Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4, 1309-1323 2024 Publisher: Learning Gate DOI: 10.55214/25768484.v8i4.1508 © 2024 by the authors; licensee Learning Gate © 2024 by the authors; licensee Learning Gate * Correspondence: farikihn@lecturer.undip.ac.id An alternative fuzzy time series forecast model combined generalized fuzzy logical relationship, natural partitioning, and adaptive defuzzification Farikhin1*, Muhammad Sam’an2, Bayu Surarso1, Bambang Irawanto1, Bibit Waluyo Aji1 1Departments of Mathematics, Universitas Diponegoro, Indonesia; farikihn@lecturer.undip.ac.id (F.) bayus@lecturer.undip.ac.id (B.S.) b_irawanto@yahoo.co.id (B.I.) bibitwaji@gmail.com (B.W.A.) 2Universiti Muhammadiyah Malaysia, Malaysia. p52400012@student.umam.edu.my (M.S.). Abstract: This paper, we propose generalized fuzzy logical relationships based on natural partitioning and adaptive defuzzification. The proposed method provides a better approach to improve performance by producing a good evaluation of the forecasted value. This study aims to minimize forecasting errors for each data series. The general suitability of the proposed model was tested by implementing it in the fore- casting of student enrollments at the University of Alabama. In order to show the superiority of the proposed model over existing methods, the results obtained have been compared with evaluations such as MSE, RMSE, MAE, MAPE, and forecasting error errors for each time series data. Comparative studies show that the proposed method is superior to existing methods for all evaluations provided.. Keywords: Adaptive defuzzification, Forecasting, Fuzzy time series, Generalized fuzzy logical relationships, Natural partitioning; 1. Introduction Forecasting continuously sequenced data that changes over time or time series is an important and exciting problem in various applications, such as predicting stock prices in the stock market, monitoring weather or air pollution in environmental protection, and estimating the number of student enrollments in universities. These problems are frequent. This problem has been extensively studied and studied comprehensively in the fields of statistics, signal processing and neural networks in the last decade. Unclear and incomplete data phenomena make forecasting problems difficult to solve. [1], [2] introduced fuzzy time series to deal with the un- certainty of the data by representing the data as linguistic values in an uncertain environment. They predict the fuzzy time series for enrollment at the University of Albama with 4 procedures: (1) partitioning the universe of discourse into intervals of equal length. (2) define the universe of discourse, fuzzify the time series and model the fuzzy relations on each time series data. (3) forecasting (4) defuzzification of fore- casted output. Modeling fuzzy equation equations on each data series and reasoning estimates