INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
Online ISSN 1841-9844, ISSN-L 1841-9836, Volume: 18, Issue: 2, Month: April, Year: 2023
Article Number: 5320, https://doi.org/10.15837/ijccc.2023.2.5320
CCC Publications
Optimization-Based Fuzzy Regression in Full Compliance with the
Extension Principle
B. Stanojević, M. Stanojević
Bogdana Stanojević*
Mathematical Institute of the Serbian Academy of Sciences and Arts
Kneza Mihaila 36, 11000 Belgrade, Serbia
*Corresponding author: bgdnpop@mi.sanu.ac.rs
Milan Stanojević
Faculty of Organizational Sciences, University of Belgrade
Jove Ilića 154, 11000 Belgrade, Serbia
milan.stanojevic@fon.bg.ac.rs
Abstract
Business Analytics – which unites Descriptive, Predictive and Prescriptive Analytics – represents
an important component in the framework of Big Data. It aims to transform data into information,
enabling improvements in making decisions. Within Big Data, optimization is mostly related to the
prescriptive analysis, but in this paper, we present one of its applications to a predictive analysis
based on regression in fuzzy environment.
The tools offered by a regression analysis can be used either to identify the correlation of a
dependency between the observed inputs and outputs; or to provide a convenient approximation
to the output data set, thus enabling its simplified manipulation. In this paper we introduce a
new approach to predict the outputs of a fuzzy in – fuzzy out system through a fuzzy regression
analysis developed in full accordance to the extension principle. Within our approach, a couple of
mathematical optimization problems are solve for each desired α−level. The optimization models
derive the left and right endpoints of the α−cut of the predicted fuzzy output, as minimum and
maximum of all crisp values that can be obtained as predicted outputs to at least one regression
problem with observed crisp data in the α−cut ranges of the corresponding fuzzy observed data.
Relevant examples from the literature are recalled and used to illustrate the theoretical findings.
Keywords: fuzzy regression, extension principle, optimization
1 Introduction
Nowadays, the importance of handling Big Data is continuously increasing. Managing Big Data by
providing better data insights is a focal point to majority of the research fields. A fuzzy representation
of information is one way to handle the uncertainty.
The fuzzy set theory introduced in [20] is widely applied in many research fields, from industry
to management and education. Fuzzy logic was developed to generalize the classic logic, and found
https://doi.org/10.15837/ijccc.2023.2.5320 2
a wide applicability in Decision Support (see [18] for more details on its methods, applications and
future trends) and Data Processing (see for instance [1] that presents particular fuzzy methods used
in Social Sciences Researches). A general view on how fuzzy logic evolved from the classic logic, and
where it meets the quantum logic can be found in [10]. The concept of linguistic variable was proposed
in [21] emphasizing its effective applications to approximate reasoning. The extension principle was
introduced to formalize the arithmetic on fuzzy quantities.
The fuzzy linear regression model was firstly proposed by Tanaka et al. [15], and extended to
possibilistic linear regression in [16]. Since then many variants of fuzzy regression models were pro-
posed in the literature. Kahraman et al. [8] surveyed the relevant fuzzy regression approaches, and
reported many of their practical applications as: forecasting models for predicting sales of computers
and peripheral equipment; the models revealing the relationship of cumulative trauma disorders risk
factors, predicting the injuries, and evaluating the risk levels of individuals; the models forecasting
the production value of mechanical industry. A systematic review and a huge amount of bibliographic
references to fuzzy regression analysis can be found in [5]. The key open questions in the field and
important research directions were also indicated in [5].
In the recent literature, Chachi et al. [3] discussed the fuzzy regression based on M-estimates,
providing robust estimators of the parameters to avoid undesired effects; Bas [2] proposed a robust
fuzzy regression functions approach whose forecasting performance is not affected by the presence
of outliers; Wang et al. [17] introduced a fuzzy regression model that uses approximate Bayesian
computation instead of usual optimization techniques; Hose and Hanss [7] presented a fuzzy regression
approach that took into consideration the worst case variation of the parameters aiming to encode
the whole relevant observed information in their membership functions. A quadratic least squared
regression analysis was carried out in [12]. The results showed that procedures which deviate from a
full compliance with the extension principle can derive misleading solutions.
