International Journal of Fuzzy Logic Systems (IJFLS)

International Journal of Fuzzy Logic Systems (IJFLS) Vol.10, No.1, January 2020
DOI: 10.5121/ijfls.2020.10101 1
INTERVAL TYPE-2 INTUITIONISTIC FUZZY LOGIC
SYSTEM FOR TIME SERIES AND IDENTIFICATION
PROBLEMS - A COMPARATIVE STUDY
Imo Eyoh1
, Jeremiah Eyoh2
and Roy Kalawsky2
1
Department of Computer Science, University of Uyo, Akwa Ibom State, Nigeria
2
School of Electrical, Electronics and Systems Engineering, AVRRC Research Group,
Loughborough University, UK
ABSTRACT
This paper proposes a sliding mode control-based learning of interval type-2 intuitionistic fuzzy logic
system for time series and identification problems. Until now, derivative-based algorithms such as gradient
descent back propagation, extended Kalman filter, decoupled extended Kalman filter and hybrid method of
decoupled extended Kalman filter and gradient descent methods have been utilized for the optimization of
the parameters of interval type-2 intuitionistic fuzzy logic systems. The proposed model is based on a
Takagi-Sugeno-Kang inference system. The evaluations of the model are conducted using both real world
and artificially generated datasets. Analysis of results reveals that the proposed interval type-2
intuitionistic fuzzy logic system trained with sliding mode control learning algorithm (derivative-free) do
outperforms some existing models in terms of the test root mean squared error while competing favourable
with other models in the literature. Moreover, the proposed model may stand as a good choice for real time
applications where running time is paramount compared to the derivative-based models.
KEYWORDS
Interval type-2 intuitionistic fuzzy set, Sliding mode control algorithm, Intuitionistic fuzzy set,
Intuitionistic fuzzy index.
1. INTRODUCTION
The classical FS including both the T1FS [1] and T2FS [2]are defined using the membership
functions. Literature has it that classical FSs are complementary sets [3]. This implies that to
compute the non-membership of a classical FS, one has to take the complement of the set so
defined. This may not always be the case in real life contexts because people are often times
hesitant to pin-point or specify a single numerical value as doing so indicate strong commitments
or evidence.
This brings the idea of intuitionistic FS (IFS) introduced by Atanassov in 1986 [4]. With IFS, a
set can be described by three components namely: membership function, non-membership
function and hesitation index. With these three representations, IFS becomes a more appropriate
tool for dealing with imprecise and vague information. While the non-membership function
captures additional information, the hesitation index makes the set description very intuitive and
close to human intelligence than classical FS.