require a large processing time in fuzzy relations. Since launching Song and Chissom’s method, many researchers have competed to reduce forecasting errors and computational burden. Standard matrix multiplication operations on the Markov model [3], Simplified arithmetic operations in the forecasting process[4]. High-order fuzzy time series models [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. A heuristic approach for forecasting fuzzy time series[24], [25], [26], [27], [28]. The definition of a universe of discourses with arbitrarily selected parameters and was decomposed into the same interval length in step (1) greatly influences forecasting performance significantly[24]. The partitioning approach of the universe of discourse has been widely proposed. Interval length based on partition density [10], [29], [30], [31], [32], [33], [34], Automatic clustering techniques [8], [35], [36], [37], [38, p. 201], [39], [40], K-Means Clustering [41], [42], [43], [44]interval 1310 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1309-1323, 2024 DOI: 10.55214/25768484.v8i4.1508 © 2024 by the authors; licensee Learning Gate ratio[24], [45], [46], [46], [47]. Low forecasting error has been demonstrated through all partitioning approaches of the universe of discourse. All of these approaches identify a very large number of intervals. The more intervals identified, the smaller the forecast error achieved. However, too many intervals could result in fewer fluctuations in the fuzzy time series and complicate defuzzification. [48] proposed natural Partitioning (NP) to tackle the issue of interval length efficiently. The 3-4-5 natural partition rule based on the value range at the most significant digit (MSD) is applied to divide the universe of discourse naturally. NP produces small intervals compared to partition density, automatic clustering, K-Means Clustering, and interval ratio (University of Alabama enrollment data experiment 10 interval values for NP and 18 to 24 for other approaches). This paper used NP for partitioning the universe of discourses. The formation of FLR by optimizing the intervals segmentation on the partition of a universe of discourse. FLR is one of the most critical factors affecting forecasting accuracy in fuzzy time series [31]. Generalized FLR or GTS (M, N) with M is a number of orders, and N is the hierarchies of principal fuzzy relationship applied to data forecasting at the University of Alabama, resulting in performance that GTS (M, N) is better than three conventional fuzzy time series models [37], [49]. In addition, forecasting with defuzzification. [50]proposed a Weighting Fuzzy Time Series (WFTS) based on chronological order in the fuzzy logical group to deal with repeated FLR problems in defuzzification.