertain information about the size of enemy unit Generally some correlation between these Joining them gives some additional information that reduces original uncertainty Sklar s s Theorem Assume U and V are random variable with CDF's of F U and F V and joint CDF denoted as F U, V. Then there exists a copula C: [0, 1 2 [0, 1 such that F U, V ((x, y) = Prob(U x, V y) = C(F U (x), F V (y)) Provides a direct way of joining random variables Choice of copula reflects correlation between joint variables Min (M): C(a, b) = Min(a, b) = 1 positive Product ( ): C(a, b) = ab = 0 independent Lukasiewicz (W): C(x, y) = Max(x + y - 1, 0) = -1 Neg General: -1 1 C, (a, b) = M(a, b) + (1 - - ) (a, b) + W(a, b) If 0 then = and = 0 If 0 then = 0 and = -

4 Extend use of Sklar theorem to joining hard and soft information Dempster-Shafer Belief Structures for Joining Possibility and Probability Uncertain Information Uncertainty Representation in D-S Framework Use belief structure 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. 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

5 Representation of Possibility Distribution in D-S Nested belief structure m 2 Assume elements indexed with i j if i < j Focal elements F j = {x 1,, x j }, for j = 1 to n (Nested: F j F j+1) ) Joining Possibilistic and Probabilistic Variables 1 Represent each as D-S Belief Structure 2. Obtain CDF of Belief Structues 3. Apply Sklars theorem to these Belief Structure m 2 (F j ) = j - j + 1 Learning from Soft Linguistic Observations Using Trapezoidal Fuzzy Set Representations Trapezoidal Fuzzy Set Representation a b c d Membership grade easily obtained from the four parameters a, b, c, d

6 Level Sets and Trapezoidal Fuzzy Sets -level set of F is a crisp set F = {x/f(x) } Level Sets are intervals: F = [L, U Linearity of Trapezoids Useful Can get every level set from any two level sets Trapezoidal Preserving Operations If A and B are trapezoids and G is arithmetic operation via extension principle F = G(A, B) G is a trapezoidal preserving operation if F is also a trapezoid F = w 1 A + w 2 B is trapezoidal preserving Level Sets and Weighted Sums If F = w 1 A + w 2 B this operation easily performed on level sets if w 1 & w 2 non negative A = [L _ A,U_ A B = [L _B,U_B F = [w 1 L _ A + w 2 L _B, w 1 L _ A + w 2 L _B To get F all we need is two F Trapezoidal representation very efficient for processes involving these operations Learning From Observations V is variable whose domain is [a, b E is the current estimate of the value of V D is a new observation of the value of V F new estimate of value of V F = E + (D - E) = D + (1 - ) E [0, 1 is learning rate

7 Learning From Soft Linguistic Observations Represent observations and estimates using trapezoidal representations Take advantage of efficiency of trapezoids in this linear environment Soft Linguistic Learning Calculation F = D + (1 - ) E E = [L _E,U_E D = [L _D,U_D F = [ L _D + L _E, U_D + U_E F can be obtained from any two level sets Uncertainty in Fuzzy Estimates Three examples of estimates of value of V A: A 1 = [5, 5 and A 0.5 = [5, 5 B: B 1 = [4, 8 and B 0.5 = [3, 10 C: C 1 = [0, 10 and C 0.5 = [0, 10 A provides most information about V, it says V = 5 B provides less information than A but it is better than that provided by C Measuring Information in Estimates Specificity measures information in fuzzy subset Assume V is variable with domain [a, b Let F = [c, d be an interval Sp(F )= 1 Length(F ) = 1 d c b a b a Let F be a trapezoidal fuzzy set Sp(F) = Sp(F 0.5 ) Bigger F 0.5 the less information

8 Effect of Learning on Specificity New Estimate F = [ L _D + L _E, U_D + U_E New Estimate Level Set Length Length(F )= Length(D )+ Length(E ) Specificity of New Estimate Sp(F) = Sp(D) + (1 - ) Sp(E) Specificity of new estimate is weighted average of specificity of observation and current estimate Very uncertain/imprecise soft linguistic observations, those with small specificity, will tend to decrease the specificity of our estimate WE MUST FIX THIS!!! F = E + (D - E) Modified Learning Rule [0, 1 term relating specificity of observation with specificity of current estimate Smaller Sp(D) relative to Sp(E) smaller Smaller less effect on observation F = D + (1 - ) E Sp(F) = Sp(D) + (1 - ) Sp(E) = 1 Possible Forms for = Sp(D)/Sp(E) = 1 = (Sp(D)/Sp(E)) r if Sp(D) Sp(E) if Sp(D) < Sp(E) if Sp(D) Sp(E) if Sp(D) < Sp(E) Obtain using a fuzzy systems model If Sp(D) is A j and Sp(E) is B j then = g j

9 Research 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 END!!!!!!!