2
However, using single membership and non-membership values to model a concept may not
capture enough information and uncertainty in the intuitionistic set definition. This is because
changes in the perception of the same linguistic term will lead to frequent re-tuning of the
membership and non-membership functions of IFSs. This may result to waste of resources and
sub-optimal performance of the system under some environmental and operation conditions.
Atanassov and Gargov [5] extended the concept of the IFS to interval valued intuitionistic fuzzy
set (IVIFS),with interval values similar to the notion of classical interval valued fuzzy set (IVFS)
[6]. To obtain an IVIFS, the sum of upper membership and upper non-membership must not
exceed 1. Based on the salient points discussed above for IFS and IVIFS, Eyoh et al., [7]
introduced a rule-based IT2IFLS that merges the capabilities of IT2FLS with those of IFS. The
IT2IFLS relaxes the single restriction of the IVIFS by allowing two constraints namely that the
sum of upper membership and lower non-membership must not exceed 1, similarly the sum of
lower membership and upper non-membership must not exceed 1. This makes IT2IFS more
flexible than IVIFS.
RELATED WORK
Several methods such as gradient descent (GD), extended Kalman filter (EKF), decoupled EKF
(DEKF) and the hybrid (DEKF and GD) have been applied so far for the optimization of
IT2IFLS. The optimization may be structure or parameter optimization.In [7] IT2IFLS is
proposed and the parameters of the model are updated using GD back-propagation method (a first
order derivative-based learning algorithm). The proposed model is applied to non-linear system
prediction with good results. The same model is also applied to time series prediction [8].
Recently, Luo et al., [9], proposed an evolving recurrent interval type-2 intuitionistic fuzzy
neural network (IT2IFNN) and applied the model for online learning and time series
prediction. The parameters of the IT2IFNN in [9] are optimized using EKF (a second-order
derivative-based method). In [10], the DEKF is adopted for the optimization of the parameters of
the IT2IFLS. Using the DEKF allows the parameters of the model to be grouped into vectors such
as antecedent and consequent vectors so that interactions are allowed at the second order.
Experimental results reveal that EKF-based learning models perform better than the GD-based in
terms of prediction accuracy. In [11], a hybrid model of GD back-propagation and DEKF is
employed for the adjustment of the parameters of IT2IFLS and the model applied to system
identification problem.Recently also, Yuan and Luo [12] proposed an online evolving
interval type-2 intuitionistic fuzzy LSTM-Neural Networks (eIT2IF-LSTMNN). In [12]
and [13], the parameters of the models are optimized using GD-back-propagationand
applied for regression problems.Other studies on IT2IFS is the one reported in [14], where
arithmetic operations are defined for IT2IFS using generalized trapezoidal type-2 intuitionistic
fuzzy numbers. The proposed model is used to solve a transportation problem. In this work, only
the parameters of the IT2IFLS are optimized. According to [15], membership function (non-
membership function) parameters of FLSs are very important in deciding the overall performance
of the system.
The weakness of GD and EKF-based models is that the GD and EKF-based methods are both
derivative-based approaches and involve the computation of partial derivatives for both the
membership and non-membership functions which is tedious and time consuming. For large scale
computation, these derivative-based approaches may not be the best choice.Unlike the derivative-
based models, using SMC learning algorithm to update the IT2IFLS parameters is straightforward

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and relieves the system of the computational burden of calculating the partial derivatives of these
parameters.
This paper proposes the optimization of the IT2IFLS parameters using derivative-free
methodology with application to time series and prediction problems. In this study, a sliding
mode control learning of the parameters of IT2IFLS is developed and used to solve time series
and identification problems for the first time. The antecedent and the consequent parameters
update benefits from the proposed learning algorithm.
The rest of the paper is structured as follows: In section 2, some underlying concepts are defined.
Section 3 takes a view of interval type-2 intuitionistic fuzzy logic system. In Section 4, sliding
mode control learning algorithm is defined with the update rule using sliding mode algorithm
derived in Section 5. Experimental evaluation is carried out in Section 6 with the conclusion
drawn in Section 7.
2. DEFINITIONS
DEFINITION 1: An IFS is totally defined by membership function, non-membership function and
hesitation index (π) of element, x ∊ X such that 0 ≤ A*( ) + νA*( ) + ( ) = 1[4].
When the intuitionistic index is 0, the IFS changed to classical FS. Many approaches have been
adopted in the literature for constructing membership and non-membership functions of IFS.
Some of the approaches include those reported in [16][17][18]. The IFS Gaussian membership
and non-membership function in definitions proposed in [17] are adopted in this paper and are
defined as follows:
μ(x) = (1 - ( )) * exp (− )2
(1)
ν(x) = (1 - ( )) - μ(x) (2)
DEFINITION 2: An IT2IFS, Ã* is a variant of IFS but membership and non-membership functions
are themselves fuzzy and defined as {μÃ* (x), μÃ* (x)} and {vÃ* (x), vÃ* (x)} respectively for all x
∈X with constraints: 0 ≤ μÃ*(x) + vÃ* (x) ≤ 1 and 0 ≤ μÃ* (x) + vÃ* (x) ≤ 1 [19]
For IT2IFS, two footprints of uncertainties (FOUs) are utilized which are membership function
FOU and non-membership function FOU (see Figure 1).
FOUμ (Ã*) = ⋃ [μÃ∗, μÃ∗]
∀ ∈' (3)
FOUν (Ã*) = ⋃ [vÃ∗, vÃ∗]
∀ ∈' (4)
As shown in Figure 1, for designing the non-membership function FOU, the lower membership
function becomes the upper non-membership function while the upper membership function
becomes the lower non-membership function [20].