[51] adopted WFTS for enrollment forecasting resulting in competitive errors. Defuzzification of GTS (M, N) with WFTS for enrollment forecasting produces a small error. However, the forecasting results for each data series still found high errors. There exists defuzzification of fuzzy time series data is inappropriate when using WFTS. Therefore, this paper proposes an adaptive rule based on a minimum distance as an additional defuzzification rule The proposed method selects the shortest distance between the actual data and the forecasting value based on the FLR. Adaptive rules based on a minimum distance can handle high error issues in each fuzzy time series defuzzification, especially the enrollment data for the University of Alabama, which is the benchmark for testing the fuzzy forecasting model. 2. Brief of Some Concepts 2.1. Fuzzy Time Series The basic concept of fuzzy time series (FTS) was introduced by [1], [2], where the values of FTS are represented by fuzzy sets. Let 𝐿 = 𝑝1, 𝑝2, 𝑝3, . . . 𝑝𝑛 be the universe of discourse. The fuzzy set 𝐾𝑖 of L is defined by 𝐾𝑖 = 𝑓𝐾𝑖 (𝑝1) 𝑝1 + 𝑓𝐾1(𝑝2) 𝑝2 +. . . + 𝑓𝐾1(𝑝𝑛) 𝑝𝑛 where 𝑓𝐾𝑖 is the membership function of fuzzy 𝑓(𝐾𝑖)(𝑝𝑖) degree of 𝑝𝑖 in the fuzzy set 𝐾 and 1 ≤ 𝑎 ≤ 𝑛 . Let 𝑍(𝑡) (𝑡 =. . . ,0,1,2, . . . ) is the universe of discourse from the predetermined fuzzy set 𝑓𝑖(𝑡) if 𝐹(𝑡) is a set of 𝑓1(𝑡), 𝑓2(𝑡) then 𝐹(𝑡) is a fuzzy time series defined by 𝑍(𝑡) [5]. If 𝐹(𝑡) is caused only by 𝐹(𝑡 − 1) then the fuzzy logical relation is represented by 𝐹(𝑡) = 𝐹(𝑡 − 1) ∗ 𝑅(𝑡, 𝑡 − 1) which is a fuzzy relation between 𝐹(𝑡) and 𝐹(𝑡 − 1) where is the operator. For simplicity, given 𝐹(𝑡 − 1) = 𝐾𝑖 and 𝐹(𝑡) = 𝐾𝑗 The fuzzy logic relation between 𝐹(𝑡) and 𝐹(𝑡 − 1) can be expressed as 𝐾𝑖 → 𝐾𝑖 where 𝐾𝑖 is called the Left-Hand Side (LHS) and 𝐾𝑗 is called the Right-Hand Side (RHS) of the Fuzzy Logic Relationship (FLR). Furthermore, fuzzy logic relations can be grouped into Fuzzy Logic Relationship Groups (FLRG) to build different fuzzy relations. Given 𝐹(𝑡) fuzzy time series. If 𝐹(𝑡) is caused by 𝐹(𝑡 − 1), 𝐹(𝑡 − 2) , . . . , 𝐹(𝑡 −𝑚) , then the fuzzy logic relations are expressed as 𝐹(𝑡 − 𝑚), . . . , 𝐹(𝑡 − 2)𝐹(𝑡 − 1) → 𝐹(𝑡) , and is called the m- order fuzzy time series forecasting model. Given 𝐺(𝑡) (𝑡 =. . . . ,0,1,2, . . . ) a fuzzy time series, where a fuzzy set represents the value of 𝐺(𝑡) If 𝐺(𝑡) is caused by 𝐺(𝑡 − 1), 𝐺(𝑡 − 2) , . . . , 𝐺(𝑡 − 𝑚), then 1311 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1309-1323, 2024 DOI: 10.55214/25768484.v8i4.1508 © 2024 by the authors; licensee Learning Gate FLR will be represented as 𝐺(𝑡 − 𝑚), . . . , 𝐺(𝑡 − 2), 𝐺(𝑡 − 1) → 𝐺(𝑡) and is called the m-order FLR, 𝑚 = 1 and 𝑚 ≥ 2 are one order and high order, respectively. 