IEEE Systems, Man and Cybernetics Magazine - April 2021 - 15

With an interaction between x 1 and x 2, (3) becomes
yt = b 0 + b 1 x 1 + b 2 x 2 + p 1,2 x 1 x 2 + g + b n x n, (4)

	

where p 1,2 is the coefficient of the interaction term.
NLR
Because many real data sets have high nonlinearity
between variables, the previous expressions are not
always preferable choices to mathematically model systems. If the nonlinearity issue is simple, NLR can be used
by reconfiguring the coefficients to be in a nonlinear form
and by using nonidentity analytic functions. Examples are
given in [2, Appendix A].
ANNs and SVMs
What if a data scientist or engineer does not have any clue
about how to, or from where, to start? This dilemma can
be effectively resolved by employing modern ML/DL tools,
such as SVMs and ANNs. However, everything has a cost.
These modeling-free ML computing systems have the
ability to provide highly precise approximations without
referring to any mathematical equations. This property
is good for some applications that require quick decisions without referring to any expensive information. On
the opposite side, these algorithms are known as black
boxes because it is hard to understand their internal
models. This major drawback makes them unsuitable for
certain applications, such as expressing objective functions of optimization algorithms, and where it is necessary to have simple/readable approximators, use little

memory to store estimated models, and easily deploy systems in real-world functions.
Furthermore, authorities, agencies, and insurance companies need explanations for many artificial intelligencebased products and applications. This could be mandatory
if human life is at risk. How can user trust be built if decisions made by mystery connections between digital neurons are unknown? This is why understanding the
mechanisms behind neural networks' final decisions is so
important. It could be a legal issue if there is a mistake in a
medical diagnosis or if there is an accident where self-driving cars are involved [3], [4].
SR
As can be seen, LR and NLR are easy to use for explaining the variability of data sets, but their accuracy is
weak. ANNs and SVMs could enhance the accuracy level
but at the expense of explainability. What about FigureĀ  2(b)? To compromise between explainability and
accuracy, SR could be used. Compared with LR and NLR,
SR optimizes both the model structure and its parameters to find mathematical equations that best fit the data
set variability [5], [6]. However, building SR from scratch
and modifying existing frameworks are not easy tasks.
This technique uses special optimization algorithms
to mimic chromosome trees, such as genetic programming (GP), which was developed by Koza [7] based on the
genetic algorithm [8].
To explain how SR works, let's consider the trees
shown in Figure 3. The structures represent the following
mathematical equations.

sub
x1

log

div

mul

div

sub
x1

-1

log

x2

-5

10
(a)

sub: Subtract

-5

mul
x1

x2

mul: Multiply

mul
x4

mul
x1

x1

(b)
div: Divide

sub

add

add

div

x2

cos

cos

(c)

log
x5

x1

div
10

x2

(d)

cos: Cosine

Figure 3. A tree structure chromosome representation of computer programs in GP [2]: (a) the first tree, (b) the

second tree, (c) the third tree, and (d) the fourth tree.

	

Ap ri l 2021

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE	

15



IEEE Systems, Man and Cybernetics Magazine - April 2021

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