255 http://sosains.greenvest.co.id
JURNAL
SOSAINS
JURNAL SOSIAL DAN SAINS
VOLUME 4 NOMOR 3 2024
P-ISSN 2774-7018, E-ISSN 2774-700X
SINGH'S FUZZY TIME SERIES FORECASTING MODIFICATION
BASED ON INTERVAL RATIO
Erikha Feriyanto, Farikhin, Nikken Prima Puspita
Universitas Diponegoro, Indonesia
Keywords:
Fuzzy time series,
interval ratio,
Alabama
enrollment
forecasting.
ABSTRACT
Background: One forecasting method that is often used is time series forecasting. The
development of applied mathematics has encouraged new mathematical findings that
led to the birth of new branches of mathematics, one of which is fuzzy.
Purpose: The objectives of the study, namely forecasting, fuzzy set, time series, fuzzy
time series, fuzzy time series Singh, interval ratio and measurement of accuracy level.
Method: This research method applies Chen's fuzzy time series in the section of
determining the universe of talk you to the fuzzification of historical data and in the part
of forecasting results obtained through a heuristic approach by building three
forecasting rules, namely Rule 2.1, Rule 2.2, and Rule 2.3 to obtain better results and
affect very small AFER values. As well as making modifications to the interval partition
section using interval ratios to be able to reflect data variations.
Results: Based on the calculation of AFER values for order 2, order 3, and order 4
respectively obtained at 1.06389%, 0.689368%, and 0.711947%. Therefore, it can be
said, Singh's fuzzy time series forecasting method based on the ratio of 3rd-order
intervals is better than that of 2nd-order and 4th-order.
Conclusion: Based on the results of research and discussion that has been carried out,
it can be concluded that Singh's fuzzy time series forecasting method has the same
algorithm as fuzzy time series forecasting. Singh's fuzzy time series forecasting method
based on interval ratios applies fuzzy time series and Singh forecasting. Singh's fuzzy
time series forecasting modification accuracy rate based on interval ratios produces
excellent forecasting values according to evaluator average forecasting error rate
(AFER).
INTRODUCTION
One forecasting method that is often used is time series forecasting. Time series
forecasting based on values observed in the past is subsequently used to predict future data.
The relationship between mathematical theory and real-world problems gave rise to the
terms pure mathematics and applied mathematics. According to Bell, (2012) The
development of applied mathematics has encouraged new mathematical discoveries that
Singh's Fuzzy Time Series Forecasting Modification
Based on Interval Ratio
2024
Erikha Feriyanto 256
led to the birth of new branches of mathematics. One branch of mathematics that continues
to grow is fuzzy (Bělohlávek, Dauben, & Klir, 2017).
Inaccuracies and incompleteness of past data stemming from a rapidly changing
environment (Morice, Kennedy, Rayner, & Jones, 2012). In addition, the decisions made
by experts are subjective and depend on the competence of each one. Therefore, it is more
appropriate when the data is presented in fuzzy numbers rather than firm numbers. Fuzzy
has a vague or vague meaning. The fuzzy set was first discovered by Zadeh in 1965. The
use of fuzzy allows a problem formulation to be solved with an accurate solution. Along
with the times, a new method emerged that combines fuzzy with time series analysis,
namely fuzzy time series. The way fuzzy time series works is that crips data is converted
first into a form of linguistic data commonly called fuzzy sets. The first fuzzy time series
introduced by Song & Chissom (Song and Chissom, 1993a) is a method of data based on
fuzzy principles. Establishing fuzzy relationships and fuzzification of time series is the top
priority of a model to predict fuzzy time series.
The application of fuzzy time series to forecasting the number of University of
Alabama registrants was first investigated by Song and Chissom in 1993 (Song &
Chissom, 1993). That same year Song & Chissom, (1993), developed the FTS method into
a time-variant fuzzy time series model using a 3-layer back propagation neural network to
defuzzify and apply it to the University of Alabama enrollment dataset. Then Chen, (1996)
proposed a method that is more efficient than Song & Chissom, (1993) which is to use
simplified arithmetic operations also apply to the University of Alabama enrollment
dataset. The same application was also made Singh, (2007) by proposing a better and more
versatile forecasting method based on the FTS forecasting concept of developing a form of
simple computational algorithm. Later that same year Singh generalized from previous
research with the aim of making it a powerful forecasting method (S. R. Singh, 2007). The
FTS application to forecast the number of University of Alabama registrants was also
carried out by Chen, Zou, & Gunawan, (2019) with interval proportioning methods and
particle swarm optimization (PSO) techniques.
