IEEE Systems, Man and Cybernetics Magazine - July 2022 - 56

x =t 4h
11 h^ t 4
-- --
_
t
x ,3h
t
x4h
t
Bh yB
xBy
tri
^
After the calculation of x ,1h
x ,2h
t hi
+ Byi
x ,3h
x4h
.
(15d)
and ,x1h
t
x ,2h
t
we have an IT2FR problem that can be solved
with the previous IT2FR model. It is only enough to
replace the new parameters in the IT2FR model. With the
replacement of
x ,1h
and
x ,1h
t
x ,2h
t
x ,3h
t
x ,2h
x4h
t
x ,3h
with Ay ,i
x4h with p ,i
Ay ,i
the T2FR can be formulated as follows:
.
MinII II I 1234
=+-+
subject to
By 11 1
^ hCyByq hx qi
h hh .
h hh .
ii i ii ih
-- -- --t
^
i 11 4
i ii
hByAyq hq x
h^
h^
# ^^
By +- -+ --t
h^
By 11 2
^
ii i ii ih
-- -- --t
^
i 11i ii
hByAyq hq x
h^
$ ^^ ih
$ ^^
By +- -- --3t
## ## #rr j jj j
hCyByq hx qi
0 aa bc cj
# ^^ ih
Experimental Results
TAIEX forecasting data are utilized to measure the
function of our suggested model. The TAIEX data in
the year 2000 have been employed to show the ability
of the TT2FR model. Data from 11/2 to 12/15 are
employed to train the model (training data). The data
from 12/16 to 12/29 are used to test the performance of
the model (testing data). H in TT2FR is adopted as 0.4
to obtain the optimal performance. Data are fuzzified
into 58 TT2F sets based on main, low, and high values
obtained for many days. Model efficiency is evaluated
by comparing root-mean-square error (RMSE) and is
given by [14]-[16], [22], [33]:
h hh.
h hh .
p ,i
r
By ,i
q ,i
Cy ,i
r ,i
ri
r
Cy ,i
(16)
Here, n is the number of data employed.
Figure 7 and Table 1 show results obtained using our
proposed model. Figure 7 shows the results obtained using
several T2FR models. Qualitatively, our suggested model
can predict the unknown data more closely. Table 1 represents
the summary of comparison with other state-of-theart
methods done using the same database (TAIEX
forecasting data). It can be noted from Figure 7 and Table 1
that for both the training and the testing data, our proposed
TT2F model yielded more proper outcomes with
minimum RMSE value compared to other T2FR models.
Our TT2FR model is able to handle the uncertainties in the
data much better; hence, it has yielded the lowest number
of prediction errors.
Figure 7 and Table 1 show that the suggested model's
RMSE is lower than the RMSE of the model posited by
Poleshchuk et al. [25], the WGP model [27], and the
IT2FLR model [30]. Table 1 shows that this model also
performed better than the classic and type-1 fuzzy models.
It shows that the predictive capability of this suggested
model is more effective than that of earlier T2FS
models [15], [34].
Due to the uncertainty in the nature of the problem
data (like the stock market), a forecasting method based
on modeling uncertainty will be more useful. T2F models
are more suitable to handle uncertainty compared to
other methods. Such models are optimized with different
criteria to attain the maximum output with real data.
Also, they obtain lower RMSE values for both the training
and the testing models. Our proposed model has obtained
the highest results and better accuracy in forecasting.
Due to the uncertainty in the nature of problem data
(TAIEX), a forecasting method that is based on modeling
RMSE=
| ^^hh^ h2
-
n
11
=
actual value forecasted value
n
tt
.
(17)
Actual Time Series and Its Prediction
H
y1h
y1
B
k
y2h
a
y2
6,200
6,000
5,800
5,600
5,400
5,200
5,000
4,800
4,600
Figure 6. The secondary membership function in
an H-cut.
56 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE July 2022
Actual
Poleshchuk et al. [25] Approach Model
WGP Model
IT2F Regression Model
Proposed TT2FR Model
Test
Train
05 10 15 20 25
Date
30 35 40 45 50
Figure 7. TAIEX forecasting results.
TAIEX
IEEE Systems, Man and Cybernetics Magazine - July 2022

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