Kao and Chyu [9] discussed the fuzzy linear regression based on least squares estimates and ex-
tension principle. They adapted the extension principle to the problem they solved, and proposed an
approximated solution to surmount the complexity of the formulation. Our approach is based on the
same idea, but we succeed to derive the exact values of the endpoints of any α−cut of the regression’s
coefficients and predicted outputs. Kao and Chyu [9] used constraint programming to optimize the
index for ranking fuzzy numbers. That index was formulated using several α−cuts of the membership
function of the regression error estimation, and a triangular fuzzy number for each coefficient of the
regression was derived. Our approach will use optimization with polynomial objective functions and
constraints to derive the supports of the regression’s fuzzy number coefficients.
Hojati et al. [6] proposed a more simple method for computing the coefficients of the fuzzy
regression, but provided solutions that were far away from those based on the extension principle.
They compared their results with those reported in [15] and [11], emphasizing that their approach
provides reasonably narrow fuzzy bands. However, a narrow fuzzy band, although desirable, might be
misleading, when the observations do not properly fit inside.
Wu [19] proved that the α−cuts of fuzzy-valued functions can be computed with the help of the
α−cuts of its coefficients and arguments. Their approach to fuzzy regression followed the extension
principle until they introduced a simplified computational procedure by replacing the variables ranging
within the α−cut intervals of the observed data with their corresponding endpoints.
Chen and Nien [4] simplified the approaches based on the extension principle by reducing the
amount of information extracted from the fuzzy observations and included in their regression model.
As usual, using less accurate models simplifies the solving procedure but increases the possibility to
obtain misleading results.
The rest of the paper is organized as follows. Section 2 provides details on the needed notation and
terminology related to the crisp regression and fuzzy concepts. In Section 3, we formally describe the
problem we aim to solve. We describe our novel approach in Section 4, providing detailed description
of the optimization models involved in a fuzzy regression analysis with a polynomial predictor function.
Section 5 is devoted to experiments: we report our numerical results and their comparative illustrations
with relevant results from the literature. Final conclusion and directions for further researches are
provided in Section 6.
https://doi.org/10.15837/ijccc.2023.2.5320 3
2 Preliminaries
In this section, we provide the specific notation and terminology related to crisp regression analysis,
and fuzzy numbers that we will need in the sequel.
The tools offered by a regression analysis can be used either to identify the strength of a dependency
between the observed inputs and outputs; or to provide a convenient approximation to the data
set, thus enabling its simplified manipulation. To describe the input-output relationship one uses a
predictor function whose unknown parameters must be determined from the observed data. The wide
variety of predictor functions defines a wide variety of regression models whose classification is generally
related to the norms involved in the criterion used to derive the best values of the parameters. For
instance, the least square regression model uses the l2 norm, the RIDGE regression uses a penalized
l2 norm, the LASSO regression uses a penalized l1 norm, all for evaluating the distance between the
observed and predicted outputs.
For a crisp linear regression analysis with multiple explanatory variables, there are given n ob-
served input k-tuples (xi1, xi2, . . . , xik)i=1,...,n and n output scalars (yi)i=1,...,n. The coefficients A =
(a0, a1, . . . , ak)T of the predictor function fA (x1, x2, . . . , xk) are computed such that fA (xi1, xi2, . . . , xik)
provides a good approximation to yi, for each i = 1, . . . , n.
The parametric expression of the linear predictor function is then
fA (X) = a0 + a1x1 + a2x2 + · · · + akxk. (1)
In the least square linear regression the coefficients a0, a1, . . . , ak are determined such that the sum
of the squared Euclidean distances between yi and fA (xi1, xi2, . . . , xik) is minimal, i.e. solving the
following optimization problem
min
A
k∑
i=1
(yi − fA (xi))2. (2)
Let us denote
X =
1 x11 x12 . . . x1k
1 x21 x22 . . . x2k
...
...
...
...
...
1 xn1 xn2 . . . xnk
, Y =
y1
y2
...
yn
. (3)
The well known formula for computing the coefficients A by minimizing the least squared distances
between the observed and predicted outputs is then A =
(
X T X
)−1
X T Y. We will use this formula
written in its equivalent form (
X T X
)
A = X T Y. (4)
In the next section we adapt the crisp linear regression to a regression in fuzzy environment, namely
we model the dependencies between inputs and outputs expressed by fuzzy numbers. The fuzzy sets
theory introduced by Zadeh [20] provides an efficient tool for modeling the uncertainty. The main
concepts of the fuzzy sets theory that are of interest to our study will be briefly presented below.
