Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

The orthogonal summation of belief functions
Assume that two knowledge sources KB1 and KB2 submit two frames of
discerrnments θ1 and θ2 respectively. Let m1 (.) and m2 (.) be the BPA at the
subsets of θ1 and θ2 respectively. The new BPA, m (.) can be computed based
on m1 (.) and m2 (.) by using
m(X) = K ∑ m1 (Xi) . m2 (Xj) (9.24)
X= Xi ∩ Xj
and K = 1 - ∑ m1 (Xi) . m2 (Xj)
Xi ∩Xj = φ
where Xi and Xj are focal elements of θ1 and θ2 respectively. We denote the
orthogonal summation operation, referred to above, by m = m1 ⊕ m2.
To illustrate the orthogonal summation process, let us consider
the BPAs that are assigned by two knowledge sources through a image
recognition process.
Let us assume that knowledge source 1 (KS1) claims that an unknown object
in a scene could be
a chair with m1({C}) = 0.3,
a table with m1({T}) = 0.1,
a desk with m1({D}) = 0.1,
a window with m1({w}) = 0.15,
a person with m1({P}) = 0.05,
and the frame θ, with m1 ({θ}) = 0.3.
The assignment of BPA = 0.3 to θ means that knowledge source 1 knows
that something in θ has occurred, but it does not know what it exactly is.
Analogously, knowledge source 2 (KS2) claims the same object in the scene
to be
a chair with m2({C}) = 0.2,
a table with m2({T}) = 0.05,
a desk with m2({D}) = 0.25,
a window with m2({W}) = 0.1,
a person with m2({P) = 0.2,
and the frame θ with m2({θ}) = 0.2
Now, suppose, we are interested to compute “What is the composite belief of
the object to be a chair ?”