2.2. Natural Partitioning This section, interval formation will be carried out, where the universe of discourse partition intervals into equal-length sub-intervals uses the concept of a hierarchy based on natural partitioning (NP). This paper uses the 3-4-5 rule of NP, which recursively collects continuous values into uniform, intuitive or natural intervals [48]. The rule is based on the most significant digit (MSD) in an interval; the continuous value is partitioned into 3, 4, or 5 relatively equal sub-intervals. Table 1 informs the high-level to low-level recursively. From this table, partitioning intervals can use the 3 - 4 -5 rule where the need to produce a hierarchical concept in discretization variables. The impact of the interval length on the FTS can be investigated through the 3-4-5 rule. Table 1. The 3-4-5 rule of NP [48] Distinct values of the value range at MSD Number of intervals segmented 3, 6, 9 3 equiwidth intervals 7 3 intervals in the grouping of 2-3-2 2, 4, 8 4 equiwidth intervals 1, 5, 10 5 equiwidth intervals 2.3. The Higher Order Fuzzy Time Series Based on Generalized Fuzzy Logical Relationship Higher-order forecasting is based on general fuzzy logical relationships, namely the process of creating a relationship matrix and finding out the fluctuation pattern of a time series based on basic fuzzy rules that are easy to understand. Let 𝐿 partitioned into 𝑛 equal intervals of length 𝑙𝑖𝑛 then 𝜇𝐾𝑖(𝑡) is the degree of membership defined to calculate the weight of fuzzy set 𝐾𝑖 by the following equation. 𝜇𝐾𝑖(𝑥𝑡) = { 1, 𝑖 = 1 ∧ 𝑥𝑡 ≤ 𝑚1 1, 𝑖 = 𝑛 ∧ 𝑥𝑡 ≤ 𝑚𝑛 max {0, (1 − |𝑥𝑖 − 𝑚𝑖| 2 × 𝑙𝑖𝑛 )}𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (1) at time 𝑡𝑖(𝑖 = 1, 2, 3, . . . , 𝑛), where 𝑥𝑡 is the time series value at time 𝑡 with 𝑚1 is middle value at time 𝑡1, 𝑚𝑛 is middle value at time 𝑡𝑛 and 𝑙𝑖𝑛 is interval of length. Suppose 𝐺𝐾(𝑡 − 𝑀) = (𝜇1(𝑡 − 𝑀), 𝜇2(𝑡 − 𝑀), . . . , 𝜇𝑛(𝑡 − 𝑀)) and 𝑓𝐾(𝑡) = (𝜇1(𝑡), 𝜇2(𝑡), . . . , 𝜇𝑛(𝑡)). If 𝜇�̅� (𝑡−𝑀) and 𝜇𝑗 𝑡 are the maximum values of (𝜇1(𝑡), 𝜇2(𝑡), . . . , 𝜇𝑛(𝑡)), respectively, then 𝐾𝑖 (𝑡−𝑀) → 𝜇𝑗 𝑡 is called the 𝑀- order first principal fuzzy relationship, denoted as GT 𝑆(𝑀, 1) (generalized fuzzy logical relationships). If �̅�𝑖 𝑡−𝑀 is the 𝑁𝑡ℎ maximum value of (𝜇1(𝑡 − 𝑀), 𝜇2(𝑡 − 𝑀), . . . , 𝜇𝑛(𝑡 −𝑀)), then �̅�𝑖 𝑡 → �̅�𝑗 𝑡+1 is called the relation 𝑀𝑡ℎ order 𝑁𝑡ℎ main fuzzy logic, denoted as 𝐺𝑇𝑆(𝑀,𝑁) that is grouped into matrix 𝑀 × 𝑁 and expressed as 𝑅(𝑘,𝑙)(𝑘 = 1, 2, . . . , 𝑀; 𝑙 = 1, 2, . . . , 𝑁)[49] 2.4. Weighted Fuzzy Time Series Weighted Fuzzy Time Series (WFTS) is the development of the FTS, which handles the problem of recurring fuzzy logic relations and gives weight to each fuzzy relation to describe the difference in the importance of the sequence of fuzzy relations [50]. The FLR with the same LHS can be grouped into FLRG by placing all of their RHS together as the RHS in FLGR. For example, 𝐴𝑖 → 𝐴𝑗, 𝐴𝑖 → 𝐴𝑘 . 