A fuzzy set à over the universe X is defined as a collection of pairs
(
x, µ
Ã
(x)
)
, where x ∈ X, and
µ
Ã
(x) ∈ [0, 1]. The function µ
Ã
: X → [0, 1] is the membership function of Ã, and µ
Ã
(x) is called the
membership degree of the crisp value x in the fuzzy set Ã. Let us denote by Supp
(
Ã
)
the support of
the fuzzy set à defined as the set of values with non-zero membership degree.
Let
[
Ã
]
α
denote the α-cut of the fuzzy set Ã, that is the set of those values x whose membership
degrees are greater or equal to α, α > 0, i.e.[
Ã
]
α
=
{
x ∈ X|µ
Ã
(x) ≥ α
}
. (5)
A fuzzy set à of the universe R of real numbers is called fuzzy number (FN) if and only if: (i) it is
a fuzzy normal and fuzzy convex set; (ii) its membership function µ
Ã
is upper semi-continuous; and
(iii) its support is bounded.
https://doi.org/10.15837/ijccc.2023.2.5320 4
Through the paper, we use triangular fuzzy numbers à that are expressed by triples of real numbers(
aL, aC , aU
)
, aL ≤ aC ≤ aU . The interval
(
aL, aU
)
is the support of Ã; and the interval
[
Ã
]
α
=
[
(1 − α) aL + αaC , αaC + (1 − α) aU
]
, (6)
is its corresponding α−cut.
The extension principle, detailed in [21] in the context of linguistic variables, is widely used to
complete the fuzzy arithmetic on fuzzy numbers. We will use it in a wider context, namely to define
the predicted fuzzy outputs with respect to the crisp observed data that belongs to the α−cut intervals
of the original fuzzy observed data. The formal definition of the extension principle given in (7), i.e.
µ
B̃
(y) =
sup(x1,...,xr )∈f −1(y)
(
min
{
µ
Ã1
(x1) , . . . , µÃr (xr)
})
, f −1 (y) ̸= Ø,
0, otherwise,
(7)
provides the membership degree of y in the fuzzy set B̃ of the universe Y , where B̃ is the result of
evaluating the function f at the fuzzy sets Ã1 ,Ã2, . . . , Ãr over their corresponding universes X1,
X2, . . . , Xr. In other words, (7) generalizes the crisp evaluation y = f (x1, x2, . . . , xr) to the fuzzy
evaluation B̃ = f
(
Ã1, Ã2, . . . , Ãr
)
.
3 Problem formulation
We focus on linear regression applied to observed fuzzy data with multiple explanatory variables,
whose parameters will be derived using the l2 norm, and whose predictions will be in full accordance
to the extension principle. Both observed inputs and outputs are triangular fuzzy numbers, while the
predicted outputs do not have an a priori imposed shape type.
By analogy to the crisp case, function f
Ã
(x̃) = ã0 + ã1x̃1 + ã2x̃2 + . . . + ãkx̃k predicts the fuzzy
output ̂̃y = f
Ã
(x̃) with respect to the parameters ã0, ã1, . . ., ãk at fuzzy input x̃. Applying the fuzzy
regression in a complete accordance to the extension principle means finding all sets of crisp values
a0, a1, . . ., ak that are parameters of at least one crisp regression derived from at least one set of crisp
input/output observations xij ∈ [x̃ij ]α, yi ∈ [ỹi]α, i = 1, . . . , n and j = 1, . . . , k. The membership
degree of such set of values a0, a1, . . ., ak in the fuzzy sets ã0, ã1, . . ., ãk respectively, must be
max︷ ︸︸ ︷
x, y that yields a
(
min
({
µx̃ij (xij ) |i = 1, n, j = 1, k
} ⋃ {
µỹi (yi) |i = 1, n
}))
, (8)
where a stands for {a0, a1, . . . , ak}, x stands for
{
xij |i = 1, n, j = 1, k
}
, and y stands for
{
yi|i = 1, n
}
.
Within this study we aim to determine the accurate fuzzy-valued coefficients a of the regression
and predicted outputs ̂̃y. In the next section we reach the goal by formulating and solving certain
pairs of optimization models.
4 Our approach
The involvement of the extension principle in defining the fuzzy regression is not new in the
literature (see for instance the recent survey [5]). However, so far, the complexity of (8) determined
the authors to search for simplified approaches that introduced certain deviations form a complete
compliance with the extension principle.