Many applications of FTS forecasting in particular focus on interval partitioning. In
1996, Chen first conducted FTS forecasting research using the Average-Based length
method to determine effective partitions (Chen, 1996). Then Huarng, (2001) found the
distribution-based length interval partitioning method, and the results of his research were
quite effective compared to the Average-Based length method discovered by (Chen, 1996).
Furthermore, research on determining interval pertiation based on frequency density (Chen
& Hsu, 2004; Jilani, Burney, & Ardil, 2007). Then in 2006, Kunhuang Huarng proposed a
new method of determining the length of intervals based on ratios (Huarng & Yu, 2006).
After that Chen et al., (2019), proposed a new fuzzy time series (FTS) forecasting method
based on interval proportions and particle swarm optimization (PSO) techniques.
Singh's fuzzy time series research Singh, (2007) is a development and simplification
of the fuzzy time series Chen, 1996; Song & Chissom, (1993) in forecasting University of
Alabama enrollment. The development of Singh's fuzzy time series forecasting method lies
in the forecasting part, which uses a simple computational algorithm using difference
parameters as fuzzy relations. While simplifying Singh's fuzzy time series forecasting
method, because it is able to minimize the complexity of calculating fuzzy relational
Volume 4, Nomor 3, Maret 2024
p-ISSN 2774-7018 ; e-ISSN 2774-700X
257 http://sosains.greenvest.co.id
equation matrices that use complex min-max composition operations and the time
consumed by various defuzzification processes. However, the weakness or rather the part
that can be explored to obtain better forecasting performance values from Singh's fuzzy
time series research is the interval partition section that uses the 7 interval partition rule.
Therefore, the author aims to propose Singh's fuzzy time series forecasting method based
on interval partitioning that is more effective, namely interval ratio. The method proposed
on the thesis was applied to the University of Alabama enrollment dataset. Next, measure
the performance of forecasting results using Average Forecasting Error Rate (AFER).
RESEARCH METHODS
Figure 1. Research procedure flowchart
The data used in this study were secondary data obtained from journals (Song &
Chissom, 1993). This data is data on the number of applicants for the University of
Alabama. This study used data on the number of University of Alabama applicants from
1971 to 1992. By selecting data on the number of University of Alabama registrants, it is
expected to provide more accurate estimates than forecasting (Chen, 1996; Chen & Hsu,
2004; Chen et al., 2019; Huarng, 2001; Huarng & Yu, 2006; Jilani et al., 2007; S. R. Singh,
2007; Song & Chissom, 1993).
Singh's Fuzzy Time Series Forecasting Modification
Based on Interval Ratio
2024
Erikha Feriyanto 258
The research method used in this study is a literature review, namely by collecting
references in the form of books, journals and writings published on the website. From this
method (Kumar & Gangwar, 2015), Singh's fuzzy time series forecasting algorithm can be
determined based on interval ratio partitions to solve forecasting problems.
RESULTS AND DISCUSSION
After getting the forecasting results, the next step is to evaluate the forecasting results
using the average forecasting error rate (AFER). The following is given an example of
AFER calculation for order 3:
AFER=t=1n|At-Ft|Atn×100%
AFER=0,009275+0,008253+0,000331+…+0,00945519×100%
AFER=0,0068937×100%
AFER=0,68937%.