𝐴𝑖 → 𝐴𝑘 , 𝐴𝑖 → 𝐴𝐹 be grouped 𝐴𝑖 → 𝐴𝑗 , 𝐴𝑘 , 𝐴𝑘 , 𝐴𝑝 The second FLR in which there is an intermediate 1312 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1309-1323, 2024 DOI: 10.55214/25768484.v8i4.1508 © 2024 by the authors; licensee Learning Gate chronological order 𝐴𝑖 and 𝐴𝑗, 𝐴𝑘 , . . . , 𝐴𝑝 Next, the computation to determine the weight is formulated as follows [51]: 𝜔 = [ 𝜔1, 𝜔2 , … , 𝜔𝑛 ] = [ 𝑗 𝑗 + 𝑘 + 𝑙 + 𝑚 , 𝑘 𝑗 + 𝑘 + 𝑙 + 𝑚 ,⋅⋅⋅, 𝑝 𝑗 + 𝑘 + 𝑙 +𝑚 ] = [ 𝑐1 𝑐1 + 𝑐_2 + 𝑐3 + 𝑐4 , 𝑐2 𝑐1 + 𝑐_2 + 𝑐3 + 𝑐4 ,⋅⋅⋅, 𝑐4 𝑐1 + 𝑐_2 + 𝑐3 + 𝑐4 ] = [ 𝑐1 ∑ 𝑐ℎ 𝑛 ℎ = 1 , 𝑐2 ∑ 𝑐ℎ 𝑛 ℎ = 1 ,⋅⋅⋅, 𝑐𝑛 ∑ 𝑐ℎ 𝑛 ℎ = 1 ] (2) 3. The Proposed Method Forecasting results on each time series data found a high error. The weighting based on chronological order in the fuzzy logical group to proposed by [50] has yet to provide the best solution for defuzzification. This paper proposes a weighted generalized fuzzy logic relationship based on natural partitioning and adaptive defuzzification on high-order fuzzy time series. The workflow of the proposed method is shown in Figure 1. This paper applied the proposed method to forecasting enrollments at the University of Alabama is as shown in [4]. The following is the proposed method algorithm: Figure 1. Workflow of proposed method. Step 1: Define the universe of discourse and partition intervals. Supposed 𝑆𝑚𝑎𝑥 and 𝑆𝑚𝑖𝑛 are actual maximum and minimum enrollment data, respectively. Select 𝜀1 and 𝜀2 are appropriate positive integer numbers such that ℓ = (𝑆𝑚𝑎𝑥 + 𝜀1 − (𝑆𝑚𝑖𝑛 − 𝜀1)) partitioned with the digital position of ℓ MSD. Next, the universe of discourse is defined to be 𝑆 = [𝑆𝑚𝑖𝑛 − 𝜀1, 𝑆𝑚𝑎𝑥 + 𝜀2]. In Table 2, 𝑆𝑚𝑎𝑥 = 19337, then 𝜀1 = 3055 and 𝜀2 = 663. Hence, 𝑆 = [10000, 200000]. Step 2: the 3-4-5 rule of NP is implemented to S and five equiwidth intervals at the first level are obtained due 𝑡𝑜 (20000 − 10000) 10000 = 1: [10000, 12000, [12000 − 14000], [14000 − 16000], [16000 − 18000], and [18000 − 20000]. However, [10000 − 12000] needs to be deleted due to without covering any historical value. In the following, play the NP rule to the remaining four intervals, respectively. For example, [12000, 14000] will be partitioned into four equal subintervals with (14000 − 12000) 10000 = 2. Thus, a second level interval partition can be obtained as shown in 1313 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1309-1323, 2024 DOI: 10.55214/25768484.v8i4.1508 © 2024 by the authors; licensee Learning Gate Table 3. Three Intervals, i.e. [12000,12500], [12500,13000], and [195000 − 20000] are deleted because of there exist no actual data are covered. Thus, the obtaining intervals are Step 3: Build fuzzy sets. From Table 4, the triangular fuzzy sets is presented. Table 2. The interval partition of actual enrollment data using the NP Rule. The one-level interval partition The 2nd -level interval partition [12000, 14000] [12000, 12500] [12500, 13000] [13000, 13500] [13500, 14000] [14000, 16000] [14000, 14500] [14500, 15000] [15000, 15500] [15500, 16000] [16000, 18000] [16000, 16500] [16500, 17000] [17000, 17500] [17500, 18000] [18000, 20000] [18000, 18500] [18500, 19000] [19000, 19500] [19500, 20000] Table 3. Partition of the universe of discourse. Index Interval Index Interval s1 [13000,13500] s8 [16500,17000] s2 [13500,14000] s9 [17000,17500] s3 [14000,14500] s10 [17500,18000] s4 [14500,15000] s11 [18000,18500] s5 [15000,15500] s12 [18500,19000] s6 [15500,16000] s13 [19000,19500] s7 [16000,16500] From Table 2 and 4, the value of membership degree by Eq.