Building around (8), we adapt a methodology already used within fuzzy mathematical program-
ming problems ([14]), and focus on how to find the values a0, a1, . . ., ak with a membership degree at
least α, α arbitrary fixed in the interval [0, 1]. Wu [19] proved that the α−cuts of fuzzy-valued func-
tions can be computed with the help of the α−cuts of its coefficients and arguments. In accordance
https://doi.org/10.15837/ijccc.2023.2.5320 5
to Proposition 3.3 [19], we impose the following box constraints (9) on variables x and y to keep them
varying within the ranges of their corresponding α−cuts.
(1 − α) (x̃ij )L + α (x̃ij )C ≤ xij ≤ α (x̃ij )C + (1 − α) (x̃ij )U , i = 1, . . . , n, j = 1, . . . , k
(1 − α) (ỹi)L + α (ỹi)C ≤ yi ≤ α (ỹi)C + (1 − α) (ỹi)U , i = 1, . . . , n.
(9)
We use the upper indexes L, C, U to identify the lower, center and upper values of each involved
triangular fuzzy number.
Further on, we use the relation between the crisp x, y and a provided by the crisp linear regression
theory (4) as additional constraint; and propose the pair of models
min (max) aq
s.t.
X T XA = X T Y,
(1 − α) (x̃ij )L + α (x̃ij )C ≤ xij ≤ α (x̃ij )C + (1 − α) (x̃ij )U , i = 1, . . . , n,
j = 1, . . . , k,
(1 − α) (ỹi)L + α (ỹi)C ≤ yi ≤ α (ỹi)C + (1 − α) (ỹi)U , i = 1, . . . , n,
aj free variable, j = 0, . . . , k,
(10)
able to derive the α−cuts of the fuzzy parameter ãq, q = 0, . . . , k of the fuzzy regression. Therefore,
models (10) can be used to construct the fuzzy sets ã0, ã1, . . ., ãk; and based on them, we propose
Algorithm 1 (EPBRC) that derives numerically the left and right sides of the membership functions
of the fuzzy linear regression’s coefficients in full accordance with the extension principle. The input
parameters for EPBRC are as follows: the sequence α1, α2, . . . , αp of values from [0, 1] representing
the desired membership levels; the fuzzy numbers representing the observed inputs x̃ij and outputs ỹi,
i = 1, . . . , n, j = 1, . . . , k. The outputs of EPBRC are the matrices that describe the α−cut intervals
of the membership functions of the coefficients Ã.
Algorithm 1 Extension-Principle-Based algorithm for Regression Coefficients (EPBRC)
Input: α1, α2, . . . , αp ∈ [0, 1]; x̃ij , ỹi, i = 1, . . . , n, j = 1, . . . , k.
1: Define the matrices X and Y using (3).
2: for s = 1, p do
3: for q = 0, k do
4: Solve min problem (10) and denote by aLqs the minimal value of the objective function.
5: Solve max problem (10) and denote by aUqs the maximal value of the objective function.
6: end for
7: end for
Output: Matrices aL =
(
aLqs
)s=1,p
q=0,k
, aU =
(
aUqs
)s=1,p
q=0,k
.
However, the fuzzy sets ã0, ã1, . . ., ãk defined by Models (10) and derived by EPBRC cannot be
used directly (i.e. through formal arithmetic) to derive the predicted outputs at a given input ṽ.
Another pair of models, namely
min (max) a0 + a1v1 + a2v2 + · · · + akvk
s.t.
X T XA = X T Y,
(1 − α) (x̃ij )L + α (x̃ij )C ≤ xij ≤ α (x̃ij )C + (1 − α) (x̃ij )U , i = 1, . . . , n,
j = 1, . . . , k,
(1 − α) (ṽj )L + α (ṽj )C ≤ vj ≤ α (ṽj )C + (1 − α) (ṽj )U , j = 1, . . . , k,
(1 − α) (ỹi)L + α (ỹi)C ≤ yi ≤ α (ỹi)C + (1 − α) (ỹi)U , i = 1, . . . , n,
aj free variable, j = 0, . . . , k,
(11)
must be utilized for this purpose. For a fixed α, Models (11) determine the left and right endpoints
of the α−cut of the predicted fuzzy output at the fuzzy input ṽ. When the given fuzzy input is one
https://doi.org/10.15837/ijccc.2023.2.5320 6
of the observed inputs, for instance x̃h, then the variables vj , j = 1, . . . , k and their box constraints
can be ignored in Models (11); and the variables xhj , j = 1, . . . , k, are used instead of vj , j = 1, . . . , k
in the objective function.