Detailed calculations of average forecasting error rate (AFER) for order 2, order 3,
and order 4 are presented in Table 4.11, Table 4.12, and Table 4.13. The following table
evaluates forecasting results using the average forecasting error rate (AFER) for order 2:
Table 1 Evaluation of forecasting results for order 2
Years
Total of registration
|At-Ft|
|At-Ft|At
1971
13055
1972
13563
1973
13867
2,32
0,000167
1974
14696
136,31
0,009275
1975
15460
96,3
0,006229
1976
15311
153,43
0,010021
1977
15603
183,71
0,011774
1978
15861
59,55
0,003754
1979
16807
118,93
0,007076
1980
16919
794,76
0,046974
1981
16388
75,13
0,004584
1982
15433
154,59
0,010017
1983
15497
90,59
0,005846
1984
15145
92,14
0,006084
1985
15163
247,12
0,016298
1986
15984
37,89
0,00237
1987
16859
112,47
0,006671
1988
18150
445,74
0,024559
1989
18970
157,57
0,008306
1990
19328
200,43
0,01037
1991
19337
395,73
0,020465
1992
18876
36,55
0,001936
Average forecasting error rate (AFER) %
1,06389%
The following table evaluates forecasting results using the average forecasting error
rate (AFER) for order 3:
Volume 4, Nomor 3, Maret 2024
p-ISSN 2774-7018 ; e-ISSN 2774-700X
259 http://sosains.greenvest.co.id
Table 2. Evaluation of forecasting results for order 3
Years
Total of registration
|At-Ft|
|At-Ft|At
1971
13055
1972
13563
1973
13867
1974
14696
136,31
0,009275
1975
15460
127,59
0,008253
1976
15311
5,07
0,000331
1977
15603
0,05
0,000003
1978
15861
85,11
0,005366
1979
16807
118,93
0,007076
1980
16919
192,45
0,011375
1981
16388
75,13
0,004584
1982
15433
154,59
0,010017
1983
15497
119,3
0,007698
1984
15145
92,14
0,006084
1985
15163
100,07
0,0066
1986
15984
37,89
0,00237
1987
16859
121,47
0,007205
1988
18150
127,15
0,007006
1989
18970
157,57
0,008306
1990
19328
161,46
0,008354
1991
19337
224,71
0,011621
1992
18876
178,47
0,009455
Average forecasting error rate (AFER) %
0,689368%
The following table evaluates forecasting results using the average forecasting error
rate (AFER) for order 4:
Table 3. Evaluation of forecasting results for order 4
Years
Total of registration
Forecasting Results
|At-Ft|
|At-Ft|At
1971
13055
1972
13563
1973
13867
1974
14696
1975
15460
15587,59
127,59
0,008253
1976
15311
15201,21
109,79
0,007171
1977
15603
15587,59
15,41
0,000988
1978
15861
15929,8
68,8
0,004338
1979
16807
16688,07
118,93
0,007076
1980
16919
17093,3
174,3
0,010302
1981
16388
16342,93
45,07
0,00275
1982
15433
15587,59
154,59
0,010017
1983
15497
15640,53
143,53
0,009262
1984
15145
15237,14
92,14
0,006084
1985
15163
15237,14
74,14
0,00489
1986
15984
15946,11
37,89
0,00237
1987
16859
16688,07
170,93
0,010139
1988
18150
18277,15
127,15
0,007006
1989
18970
19127,57
157,57
0,008306
Singh's Fuzzy Time Series Forecasting Modification
Based on Interval Ratio
2024
Erikha Feriyanto 260
Years
Total of registration
Forecasting Results
|At-Ft|
|At-Ft|At
1990
19328
19127,57
200,43
0,01037
1991
19337
19127,57
209,43
0,010831
1992
18876
18725,01
150,99
0,007999
Average forecasting error rate (AFER) %
0,711947%
Based on Table 1, Table 2, and Table 3, AFER for order 2, order 3, and order 4
respectively amounted to 1.063891%, 0.689368%, 0.711947%. Furthermore, based on
Table 1, because the three AFER values are less than 10%, it can be concluded that the
forecasting results have very good criteria. Then from the AFER results obtained, it can be
concluded that Singh's fuzzy time series forecasting method based on the ratio of intervals
to order 3 is the best method compared to order 2 and order 4.
A comparison chart visualization of Singh's fuzzy time series forecasting method
based on the interval ratio between order 2, order 3, and order 4 can be seen in the following
graph:
Figure 2. Comparative graph between actual data and order 2, order 3, and order 4
The blue graph shows the actual number of University of Alabama registrants, the
green graph shows Singh's fuzzy time series forecasting results based on the 2nd-order
interval ratio, the black graph shows Singh's fuzzy time series forecasting results based on
the 3rd-order interval ratio, and the red graph shows the fuzzy time series forecasting
resultsSingh based on the ratio of 4th-order intervals. From the graph above, it shows that
the pattern of Singh's fuzzy time series forecasting results based on interval ratios is almost
the same as the actual data on the number of University of Alabama registrants, although
the resulting forecasting value is not the same as the actual data on the number of University
of Alabama registrants.
