IEEE Systems, Man and Cybernetics Magazine - April 2021 - 17

An Illustrative Example
To clarify how the UFO builds mathematical equations,
let's suppose the following: v = 3, n = 4, f = [versin, csc,
], 9 1 = [- , + , + , '], 9 2 = [# , - , # , +], 9 3 = [' , + , # , -],
V1 = + , and V2 = ' . The actual response (y) can be
approximated as follows:

max
weight " assigned to B j, where w j ! [w min
j , w j ]. It has to
be remembered that all the coefficients are not normalized. They could be normalized, but this action will reduce
the explainability level of the UFO. The overall mathematical model expressed by the UFO can, then, be expressed
as follows:

	

yt (X ) = g 1 (X ) V 1 g 2 (X ) V 2 g V v - 1 g v (X ), (14)

where {V1, V2, f, Vv - 1} represent the interactions between
the v blocks.
From the previous two equations, 9 and V are identical. The first, 9, is used between predictors and called
the internal universal arithmetic operator, and the second, V, is used between blocks and called the external
universal arithmetic operator. Although {a 0,1, a 0,2, f, a 0,v}
give the UFO the ability to explain the constant term of
the model, these internal intercepts sometimes serve as
weights when {9 1,1, 9 1,2, f, 9 1,v} ! [# , ']. Thus, an " initial block " (B 0), which is also called the global block, is
included in the UFO str ucture to have the following estimation:
	

g 2 (X )
yt (X ) = w 0 + g 1 (X ) + g (X ) , (16)
3

	

which ends up with the final mathematical expression:
yt (X ) =
	
a 3,1 $ x b3 c
nF
w 0 + w 1 $ 


IEEE Systems, Man and Cybernetics Magazine - April 2021

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