Algorithm 2 (EPBRO) uses Models (11), and derives the estimated fuzzy outputs in full accordance
with the extension principle. The values of ṽj , j = 1, . . . , k are inputs to Algorithm 2 together with all
inputs of Algorithm 1. The outputs of Algorithm 2 are the vectors ŷL =
(
ŷLs
)
s=1,p
, ŷU =
(
ŷUs
)
s=1,p
representing the left and right sides of the membership function of the fuzzy set that is the evaluation
of the predictor function at ṽ.
Algorithm 2 Extension-Principle-Based algorithm for Regression Outputs (EPBRO)
Input: α1, α2, . . . , αp ∈ [0, 1]; x̃ij , ỹi, ṽj , i = 1, . . . , n, j = 1, . . . , k.
1: Define the matrices X and Y using (3).
2: for s = 1, p do
3: for i = 1, n do
4: Solve min problem (11) and denote by ŷLs the minimal value of the objective function.
5: Solve max problem (11) and denote by ŷUs the maximal value of the objective function.
6: end for
7: end for
Output: Vectors ŷL =
(
ŷLs
)
s=1,p
, ŷU =
(
ŷUs
)
s=1,p
.
Neither the coefficients, nor the predicted outputs derived by running Algorithms 1 and 2, re-
spectively are triangular fuzzy numbers. However, their accurate shapes can be approximated by the
triangular fuzzy numbers aq =
(
aLq1, a
L
qp, a
U
q1
)
, q = 0, k, and y =
(
yL1 , y
L
p , y
U
1
)
, i = 1, n.
4.1 Generalizations to fuzzy polynomial regression analysis
Models (10) and (11) can be generalized to derive the coefficients and the predicted outputs of
an fuzzy polynomial regression. Models fully complying to the extension principle and needed in
the quadratic regression analysis of a crisp in – fuzzy out system with a single observed explanatory
variable were provided in [12]. In this section we generalize the crisp formula provided in [12] for a
quadratic regression to the formula needed to model the fuzzy polynomial regression analysis of degree
p, i.e. we propose the constraint system
n∑
i=1
x
2p
i
n∑
i=1
x
2p−1
i . . .
n∑
i=1
x
p+1
i
n∑
i=1
x
p
i
n∑
i=1
x
2p−1
i
n∑
i=1
x
2p−2
i . . .
n∑
i=1
x
p
i
n∑
i=1
x
p−1
i
...
...
. . .
...
...
n∑
i=1
x
p+1
i
n∑
i=1
x
p
i . . .
∑n
i=1 x
2
i
∑n
i=1 xi
n∑
i=1
x
p
i
n∑
i=1
x
p−1
i . . .
n∑
i=1
xi n
ap
ap−1
...
a1
a0
=
n∑
i=1
x
p
i yi
n∑
i=1
x
p−1
i yi
...
n∑
i=1
xiyi
n∑
i=1
yi
, (12)
that we will further use within our optimization models. Observing (12) from the point of view of a
crisp regression analysis, the scalars a0, a1, . . ., ap are the coefficients of the predictor function
f (x, a0, a1, . . . , ap) = a0 + a1x + . . . + apxp,
that is a polynomial of degree p in x; and the scalars xi and yi, i = 1, . . . , n used in (12) are the crisp
observed data. Observing the same matrix equality (12) from the point of view of a fuzzy polynomial
regression, all scalar quantities aq, q = 0, . . . , p, xi and yi, i = 1, . . . , n are seen as variables that range
within their corresponding α−cuts.