Based on Figure 2, it can be seen that the green graph has a considerable error
because the forecasting results are not close to the actual data, for example in 1980 and
1991. Then for the red chart is better than the green chart because the pattern is closer to
the actual data. However, compared to the black color chart, for example, in 1976 and 1990,
it is not good enough because the black color graph is closer to the actual data. From the
description above, it can be concluded that Singh's fuzzy time series forecasting results
based on the ratio of 3rd order intervals are closer to the actual data compared to order 2
and order 4. This is because there are differences in the formation of fuzzy logical
relationships that affect forecasting results.
Volume 4, Nomor 3, Maret 2024
p-ISSN 2774-7018 ; e-ISSN 2774-700X
261 http://sosains.greenvest.co.id
Singh's fuzzy time series forecasting based on interval ratios for Year 1993
Singh's fuzzy time series forecasting method based on interval ratios was used to
forecast data on the number of University of Alabama enrollees in 1993. Based on Table
4.6, obtained fuzzy logical relationship order 2 in 1993 is A21,A20. Because fuzzy logical
relationship (A21, A20→) is not the same as fuzzy logical relationship (AiAj) so that the
forecasting results in 1993 for order 2 do not follow Rule 2.1, but in the form of average
forecasting results corresponding to A21 and A20 namely 18941.27+18912.552=18926.91.
As for the 3rd order fuzzy logical relationship in 1993 it was A21,A21,A20. Because fuzzy
logical relationship (A21,A21,A20→) is not the same as fuzzy logical relationship (AiAj)
so that the forecasting results in 1993 for order 3 do not follow Rule 2.2, but in the form of
average forecasting results corresponding to A21,A21 and A20 namely
19166,54+19112,29+18697,533=18992,12. As well as for fuzzy logical relationships of
order 4 in 1993 were A21,A21,A21,A20. Because fuzzy logical relationship
(A21,A21,A21,A20→) is not the same as fuzzy logical relationship (AiAj) so that the
forecasting results in 1993 for order 4 did not follow Rule 2.3, but in the form of average
forecasting results corresponding to A21,A21,A21 and A20 namely
19127.57+19127.57+19127,57+18725,014=19026.93.
The method proposed by the researcher is Singh's fuzzy time series forecasting
method based on interval ratios. Singh's fuzzy time series forecasting method based on
interval ratios applies fuzzy time series (Chen, 1996) in the section of determining the
speech universe you to fuzzification of historical data which includes fuzzy logical
raltionship (FLR) and fuzzy logical raltionship group (FLRG). Starting from the
determination of the universe of U-talk to the fuzzification of historical data which includes
FLR and FLRG, there is one step that plays an important role in forecasting fuzzy time
series, namely determining the interval partition that will affect FLR, forecasting results,
and the level of forecasting accuracy.
Fuzzy time series forecasting (Chen, 1996) still uses classical interval partition
determination by dividing the U-talk universe into equal-length intervals. The disadvantage
of determining interval partitions of equal length is that it may not reflect data variations
precisely because time series data have a tendency to fluctuate (up or down). Therefore, to
correct weaknesses, the determination of interval partitions using interval ratios (Huarng
&; Yu, 2006) with the aim of being able to reflect data variations and assist in forecasting
time series data and determining intervals based on ratios is considered better than the
same interval length.
The next problem that arises and becomes a concern after knowing that ratio is a
suitable approach to determining interval length is how to determine the percentile sample
ratio. First, if the percentile sample ratio is set too large then there will be no fluctuations
in the fuzzy time series. Suppose the percentile sample ratio is determined 200% of the
first interval of the time series data 5000, 5100, ..., 10000, 10100. Causes the first interval
between 5000 and 15000 (5000+5000×200%). Indicates that the result is undesirable
because it is unable to describe fluctuations. Second, if the percentile sample ratio is set
too small, then the fuzzy time series becomes trivial, maybe even the same as the original
time series data . Based on the same time series data , for example the ratio is too small,
say 0.02%, the first interval. Causes the first interval between 5000 and 5001
(5000+5000×0.02%), the second interval between 5001 and 5002, and so on. From this
case, it is possible that the fuzzy time series will be very close to the original time series
data , which is also undesirable. Therefore, guidelines appear in determining the percentile
sample ratio , which must be large enough so that the length of the interval will not be
trivial. Intuitively the percentile sample ratio is set to 50%.