(1). Next, the fuzzification process is carried out by selecting 2 values of the maximum degree of membership (Jilani et al., 2010). In detail, The value of the membership degree and the fuzzification results are shown in Table 5. Table 4. Triangular fuzzy sets. Fuzzy Interval fuzzy A1 [13000, 13250, 13500] A2 [13500, 13750, 14000] A3 [14000, 14250, 14500] A4 [14500, 14750, 15000] A5 [15000, 15250, 15500] A6 [15500, 15750, 16000] A7 [16000, 16250, 16500] A8 [16500, 16750, 17000] A9 [17000, 17250, 17500] A10 [17500, 17750, 18000] A11 [18000, 18250, 18500] A12 [18500, 18750, 19000] A13 [19000, 19250, 19500] 1314 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1309-1323, 2024 DOI: 10.55214/25768484.v8i4.1508 © 2024 by the authors; licensee Learning Gate Step 4 Build FLR based on generalized fuzzy logical relationships or GTS(M,N) with 𝑀 = 3 and 𝑁 = 2. The fuzzy logical relationships of the enrollments as shown in Table 7. Next, create the FLRG for each GTS(M,N). The groups identified for the enrollments is shown in Table 8. Step 5: Determine weight by using weight matrix proposed by using Eq. (2). The weight matrix identified for the enrollments is presented in Table 9. Step 6: Play adaptive defuzzyfication. Let 𝐴𝑖 → 𝐴𝑗 , 𝐴𝑘 , . . . , 𝐴𝑝 is FLRG and the corresponding weight for 𝐴𝑗 , 𝐴𝑘 , . . . , 𝐴𝑝 are 𝜔1, 𝜔2, . . . , 𝜔𝑛. Mid value of 𝐴𝑗, 𝐴𝑘 , . . . , 𝐴𝑝 are 𝑚𝑗, 𝑚𝑘 , . . . , 𝑚𝑝. The final defuzzification is denoted as the multiplication of the defuzzification matrix and the transpose of the weighting matrix as follows: 𝜚(𝑡) = 𝑀(𝑡) × 𝑊(𝑡)𝑇 = [𝑚𝑗 , 𝑚𝑘, . . . , 𝑚𝑝] × [𝜔1, 𝜔2, . . . , 𝜔𝑛] (3) Thus, the adaptive defuzzyfication denoted by 𝑇 that is formulated as follows: 𝑇(𝑡) = min[(𝑚𝑗 − 𝑥(𝑡)), (𝑚𝑘 − 𝑥(𝑡)), . . . , (𝑚𝑝 − 𝑥(𝑡)), (𝜚(𝑡) − 𝑥(𝑡))] (4) Table 5. The value of the membership degree and the fuzzification results. Year(t) Actual (x) max(μ_(K_i )) Fuzzified enrollment 1 2 1971 13055 0.55 0.05 A1;A2 1972 13563 0.93 0.56 A1;A2 1973 13867 0.87 0.63 A2;A1 1974 14696 0.80 0.69 A3;A4 1975 15460 0.96 0.64 A5;A4 1976 15311 0.81 0.61 A5;A4 1977 15603 0.89 0.60 A5;A6 1978 15861 0.86 0.63 A6;A5 1979 16807 0.86 0.63 A8;A7 1980 16919 0.91 0.58 A8;A7 1981 16388 0.88 0.61 A7;A6 1982 15433 0.93 0.56 A5;A4 1983 15497 0.99 0.50 A5;A4 1984 15145 0.85 0.64 A4;A5 1985 15163 0.83 0.66 A4;A5 1986 15984 0.98 0.51 A6;A5 1987 16859 0.85 0.64 A8;A7 1988 18150 0.85 0.65 A10;A11 1989 18970 0.97 0.53 A12;A11 1990 19328 0.82 0.67 A13;A12 1991 19377 0.87 0.62 A13;A12 1991 18876 0.87 0.62 A12;A11 Based on Eq. (4), if 𝑇(𝑡) = (𝑚𝑗 − 𝑥(𝑡)), then 𝑇(𝑡) = 𝑚𝑗 . Suppose from Table 7, 𝐴1 is the