https://doi.org/10.15837/ijccc.2023.2.5320 7
Table 1: Observed and predicted fuzzy data based on regression analysis for the first example recalled
from [11]. The EPBRO predictions are reported as triangular fuzzy numbers approximations
Index Observed inputs Observed outputs HBS2 predictions EPBRO predictions
1 (1.5, 2.0, 2.5) (3.5, 4.0, 4.5) (3.75, 4.20, 4.72, 5.18) (3.113, 4.611, 5.935)
2 (3.0, 3.5, 4.0) (5.0, 5.5, 6.0) (4.50, 4.96, 5.50, 6.00) (4.062, 5.390, 6.496)
3 (4.5, 5.5, 6.5) (6.5, 7.5, 8.5) (5.25, 5.76, 6.81, 7.36) (5.160, 6.429, 7.618)
4 (6.5, 7.0, 7.5) (6.0, 6.5, 7.0) (6.25, 6.81, 7.33, 7.91) (6.087, 7.208, 8.218)
5 (8.0, 8.5, 9.0) (8.0, 8.5, 9.0) (7.00, 7.59, 8.12, 8.73) (6.844, 7.987, 8.957)
6 (9.5, 10.5, 11.5) (7.0, 8.0, 9.0) (7.75, 8.38, 9.43, 10.10) (7.750, 9.026, 10.272)
7 (10.5, 11.0, 11.5) (10.0, 10.5, 11.0) (8.25, 8.90, 9.43, 10.10) (8.072, 9.285, 10.547)
8 (12.0, 12.5, 13.0) (9.0, 9.5, 10.0) (9.00, 9.68, 10.20, 10.90) (8.735, 10.064, 11.406)
Denoting by x̃i and ỹi, i = 1, . . . , n the observed triangular fuzzy numbered inputs and outputs,
respectively, we formulate the box constraints system (13)
(1 − α) (x̃i)L + α (x̃i)C ≤ xi ≤ α (x̃i)C + (1 − α) (x̃i)U , i = 1, . . . , n,
(1 − α) (ỹi)L + α (ỹi)C ≤ yi ≤ α (ỹi)C + (1 − α) (ỹi)U , i = 1, . . . , n,
(13)
that corresponds to their α−cut intervals, α ∈ [0, 1].
The optimization models that derive the endpoints of the α−cut intervals of the regression’s
coefficient aq, q = 0, . . . , p minimize and maximize, respectively the objective function aq over the
feasible set defined by (12) and (13), with xi, yi, i = 1, . . . , n bounded, and a0, a1, . . ., ap free
variables.
In the same manner, the optimization models that derive the endpoints of the α−cut intervals
of the predicted output ̂̃yh, h = 1, . . . , n minimize and maximize, respectively the objective function
f (xh, a0, a1, . . . , ap) over the feasible set defined by (12) and (13). Moreover, a predicted output at a
given fuzzy input ṽ, other than the previously observed ones, can be determined by minimizing and
maximizing, respectively the objective function f (v, a0, a1, . . . , ap) over the feasible set defined by (12)
and (13) and the additional box constraint on v, i.e.
(1 − α) ṽL + αṽC ≤ v ≤ αṽC + (1 − α) ṽU .
5 Computation results
In this section we report the numerical results aiming to illustrate the theoretical statements and
provide a comparison with the results found in the literature. Both considered examples are recalled
from the literature and use triangular fuzzy numbers to describe the observed data. We derive the
predicted fuzzy outputs numerically, for eleven equidistant α−cuts, α ∈ {0, 0.1, 0.2, . . . , 0.9, 1}.
Our experiments include one instance with a single fuzzy explanatory variable; and one instance
with two fuzzy explanatory variables. The output estimations are derived using a linear predictor
function in both cases.
5.1 One explanatory fuzzy variable
The first example containing one explanatory fuzzy variable is taken from [11]. Hojati et al. [6]
solved the same example, and we use their results for comparison. The fuzzy observed data is reported
in Table 1 together with the estimated fuzzy outputs which are also graphed in Figure 1.
Table 1 reports the triangular fuzzy number approximations of the predicted fuzzy outputs derived
by our approach. HBS2 is the approach proposed by Hojati et al. [6]. HBS2 derived two intervals
that together define the trapezoidal fuzzy predicted outputs included in the comparison.
Figure 1 also illustrates how the application of the Monte Carlo method in the simulation of the
extension principle [13] proves that the results of Hojati et al. [6] do not fully comply to the extension
principle, thus EPBRO improves their results. More precisely, analyzing the predicted outputs y1,
y2, y7 and y8 one may notice the simulated values that are out of the membership functions of the
predicted outputs reported in [6].
https://doi.org/10.15837/ijccc.2023.2.5320 8
Figure 1: Graphic illustration of the predicted fuzzy outputs provided by Hojati et al. [6], the Monte
Carlo simulation [13], and our algorithm EPBRO
Figure 2: Hojati et al.’s approach [6] versus EPBRO through predicted fuzzy bands and squared
representations for the observed data reported in Table 1
Another way to visually compare the derived predictions is to graph the fuzzy bands covered by
the regression’s outputs. Figure 2 compares the fuzzy bands obtained in [6] and by our approach (on
the left), and the predicted outputs drown as squares whose horizontal edges represent the supports
of input data, while the vertical edges represent the supports of the predicted data (on the right). A
wider predicted fuzzy bands assure a better cover of the observed outputs.
