ronald r. yager , iona college tel. 212 249 2047, e-mail ...nagi/muri/muri/year_4_files/pdf...

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Network-based Hard/Soft Information FusionNetwork-based Hard/Soft Information Fusion:

Soft Information and its FusionSoft Information and its FusionRonald R. Yager , Iona CollegeRonald R. Yager , Iona College

Tel. 212 249 2047, E-Mail: [email protected]. 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 Methods

• Querying Under Uncertainty

• Forminng Joint Variables

• Set measure Representation

Challenges• Mixed uncertainty mode fusion

• Complexity of Soft information

Computing with WordsComputing with Words

Representation(Translation)

Fusion

Inference Reasoning

Retranslation

Soft Information

Hard

Information

Fusion

Instructions

Focus of ResearchFocus of Research

Iona CollegeIona 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



Previous AccomplishmentsPrevious 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

Publication ListPublication ListIona CollegeIona College

Journals

•Yager, R. R., "On prioritized multiple criteria aggregation," IEEE Transactions on

Systems, Man and Cybernetics: Part B 42, 1297-1305, 2012.

•Yager, R. R., "Participatory learning of propositional knowledge," IEEE Transactions

on Fuzzy Systems 20, 715-727, 2012.

•Yager, R. R. and Alajlan, N., "Measure based representation of uncertain

information," Fuzzy Optimization and Decision Making 11, 363-385, 2012.

•Yager, R. R., "Membership modification and level sets," Soft Computing 17, 391-

399, 2013.

•Yager, R. R. and Abbasov, A. M., "Pythagorean membership grades, complex

numbers and decision-making," International Journal of Intelligent Systems 28, 436-

452, 2013.

Publication List (2)Publication List (2)Iona CollegeIona College

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, 90-101, 2012.

•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, 1435-1442, 2013.

•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, 386-389, 2013.

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

Publication List (3)Publication List (3)Iona CollegeIona College

Manuscripts

•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 SummaryProject Statistics and SummaryIona CollegeIona College

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 - 3

- Manuscripts -3

- Book and book chapters - 1

Technology Transitions:

- Patents (disclosures) - none

Awards: None

-



Joint Variables with Hard and Soft

Uncertainties



Joint Probability DistributionsJoint 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



SklarSklar’’s Theorems Theorem

Assume U and V are random variable with CDF's of

FU and FV and joint CDF denoted as FU, V. Then

there exists a copula C: [0, 1]2 ! [0, 1] such that

FU, V((x, y) = Prob(U " x, V " y) = C(FU(x), FV(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 & = - #



Extend use of Sklar theoremExtend use of Sklar theorem

to joining hard and softto joining hard and soft

informationinformation



Dempster-Shafer Belief StructuresDempster-Shafer Belief Structures

forfor

Joining Possibility and ProbabilityJoining Possibility and Probability

Uncertain InformationUncertain Information



Uncertainty Representation in D-SUncertainty Representation in D-S

FrameworkFramework

• 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.



Representation of Probability DistributionRepresentation of Probability Distribution

in D-Sin D-S

• Bayesian belief structure m1

• Focal elements are singletons Ei = {xi}

• m1 (Ei) = pi



Representation of Possibility Representation of Possibility )) Distribution Distribution

in D-Sin D-S

• Nested belief structure m2

• Assume elements indexed with *i + *j if i < j

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

(Nested: Fj , Fj+1) )

• m2(Fj) = *j - *j + 1



Joining Possibilistic and ProbabilisticJoining Possibilistic and Probabilistic

VariablesVariables

1 Represent each as D-S Belief Structure

2. Obtain CDF of Belief Structues

3. Apply Sklars theorem to these Belief

Structure



Learning from Soft LinguisticLearning from Soft Linguistic

Observations Using TrapezoidalObservations Using Trapezoidal

Fuzzy Set RepresentationsFuzzy Set Representationsa b c d

Trapezoidal Fuzzy Set RepresentationTrapezoidal Fuzzy Set Representation



Membership grade easily obtained from the

four parameters a, b, c, d



Level Sets and Trapezoidal Fuzzy SetsLevel 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 UsefulLinearity of Trapezoids Useful

Can get every level set from any two level

sets



Trapezoidal Preserving OperationsTrapezoidal 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 = w1 A + w2 B is trapezoidal preserving



Level Sets and Weighted SumsLevel Sets and Weighted Sums

• If F = w1 A + w2 B this operation easily performed

on level sets if w1 & w2 non–negative

A!= [L _ A

!,U_ A

!] B

!= [L _B

!,U_B

!]

F!= [w1L _ A

!+ w2L _B

!, w1L _ A

!+ w2L _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



Learning From Soft LinguisticLearning From Soft Linguistic

ObservationsObservations

• Represent observations and estimates

using trapezoidal representations

• Take advantage of efficiency of trapezoids

in this linear environment



Soft Linguistic Learning CalculationSoft 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 EstimatesUncertainty in Fuzzy Estimates

• Three examples of estimates of value of V

A: A1 = [5, 5] and A0.5 = [5, 5]

B: B1 = [4, 8] and B0.5 = [3, 10]

C: C1 = [0, 10] and C0.5 = [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 EstimatesMeasuring 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!

)

b " a= 1"

d " c

b " a

•Let F be a trapezoidal fuzzy set

Sp(F) = Sp(F0.5 )

Bigger F0.5 the less information



Effect of Learning on SpecificityEffect 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 !!!WE MUST FIX THIS !!!



Modified Learning RuleModified Learning Rule

• F = E + -.(D - E)

• . '[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)



Possible Forms for Possible Forms for ..

• . = 1 if Sp(D) " Sp(E)

. = Sp(D)/Sp(E) if Sp(D) < Sp(E)

• . = 1 if Sp(D) " Sp(E)

. = (Sp(D)/Sp(E))r if Sp(D) < Sp(E)

• Obtain . using a fuzzy systems model

If Sp(D) is Aj and Sp(E) is Bj then . = gj

2013-2014 Research Plans2013-2014 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 !!!!!!!END !!!!!!!



ronald r. yager , iona college tel. 212 249 2047, e-mail ...nagi/muri/muri/year_4_files/pdf...

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