fuzzification of 𝑥(1971) = 13055 and 𝑥(1972) = 13563. FLR in 𝑅(1,1) used for bulid of FLRG as 𝐴1 → 𝐴1, 𝐴2. If 𝐹(𝑡 − 1) = 𝐴1, then forcasting value is 𝐴1, 𝐴2. From Eq. (3) is obtained 𝜚(1972) = 13583 and Eq. (4) is obtained 𝑇(1972) = min(313, 187, 20) = 20. Hence, 𝑇(1972) = 20, then 𝑇(𝑡) = 13583. Next, we will calculate the forecasting results at 𝑅(1,2), 𝑅(2,1), 𝑅(2,2), 𝑅(3,1) and 𝑅(3,2) in 1315 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1309-1323, 2024 DOI: 10.55214/25768484.v8i4.1508 © 2024 by the authors; licensee Learning Gate the same away analogously. The results of enrollment data forecasting using generalized FLR with 𝑀 = 3 and 𝑁 = 2 are presented in Table 10. Step 7: Evaluation of forecasting results. The calculations of rrror evaluation used Mean Absolute Error (MAE), Root Mean Squared Error (RMSE) and Mean Absolute Percentage Error (MAPE). MAE = 1 𝑛 ∑|𝐴𝑡 − 𝐹𝑡 | 𝑛 𝑡=1 (5) RMSE = √ ∑ (𝐴𝑡 − 𝐹𝑡) 2𝑛 𝑡=1 𝑛 (6) MAPE = ∑ | 𝐴𝑡−𝐹𝑡 𝐴𝑡 × 100%|𝑛 𝑡=1 𝑛 (7) From Eq. (5), (6) and (7), we calculated the evaluation error of the enrollment data forecasting. The comparison results of enrollment data forecasting errors are presented in Table 10. 1316 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1309-1323, 2024 DOI: 10.55214/25768484.v8i4.1508 © 2024 by the authors; licensee Learning Gate Table 6. The FLR of enrollments based on generalized fuzzy logical relationships. Year (t) Actual (x) Fuzzified enrollment k=1 k=2 k=3 l=1 l=2 l=1 l=2 l=1 l=2 1971 13055 A1;A2 - - - - - - 1972 13563 A1;A2 A1→ A1 A2→ A1 - - - - 1973 13867 A1;A2 A1→ A2 A2→ A2 A1→ A2 A2→ A2 - - 1974 14696 A2;A3 A2→ A3 A1→ A3 A1→ A3 A2→ A3 A1→ A3 A2→ A3 1975 15460 A3;A4 A3→ A5 A4→ A5 A2→ A5 A1→ A5 A1→ A5 A2→ A5 1976 15311 A5;A4 A5→ A5 A4→ A5 A3→ A5 A4→ A5 A2→ A5 A1→ A5 1977 15603 A5;A6 A5→ A5 A4→ A5 A5→ A5 A4→ A5 A3→ A5 A5→ A5 1978 15861 A6;A5 A5→ A6 A6→ A6 A5→ A6 A4→ A6 A5→ A6 A4→ A6 1979 16807 A8;A7 A6→ A8 A5→ A8 A5→ A8 A6→ A6 A6→ A8 A4→ A8 1980 16919 A8;A7 A8→ A8 A7→ A8 A6→ A8 A5→ A8 A5→ A8 A4→ A8 1981 16388 A7;A6 A8→ A7 A7→ A7 A8→ A7 A5→ A7 A7→ A7 A6→ A7 1982 15433 A5;A4 A7→ A5 A6→ A5 A8→ A5 A7→ A5 A6→ A5 A5→ A5 1983 15497 A5;A4 A5→ A5 A4→ A5 A7→ A5 A6→ A5 A4→ A4 A7→ A5 1984 15145 A4;A5 A5→ A4 A4→ A4 A5→ A4 A4→ A4 A4→ A4 A7→ A4 1985 15163 A4;A5 A4→ A4 A5→ A4 A5→ A4 A4→ A4 A4→ A5 A6→ A4 1986 15984 A6;A5 A4→ A6 A5→ A6 A4→ A6 A5→ A6 A5→ A8 A6→ A6 1987 16859 A8;A7 A6→ A8 A5→ A8 A4→ A8 A5→ A8 A5→ A10 A4→ A8 1988 18150 A10;A11 A8→ A10 A7→ A10 A6→ A10 A5→ A10 A7→ A12 A5→ A10 1989 18970 A12;A11 A10→ A12 A11→ A12 A8→ A12 A7→ A12 A11→ A13 A5→ A12 1990 19328 A13;A12 A12→ A13 A11→ A13 A10→ A13 A11→ A13 A11→ A13 A5→ A13 1991 19377 A13;A12 A13→ A13 A12→ A13 A12→ A13 Table 7. The generalized fuzzy logical relationship group. Group k=1 k=2 k=3 l=1 l=2 l=1 l=2 l=1 l=2 G1 A1→ A1, A2 A1→ A3 A1→ A2, A3 A1→ A5 A1→ A3, A5 A1→ A5 G2 A2 → A3 A2→ A1, A2 A2→ A5 A2→ A2, A3 A2→ A5 A2→ A3, A5 1317 