Figure 3 is similar to Figure 2 but reports our results compared to the results of Kao and Chyu
[9]. The results are quite similar, showing that Kao and Chyu’s methodology [9] respects in a high
extent the extension principle. In fact, Kao and Chyu [9] provided approximate predictions that can
be improved by increasing the number of analyzed α−cuts. The advantage of our approach is in
providing the exact support of the predicted outputs, by solving two optimization problems that uses
polynomials of degree three in defining the feasible set, and polynomials of degree two in defining the
objective functions.
The regression’s coefficients derived by HBS2 [6] and our approach EPBRC are shown in Figure
4. EPBRC derives fuzzy sets ã0, ã1 with wider supports than HBS2. However, HBS2 used their
coefficients ã0, ã1 in arithmetic expressions; and obtained output predictions with wider supports than
those derived by our approach through the EPBRO algorithm. This experiment shows once again
the importance of involving the extension principle in all steps of the analysis. The coefficients ã0, ã1
derived by EPBRC provide information about all possible crisp values of the regression’s coefficients
and their membership levels, thus one can use them to study the whole system.
https://doi.org/10.15837/ijccc.2023.2.5320 9
Figure 3: Kao and Chyu’s predictions [9] versus EPBRO predictions for the observed data reported
in Table 1. In other words, approximate versus full compliance with the extension principle
Figure 4: Visual comparison of the regression’s coefficients for data given in Table 1
5.2 Two explanatory fuzzy variables
The second example has two explanatory fuzzy variables and is recalled from [19]. The observed
fuzzy data is reported in Table 2.
The estimated fuzzy outputs obtained by our approach (EPBRO), Wu’s approach [19], and Chen
and Nien’s approach [4] are included in Table 3, and graphed in Figure 5. Wu’s approach [19] was
used for comparison since their methodology partly followed the extension principle deviating from it
to reduce the computation complexity. On the other side, we also involve in the comparison Chen and
Nien’s approach [4], to emphasize their higher deviation from the accurate results. Chen and Nien’s
approach is also relevant for the comparison since it was recently published.
In Figure 5 we additionally report the results obtained by the Monte Carlo simulation procedure
[13] that clearly illustrate the difference between the results obtained in [19] and [4]: only a few com-
binations of the parameters yielded estimated values of y10 out of the membership function provided
in [19], but many combination of the same parameters derived many predicted values for all outputs
y1 − y10 out of the membership functions reported in [4].
6 Conclusions and further researches
In this paper we addressed the problem of predicting the fuzzy outputs of a fuzzy in – fuzzy
out system through a fuzzy linear regression analysis. We developed our solution approach in full
accordance to the extension principle. We proposed pairs of optimization models to be solved for
each desired α−level degree of each estimated fuzzy output. Both the left and right endpoints of
each α−cut interval are derived as minimum and maximum, respectively of all crisp values that can
https://doi.org/10.15837/ijccc.2023.2.5320 10
Table 2: Observed fuzzy inputs and outputs for the numerical example recalled form [19]
Index Observed inputs Observed outputs
1 (151, 274, 322) (1432, 2450, 3461) (111, 162, 194)
2 (101, 180, 291) (2448, 3254, 4463) (88, 120, 161)
3 (221, 375, 539) (2592, 3802, 5116) (161, 223, 288)
4 (128, 205, 313) (1414, 2838, 3252) (83, 131, 194)
5 (62, 86, 112) (1024, 2347, 3766) (51, 67, 83)
6 (132, 265, 362) (2163, 3782, 5091) (124, 169, 213)
7 (66, 98, 152) (1687, 3008, 4325) (62, 81, 102)
8 (151, 330, 463) (1524, 2450, 3864) (138, 192, 241)
9 (115, 195, 291) (1216, 2137, 3161) (82, 116, 159)
10 (35, 53, 71) (1432, 2560, 3782) (41, 55, 71)