4
Figure 1: An IT2 intuitionistic Gaussian membership and non-membership functions
FOU
3. INTERVAL TYPE-2 INTUITIONISTIC FUZZY LOGIC SYSTEM
An interval type-2 fuzzy logic system consist of intuitionistic - fuzzifier, rulebase, inference
engine and output processing block (type reduction and defuzzifier) as shown in Figure 2. The
intuitionistic fuzzifier converts crisp exogenous inputs into IT2IFS, with exactly four components
namely lower membership, upper-membership, lower non-membership and upper non-
membership. Here, singleton fuzzification is considered. The hesitation index ensures that the
sum of lower membership and upper non-membership functions in the input partition space is less
than or equal to 1 and similarly, the sum of upper membership and lower non-membership
functions is less than or equal to 1. In this study, the approach for constructing membership and
non-membership functions reported in [17] is adopted and modified to reflect the type-2 version
as follows:
μ./( .) = exp (−
0
12,34
)2
* (1- ( .)) (5)
μ./( .) = exp (− 05,34
)2
* (1- ( .)) (6)
v./( .) = (1- π
1 ( .)) - μ./( .) (7)
v./( .) = (1-π ( .)) - μ./( .) (8)
where and π are the intuitionistic fuzzy indices for center and variance respectively [17].
The and π are small numbers in the range of 0 and 1. The intuitionistic fuzzy indices can be
chosen by the user or randomly generated. In this study, and π are randomly generated in
the interval [0,1].
The intuitionistic fuzzy indices use in this study are expressed for IT2IFLS as follows:
c( ) = max (0, (1- (µÃ* ( ) + νÃ* ( )))) (9)

5
var( ) = max (0, (1 – (μÃ* ( ) + vÃ* ( )))) (10)
var( ) = max (0, (1- (μÃ* ( ) + vÃ* ( )))) (11)
such that 0≤ c( ) ≤ 1 and 0 ≤ var ( ) ≤ 1
Figure 2: Interval type-2 intuitionistic fuzzy logic system [20]
The generic IT2IFLS IF-THEN rule is represented as follows:
7/ : IF . is Ã./
∗
and … and 8 is Ã8/
∗
THEN 9/ = ∑ ;./
8
.<= . + >/ (12)
The generic rule can be formulated for membership and non-membership functions respectively
as follows:
7/
?
: IF . is Ã∗
./
?
and … and 8 is Ã∗
8/
?
THEN 9/ = ∑ ;./
?
8
.<= + >/
?
(13)
7/
@
: IF . is Ã∗
./
@
and … and 8 is Ã∗
8/
@
THEN 9/ = ∑ ;./
@
8
.<= + >/
@
(14)
Where Ã./
∗
s, are IT2IFS, 9/are the ABC rule outputs for membership function and non-
membership function, ; and >are parameters of the consequent parts. The inference engine maps
IT2IFS inputs to IT2IFS outputs using the combinations of the formulated rules. According to
[7], the final output of a TSK-type IT2IFLS is defined as follows:
y = (1 - D) ∑ E
F
/
?
G
/<= 9/
?
+ (1 - D) ∑ E
F/
@
G
/<= 9/
@
(15)
where:
E
F
/
?
=
(H4
I
J H̅
4
I
)
∑ H4
I
L
4M5 J∑ H̅
4
I
L
4M5
(16)
E
F/
@
=
(H4
N
J H̅4
N
)
∑ H4
N
L
4M5 J∑ H̅
4
N
L
4M5
(17)

