IEEE Systems, Man and Cybernetics Magazine - April 2021 - 16

```◆◆ The first tree is parent 1, which represents

10
yt = x 1 - log a x 2 k . (5)

UFO structure containing one output stream (i.e., referring
to Figure 1, m = 1). If a data set contains n predictors
{x 1, x 2, f, x n}, the approximated response can be mathematically expressed as

◆◆ The second tree, parent 2, represents

-1
yt = x 2 \$ cos ^ x 21 - 5 h. (6)

yt = f (x 1, x 2, f, x k - 1, x k, x k + 1, f, x n - 1, x n) (9)

and by using the vector notation

◆◆ The third tree is a child of the genetic crossover opera-

tion, representing
yt = x 1 - log c

yt = f (X ) ; X = [x 1, x 2, g, x n]. (10)

cos ^ x 12 - 5 h
m. (7)
x2

◆◆ The fourth tree is a child of the subtree mutation oper-

Now, suppose that yt is decomposed into v functions:

f / { f1, f2, f, f j - 1, f j, f j + 1, f, fv - 1, fv}. (11)

ation, and it represents
10
yt = x 4 \$ x 5 - log a x 2 k. (8)

Although the UFO and classical SR share the same aim to
automatically build linear and nonlinear mathematical
equations, they are totally different. There are some difficulties to practically applying SR techniques in real-world
applications. A detailed discussion about the major differences between these two ML computing systems is given
in [2].
The UFO
The UFO is a new multipurpose ML computing system that
explains everything as pure mathematical equations. It
can act like SR to build simple and highly complicated linear/nonlinear equations. Also, it can be used in applications such as function simplification, function complication,
dimension reduction, dimension expansion, and highdimensional function visualization. The UFO has a built-in
selection process for features and analytic functions,
which enables it to remove unwanted variables and terms
from a final model. To help convey how this innovative
computing system works, Figure 4 describes the basic

x1

B1

x2

x3

f j (X ) = f j ^a 0, j 9 1, j a 1, j \$ x b1 9 2, j a 2, j \$ x b2

9 3, j a 3, j \$ x b3 9 4, j g 9 n, j a n, j \$ x bn h,
1, j

g1 g2

g j (X ) = w j \$ 6 f j (X )@c = w j \$ 6 f j (x 1, x 2, f, x n)@c
= w j \$ 6 f j ^a 0, j 9 1, j a 1, j \$ x b1 9 2, j a 2, j \$ x b2

9 3, j a 3, j \$ x b3 9 4, j g 9 n, j a n, j \$ x bn h@c ,
j

j

1, j

3, j

2, j

n, j

j

(13)

where c j is the " external exponent " assigned to f j locatmax
ed in B j, where c j ! [c min
j , c j ], and w j is the " external

B0

Bv
gv (x1, x2, ... , xn)

g1 g2 g3
g1 g2 g3

g1 g2 g3
... gv -1

Figure 4. The UFO mechanism with only one output stream [2].
16

(12)

n, j

where 9 k, j is the kth " universal arithmetic operator "
assigned to the kth predictor (x k) of the jth block (B j);
it could be +, -, ×, or ' . In addition, a 0, j is the " in--
max
a k, j is the kth
tercept " of B j, where a 0, j ! [a 0min
, j , a 0, j ];
" internal weight " assigned to x k located in B j, where
max
a k, j ! [a min
k, j , a k, j ]; and b k, j is the kth " internal exponent "
max
assigned to x k located in B j, where b k, j ! [b min
k, j , b k, j ].
If each f j has an exponent (cj) and the result is multiplied by a weight (w j), the following more general function can be attained:

g2 (x1, x2, ... , xn) g3 (x1, x2, ... , xn)
g1 (x1, x2, ... , xn)

2, j

3, j

xn

B3

B2

The interaction between {x 1, x 2, f, x n} in each jth analytic
function can be modeled in many ways. The basic
approach is mathematically expressed as follows:

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE Apri l 2021

... gv

w0
y (x1, x2, ... , xn)

"

```

IEEE Systems, Man and Cybernetics Magazine - April 2021

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