11 (307, 430, 584) (2592, 4020, 5562) (168, 252, 367)
12 (284, 372, 498) (2792, 4427, 6163) (178, 232, 346)
13 (121, 236, 370) (1734, 2660, 4094) (111, 144, 198)
14 (103, 157, 211) (1426, 2088, 3312) (78, 103, 148)
15 (216, 370, 516) (1785, 2605, 4042) (167, 212, 267)
Table 3: The predictions provided by our approach EPBRO and the approaches presented in [19]
and [4]. The predicted outputs are reported as triangular fuzzy number approximations
Index Wu’s predictions (2003) Chen - Nien predictions (2020) EPBRO predictions
1 (40.731, 161.897, 254.675) (104.24, 161.53, 177.82) (−63.676, 161.53, 375.403)
2 (22.831, 122.669, 258.107) (88.73, 122.62, 173.13) (−72.171, 122.62, 366.204)
3 (80.253, 224.431, 397.099) (155.81, 223.10, 288.00) (−4.637, 223.10, 425.660)
4 (29.779, 131.242, 246.073) (91.49, 131.17, 172.00) (−82.896, 131.17, 374.616)
5 (−3.544, 67.701, 153.452) (51.00, 68.46, 88.21) (−99.875, 68.46, 266.596)
6 (35.861, 169.687, 306.367) (102.02, 169.00, 209.95) (−53.927, 169.00, 389.486)
7 (2.056, 79.734, 184.581) (60.85, 80.24, 110.79) (−85.174, 80.24, 295.228)
8 (41.247, 189.673, 334.309) (105.30, 188.97, 243.41) (−55.396, 188.97, 402.216)
9 (22.537, 119.833, 233.108) (82.11, 120.01, 161.51) (−83.392, 120.01, 356.569)
10 (−13.997, 53.293, 132.854) (41.00, 54.18, 70.33) (−98.220, 54.18, 247.551)
11 (120.828, 253.717, 428.6082) (202.69, 252.00, 311.77) (34.408, 252.00, 465.188)
12 (111.097, 228.693, 396.296) (192.47, 227.20, 279.32) (37.340, 227.20, 459.329)
13 (28.2686, 144.9806, 291.304) (91.37, 144.77, 204.57) (−60.669, 144.77, 383.842)
14 (18.0514, 100.5342, 195.2155) (78.00, 100.95, 127.69) (−77.172, 100.95, 334.607)
15 (73.3752, 210.9386, 364.751) (143.75, 209.96, 268.30) (−25.345, 209.96, 415.196)
https://doi.org/10.15837/ijccc.2023.2.5320 11
Figure 5: Visual comparison of the results obtained by our algorithm EPBRO and the results from
the literature
be obtained as predicted outputs to at least one regression problem with observed crisp data in the
α−cut ranges of the corresponding fuzzy observed data. Relevant examples from the literature were
solved, and the carried out experiments were used to illustrate the theoretical findings.
The main advantage of the new proposed methodology is in the fact that the obtained results are
realistic, being derived in full accordance to the extension principle, and better than the approximated
results that can be found in the literature. The computation complexity of the approach is related
to the optimization solvers that have to handle mathematical programming problems with quadratic
objective functions and polynomial constraints of degree three.
We plan to continue the research, and study how to apply the crisp non linear regression models
to estimate the behavior of fuzzy in – fuzzy out systems. Aiming to solve real life problems, we will
pay attention on selecting the norm that best suits each specific problem, and incorporate within the
approach a procedure able to remove the outliers that might be identified between the fuzzy observed
data.
Acknowledgments
This work was partly supported by the Serbian Ministry of Science, Technological Development
and Innovation through Mathematical Institute of the Serbian Academy of Sciences and Arts, and
Faculty of Organizational Sciences, University of Belgrade.
Author contributions
The authors contributed equally to this work.
Conflict of interest
The authors declare no conflict of interest.
https://doi.org/10.15837/ijccc.2023.2.5320 12
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Cite this paper as:
Stanojević, B.; Stanojević, M. (2023). Optimization-Based Fuzzy Regression in Full Compliance
with the Extension Principle, International Journal of Computers Communications & Control, 18(2),
5320, 2023.
https://doi.org/10.15837/ijccc.2023.2.5320
Introduction
Preliminaries
Problem formulation
Our approach
Generalizations to fuzzy polynomial regression analysis
Computation results
One explanatory fuzzy variable
Two explanatory fuzzy variables
Conclusions and further researches