6
and E
F
/
?
and E
F/
@
are normalized firing signals for membership and non-membership functions
respectively.
4. SLIDING MODE CONTROL LEARNING
The error of IT2IFLS is defined as follows:
E(t) = 9 (t) – 9(t) (18)
The error shows the difference between the measured output and the predicted output of the
model. The zero value of the error coordinate is shown as time varying sliding surface [21] as
below:
S(E(t)) = E(t) = 9 (t) – 9(t) = 0 (19)
With this,the system which is in a sliding mode is guaranteed to be on a sliding. In this way, the
IT2IFLS output will match the actual output as close as possible for all time t >OC, where OC is the
hitting time for E(t) = 0.
DEFINITION 3: A sliding motion will be on a sliding manifold S(E(t)) = E(t) = 0 after a time OC if
the condition S(t)PQ (t) < 0 is valid ∀O in some nontrivial semi open subinterval of time of the form
[t, OC) ⊂ (-∞, OC) [22][23]
5. PARAMETER UPDATE
In this study, the SMC learning is used to update the parameters of IT2IFLS. The antecedent
parameters are center (c), lower membership function standard deviation (T) and upper
membership function standard deviation (T) while w and b are consequent parameters with D as
the user defined parameter. The update rules of the IT2IFLS parameters are adapted from [24] for
IT2FLS as follows for the antecedent parameters:
For the membership function parameter update, we have the following:
UQ./ = Q. + ( .-U./)V=sgn(E) (20)
TQ./ = - (T./ + 34
W
( 3 34)2)V=sgn(E) (21)
TQ./= - (T./ +
134
W
( 3 34)2)V=sgn(E) (22)
where V= is the learning rate and E is the learning error. Since the lower non-membership
assumes the upper membership function and vice versa, the standard deviation update for the
upper membership function applies to standard deviation update for the lower non-membership
function and vice versa.

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The consequent parameters are updated as follows:
;
Q ./ = - .
(= X)H
F
4
I
J X H
F4
N
G ((= X)H
F
4
I
J X H
F
4
N)
* Vsgn(E) (23)
Where M = ((1-D)E
F
/
?
+ DE
F/
@
)T
.
>Q./ = -
(= X)H
F
4
I
J X H
F4
N
G ((= X)H
F
4
I
J X H
F
4
N)
* Vsgn(E) (24)
In order to adjust the D values, the following vectors are defined: Y
Z?
(t) ≡ [Y
Z? (O) + Y
?
(O)]. For
the lower membership function, we obtained: Y
Z? (O) = [Y
Z=
?
(O)Y
Z?
(O) … Y
ZG
?
(t)]T
and Y
Z?
(t) =
[Y
Z
=
?
(O)Y
Z?
(O) … Y
Z
G
?
(O)]T
for upper membership function. Similarly, for the non-membership
functions: Y
Z@
(t) ≡ [Y
Z@ (O) + Y
@
(O)] is specified and defined as Y
Z@ (O) = [Y
Z=
@(O)Y
Z@(O) … Y
ZG
@
(t)]T
and Y
Z?
(t) = [Y
Z
=
@(O)Y
Z@(O) … Y
Z
G
@
(O)]T
for lower and upper non-membership functions respectively.
Y = [9= 9 … 9G ] represents rule consequent output vector. Then:
D = -
=
^(_
ZI J _
ZI J _
ZN J _
N
)
* Vsgn(E) (25)
In this study, the membership and non-membership function parameters are tuned using SMC
learning algorithm.
6. PERFORMANCE EVALUATION
This study analysis two real world datasets and two artificially generated datasets and compares
the performances with existing studies in the literature. For all experiments, two IT2IFS are
utilized in order to reduce the complexity of the model. The value of the user-defined parameter D
is kept at 0.5 for all experiments which are carried out using MATLAB© 2019.
6.1. Australian new electricity market
The Australian new electricity market otherwise known as New South Wales (NSW) electricity
market data is a real world dataset. Following from [25], four weeks corresponding to four
Australian seasons of the year are adopted. These seasons are weeks in January, May, August and
October given as in [25]:
• January 24 - January 30
• May 24 - May 30
• August 24 - August 30
• October 24 - October 30
The NSW electricity dataset was analysed in [25] using wavelet neural network. According to
[25], the NSW electricity market in the year 2008 was chosen for the analysis because it was the
largest. In [10], the same dataset was analysed using IT2IFLS trained with the decoupled
extended Kalman filter algorithm. To aid comparison with other learning algorithms, the
computational arrangement is the same as that used in [10] with 336 input data for each season,

