Extension of Inagaki General Weighted Operators and A New Fusion Rule Class of Proportional Redistribution of Intersection Masses

2 S θ = ^θrefined = 2^(2^θ) = D θcθc , when refinement is possible, where θ c = {τ(θ 1 ), τ(θ 2 ), …, τ(θ n )}. We consider the general case when the domain is S θ , but S θ can be replaced by D θ = (θ , c,1) or by 2 θ = (θ , c) in all formulas from below. Let 1 2 ( ) and ( ) m m ⋅ ⋅ be two normalized masses defined from S θ to [ ] 0,1 . We use the conjunction rule to first combine 1 ( ) m ⋅ with 2 ( ) m ⋅ and then we redistribute the mass of ( ) 0 m X Y ≠ I , when X Y = Φ I . Let's denote ( ) using the conjunction rule. Let's note the set of intersections by: is in a canonical form, and contains at least an symbol in its formula In conclusion, S 1 is a set of formulas formed with singletons (elements from the frame of discernment), such that each formula contains at least an intersection symbol Inagaki general weighted operator ( ) WO is defined for two sources as: So, the conflicting mass is redistributed to non-empty sets according to these weights ( ) In the extension of this WO , which we call the Double Weighted Operator ( ) In the free and hybrid modes, if no non-empty intersection is redistributed, i.e. In the Shafer's model, always DWO coincides with WO . For 2 s ≥ sources, we have a similar formula: For for two sources we have: , and or and where ( ) and M is a subset of S θ , for example: (10) where N is a subset of S θ , for example: These formulas are easily extended for any 2 s ≥ sources 1 2 ( ), ( ),..., ( ) Let's denote, using the conjunctive rule: ( ) ( ) ( ) Inagaki rule was defined on the fusion space (Θ, U ) . In this case, since all intersections are empty, the total conflicting mass, which is m 12∩ ( A∩B) + m 12∩ ( A∩C) + m 12∩ ( B∩C) = 0.26 + + 0.13 + 0.08 = 0.47, and this is redistributed to the masses of A, B, C, and A U B U C according to some weights w 1 , w 2 , w 3 , and w 4 respectively, depending to each particular rule, where: 0 ≤ w 1 , w 2 , w 3 , w 4 ≤ 1 and w 1 + w 2 + w 3 + w 4 = 1. Hence A B C A U B U C m Inagaki (.) 0.26+0.47w 1 0.18+0.47w 2 0.07+0.47w 3 0.02+0.47w 4 Yet, Inagaki rule can also be straightly extended from the power set to the hyper-power set. Suppose in DWO the user finds out that the hypothesis B∩C is not plausible, therefore m 12∩ ( B∩C) = 0.08 has to be transferred to the other non-empty elements: A, B, C, A U B U C, A∩B, A∩C, according to some weights v 1 , v 2 , v 3 , v 4 , v 5 , and v 6 respectively, depending to the particular version of this rule is chosen, where: Applying one or another fusion rule is still debating today, and this depends on the hypotheses, on the sources, and on other information we receive. A generalization of Inagaki rule has been proposed in this paper, and also a new class of fusion rules, called Class of Proportional Redistribution of Intersection Masses (CPRIM), which generates many interesting particular fusion rules in information fusion.