Singh's Fuzzy Time Series Forecasting Modification
Based on Interval Ratio
2024
Erikha Feriyanto 262
Furthermore, another step that plays an important role so that it affects forecasting
results in addition to determining interval partitions is the process of forming fuzzy logical
relationship groups (FLRG). In fuzzy time series forecasting (Chen, 1996), if the FLRG of
Ai is (Ai→A1,A2,...,Am) then Ft+1=A1,A2,...,Am and the crisp value is the average of
the fuzzy set corresponding to the middle value of the interval (Chen, 1996). However,
Singh 's fuzzy time series forecasting method based on interval ratios is different from
(Chen, 1996) because the forecasting concept is based on the upper and lower and midpoint
bounds. Therefore, if there is a FLRG of Ai is (Ai→A1,A2,...,Am) then
Ft+1=A1,A2,...,Am and determines the average of the fuzzy set corresponding to the upper
and lower bounds and then obtained the midpoint. Thus, the forecasting result Ft+1 is
obtained by forecasting according to Rule 2.1, Rule 2.2, or Rule 2.3. However, if there is
an empty FLRG of Ai (Ai), then the forecasting result is Ft+1=Ai.
The forecasting results in Singh's fuzzy time series forecasting method based on
interval ratios are obtained through a heuristic approach, in contrast to fuzzy time series
forecasting (Singh, 2007a) which uses computational algorithms that aim to minimize the
complexity of calculating min-max composition operations in the FLR equation and time
in the defuzzification process. In simple terms, heuristics are simple guidelines or rules that
are commonly used by humans in assessing something or used to make decisions. The
construction of three forecasting rules, namely Rule 2.1, Rule 2.2, and Rule 2.3 aims to
obtain better results and affect very small AFER values.
Rule 2.1 is used for 2nd order fuzzy time series forecasting , Rule 2.2 is used for 3rd
order fuzzy time series forecasting , and Rule 2.3 is used for 4th order fuzzy time series
forecasting . The three rules have almost the same steps, which differ only in the initial
step, namely the addition of a new difference parameter (Dt) (Yang & Shami, 2020). The
establishment of new difference parameters as fuzzy relations is applied to the current state
to estimate the value of the next state in order to better accommodate possible data
vagueness and make it a powerful method. The new difference parameter for Rule 2.1 is
Dt=|Xt-Xt-1|, Rule 2.2 i.e. Dt=|Xt-Xt-1|-|Xt-1-Xt-2|, and Rule 2.3 i.e. Dt=|Xt-Xt-1-|Xt-1-
Xt-2-|Xt-2-Xt-3|.
The formation of a new difference parameter (Dt) results in the need to add four
other parameters, namely Mt1=Nt1=Ot1=Xt+Dt, Mt2=Nt2=Ot2=Xt-Dt,
Mt3=Nt3=Ot3=Xt+Dt/2, and Mt4=Nt4=Ot4=Xt-Dt/2 where Mtp is for Rule 2.1, Ntp is for
Rule 2.2, and Otp is for Rule 2.3 with p1,2,3,4. The addition of more than one parameter
produces many forecasting results so that it is expected to be able to reflect data variations
through the average of the forecasting results obtained. In Singh's fuzzy time series
forecasting method based on interval ratios, forecasting based on upper and lower limits
is expected to have flexibility in approaching actual data (Pritpal Singh & Borah, 2013)v. .
Then if the forecasting result of the four parameters is not in the closed interval of the fuzzy
or unqualified set more than equal to the lower bound and less than equal to the upper limit,
then the forecasting result (Ft+1) is the midpoint value of the interval (Pritpal Singh, 2017).
Meanwhile, implementing into the University of Alabama enrollment data because
related fuzzy time series research has been done previously by (Song & Chissom, 1993b),
(Chen, 1996), (Huarng, 2001), (Huarng &; Yu, 2006), (Singh, 2007a), (Singh, 2007b),
(Jilani et al., 2007), and (Zou et al., 2019). And by doing forecasting on the same data, we
can find out how effective Singh's fuzzy time series forecasting method is based on interval
ratios than previous fuzzy time series forecasting methods by (Song &; Chissom, 1993b),
(Chen, 1996), (Huarng, 2001), (Huarng &; Yu, 2006), (Singh, 2007a), (Singh, 2007b),
(Jilani et al., 2007), and (Zou et al., 2019). The effectiveness of the forecasting method
obtained can be determined based on the value of the average forecasting error rate
(AFER). Research (Song & Chissom, 1993b) obtained AFER 3.22%, (Chen, 1996)