8
231 data points are used for training while 100 data points are used for testing. The input-output
vector is [x(t-4), x(t-3), x(t-2), x(t-1); x(t+1)], where x(t+1) is the desired output. For the four
inputs NSW electricity dataset, 16 rules are generated with 192 parameters. The dataset for each
season is normalized to fall within the range of zero and one and the performance computed over
30 simulations with the root mean squared error (RMSE) and mean absolute error (MAE) as the
performance metrics. The RMSE and MAE are computed as below:
RMSE = `
=
a
∑ (9 − 9b)
a
.<=
2
(26)
MAE =
=
a
∑ |9 − 9b
|
a
.<= (27)
where N is the total number of test data, 9 and 9b
are the actual and predicted outputs
respectively.
Table 1: Performance of different models during Summer season.
Table 2: Performance of different models during Autumn season.
Table 3: Performance of different models during Winter season.

9
Table 4: Performance of different models during Spring season.
As indicated in tables 1 to 4, IT2IFLS-SMC outperforms other IT2IFLS models in terms of the
test RMSE during the Summer and Spring seasons and competes favourable with other models in
the other instances. The DEKF trained models tend to exhibit better prediction accuracy in terms
of training RMSE in most instances while the type-1 version do perform better in some cases, an
indication that a type-1 variant of FLS may work well in some instances depending on the type of
data and level of uncertainty in the data.There is a loss in the performance SMC trained IT2IFLS
in terms of the training RMSE. However, the computational cost of IT2IFLS-SMC is much more
impressive than other competing models in all the cases. For example, the running times for
IT2IFLS-SMC, GD and DEKF for one instance of NSW electricity data are 25.78secs, 28.03secs
and 44.75secs respectively. Hence IT2IFLS-SMC may stands as an appropriate choice in real
time applications where running time is paramount.However, using the SMC learning algorithm
with the sign function exhibitedsome chattering effects in the system during the evaluation of the
model. This might also explain the poor performance of IT2IFLS-SMC in the training RMSE.
6.2. Canadian Lynx Dataset
The Canadian lynx time series dataset consist of the number of lynx trapped in the McKenzie
river annually in Northern Canada and is taken from period 1821 to 1934. To aid comparison
with existing studies [26][27][28][10] the logarithms to base 10 of the Canadian time series is
adopted with a periodicity of 10 years. The Canadian time series has a total of 114 sample points
where 100 instances are used for training and the rest are used for testing. For a fair comparison
with previous studies, the experiment is run for 2000 epochs. The quality of prediction is
measured using the mean squared error (MSE) and MAE. The MSE is computed as below:
MSE =
=
a
∑ (9 − 9b
)
a
.<=
2
(28)
While the MAE is as shown in (26).

10
: (a) (b)
Figure 3 (a) shows original Canadian lynx data and (b) transformed Canadian lynx data (log10)
Table 5: Comparison of IT2IFLS with other models on Canadian lynx data

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Figure 4: Actual and predicted output of Canadian lynx data
Figure 4shows the actual and predicted values of Canadian lynx data. As shown in Table 5,
IT2IFLS-SMC performs better than other existing models in the literature. The SMC-based
IT2IFLS prediction accuracy is very close to the GD-based IT2IFLS in this problem domain.
6.3. System identification problem 1
Here, comparison of IT2IFLS-SMC learning with other existing methods in the literature using a
system identification problem is conducted. In the first non-linear dynamic system identification,
we adopt a second-order time-varying system defined as:y(t+1) = f(y(t), y(t-1), y(t-2), u(t), u(t-1))
where f( =, , d, e, f ) = 5, 2, W , g ( W h) J i
J 2
2 J W
2 (29)
and a, b, and c are time-varying parameters defined as below:
a(t) = 1.2 - 0.2cos(2 O/k) (30)
b(t) = 1.0 - 0.4sin(2 O/k) (31)
c(t) = 1.0 - 0.4sin(2 O/k) (32)
Here, T = 1000 represents the total number of sample points. The parameters a, b and c take the
value 1 as reported in [9]. Similar to [38], in order to keep a manageable number of parameters,
only two inputs values are utilised which are u(t) and y(t) with y(t +1) as the desired output. The
plot of u(t) versus y(t) is as shown in Figure 5.

12
Figure 5: Plot showing the relationship between the two input u and y for system
identification 1.
The following control signal is used to test the performance of the non-linear system.
l
m
n
m
o sin r
sB
f
t O < 250
1.0 250 ≤ O < 500
− 1.0 500 ≤ O < 750
0.3 sinr
sB
f
t + 0.1 sinr
sB
d
t + 0.6 sin r
sB
=|
t 750 ≤ O < 1000
} (33)
The simulation is conducted for 1000 time steps with 100 training epochs. The performance
metrics used is the RMSE. For the non-linear system identification, 4 rules are obtained with 40
tunable parameters.
Table 6: Performance comparison of IT2IFLS with other models on system identification 1

13
Figure 6: Actual and predicted outputs for system identification problem 1
Shown in Table 6 is the performance comparison of IT2IFLS-SMC with other learning models in
the literature. As shown in Table 6, the performance of IT2IFLS is comparative to those reported
in the literature especially the evolving classical IT2FLS methods reported in [9]. Figure 6 shows
the actual and predicted outcome for system identification problem 1.
6.4. System identification problem 2
For the second identification problem, a non-linear dynamic plant with longer input delay is
adopted and defined as:
y(t+1)=0.72y(t) + 0.025y(t-1)u(t-1) + 0.01u2
(t-2)+0.2u(t-3) (34)
Equation (34) involves a one-step ahead prediction using two previous outputs and four previous
inputs. The test signal in (34) is adopted to test the quality of prediction. Similar to [12], we adopt
the same training data and time steps as in system identification 1. To reduce the complexity of
the system, two inputs u(t) andy(t)are passed into the IT2IFLS. Shown in Figure 6 is the
relationship between the two inputs u and y for system identification 2.
Figure 7: Plot showing the relationship between the u and y in system identification problem 2

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Table 7:Performance comparison of IT2IFLS on system identification problem 2 with other models in the
literature
Figure 8: Actual and predicted outputsfor system identification problem 2
Figure 7 shows the actual and predicted output for system identification problem 2. As shown in
the figure, the predicted output closely follows the actual output indicating a good learning
performance. As shown in Table 7, IT2IFLS-SMC performs well in this problem instance in
terms of the test RMSE compared to other existing models in the literature.
7. CONCLUSION
This paper analysis time series and system identification problems using IT2IFLS trained with
SMC learning algorithm. As demonstrated in the experimental studies, IT2IFLS-SMC
outperforms some existing studies in the literature and competes favourably with others in some
problem instances in terms of the test RMSE. For the training RMSE, IT2IFLS-SMC performs
poorly in many cases. A careful look at all the Australian seasons prediction, IT2IFLS-SMC
performs poorly compared to other learning models in terms of the training RMSE.

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In the future, it will be worthwhile to look at using other derivative-free learning algorithms and
possibly hybridize them and evaluate their predictive strengths. It will also be interesting to
consider other membership functions outside the conventional membership functions such as
elliptic and diamond-shaped membership functions for interval type-2 intuitionistic fuzzy set.
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AUTHORS
Eyoh, Imo Jeremiah received the B.Sc. and M.Sc. degrees in Computer Science
from the University of Uyo and Ibadan, all in Nigeria in 1999 and 2005
respectively. She obtained her PhD in Computer Science from the University of
Nottingham, United Kingdom in 2018. She is currently working in the Department
of Computer Science, University of Uyo, as a Senior lecturer. Her main research
interest is uncertainty modelling using fuzzy logic. In particular, she works in the
area of type-2 fuzzy logic (classical and intuitionistic) with application to diverse
problem domains. She has published many papers in reputable national and international journals and
conferences. She is a member of Computer Professionals Registration Council of Nigeria (CPN) and
Organisation for Women in Science for the Developing World (OWSD). She is also a member of IEEE.
Eyoh, Jeremiah Effiong received the B.Sc in Computer Engineering from the
University of Uyo in 2006, Nigeria. He received the M.Sc in Control Engineering
from the University of Sheffield, United Kingdom in 2008. He is currently working
towards the Ph.D. degree in Electrical, Electronics and Systems Engineering at the
Loughborough University, United Kingdom.
Short Biography

International Journal of Fuzzy Logic Systems (IJFLS)

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