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1309-1323, 2024 DOI: 10.55214/25768484.v8i4.1508 © 2024 by the authors; licensee Learning Gate G3 A3→ A5 A3→ A5 A3→ A5 G4 A4→ A4, A6 A5→ A8, A4, A6, A8 A4→ A6, A8 A5 → A8, A6, A8, A10 A4→ A8, A10 A5→ A5, A5, A10, A12, A13 G5 A5→ A5, A5, A6 A6→ A6, A5 A5→ A5, A6, A8, A4 A6→ A8, A5 A5→ A6, A8, A8, A4, A6 A6→ A7, A4, A6 G6 A6→ A8, A8 A7→ A8, A7, A10 A6→ A8, A10 A7→ A7, A5, A12 A6→ A7, A12 A7→ A5, A4, A13 G7 A7→ A5 A11→ A12, A13 A7→ A5 A11→ A13, A13 A7→ A4 A11→ A12 G8 A8→ A8, A7, A10 A12→ A12, A13 A8→ A7, A5, A12 A12→ A13 A8→ A13, A5, A6 - G9 A10→ A12 A10→ A13 - A10→ A13 - G10 A12→ A13 A12→ A13 - A12→ A12 - G11 A13→ A13, A12 - - - - Table 8. The weight matrix of enrollments. Group k=1 k=2 k=3 l=1 l=2 l=1 l=2 l=1 l=2 G1 (1, 3/2) (1) (2/5,3/5) (1) (3/8,5/8) (1) G2 (1) (1, 3/2) (1) (2/5,3/5) (1) (3/8,5/8) G3 (1) (5/24,5/24,5/ 24,5/24,4/24) (1) (5/24,5/24,4/ 24,6/24,4/24) (1) (6/30,8/30,8/30,8/30) G4 (4/10, 6/10) (8/26,4/26,6/26,8/26) (6/14,8/14) (8/32,6/32,8/ 32,10/32) (8/18,10/18) (5/45,5/45,10/45, 12/45,13/45) G5 (5/25, 5/25, 6/25, 5/25, 4/25) (6/11, 5/11) (5/29,6/29,8/ 29,4/29,4/29) (8/13, 5/13) (6/32,8/32,8/ 32,4/32,6/32) (7/17,4/17,6/17) G6 (8/16,8/16) (8/25,7/25,10/25) (8/18,10/18) (7/24,5/24,12/24) (7/19,12/19) (5/22,4/22,13/22) G7 (1) (12/25,13/25) (1) (13/26,13/26) (1) (1) G8 (8/25,7/25,10/25) (12/25,13/25) (7/24,5/24,12/24) (1) (13/24,5/24,6/24) - G9 (1) - (1) - (1) - G10 (1) - (1) - (1) - G11 (13/25,12/25) - (1) - - - 1318 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1309-1323, 2024 DOI: 10.55214/25768484.v8i4.1508 © 2024 by the authors; licensee Learning Gate Table 9. The results of the actual forecasting enrollment data. Year (t) Actual (x) k=1 k=2 k=3 l=1 l=2 l=1 l=2 l=1 l=2 1971 13055 - - - - - 1972 13563 13583 13583 - - - 1973 13867 13750 13750 13950 13950 - 1974 14696 14250 14250 14250 14250 14875 14875 1975 15460 15250 15250 15250 15250 15250 15250 1976 15311 15283 15250 15250 15250 15250 15250 1977 15603 15750 15250 15657 15750 15250 15250 1978 15861 15750 15750 15750 15750 15750 15750 1979 16807 16750 16750 16750 16750 16750 16750 1980 16919 17010 17010 16750 16875 16750 16750 1981 16388 16250 16250 16250 16250 16250 16250 1982 15433 15250 15523 15250 15250 15250 15250 1983 15497 15283 15250 15250 15250 15250 15250 1984 15145 15250 15167 15250 15208 14750 15250 1985 15163 15350 15250 15250 15208 14750 15250 1986 15984 15750 16212 15750 15750 16125 15750 1987 16859 16750 16750 16750 16875 16750 16750 1988 18150 17750 17750 17750 17750 17750 17894 1989 18970 18750 18990 18750 18750 18750 18750 1990 19328 19250 19250 19250 19250 19250 19250 1991 19377 19250 19250 19250 19250 19250 19250 1991 18876 18750 18750 18750 18750 18750 18750 Table 10. The comparison results of enrollment data forecasting errors. Evaluation Model Proposed Method [49] M = 1 M = 2 M = 3 M = 1 M = 2 M = 3 MAE GTS(M,1) 152.17 147.52 168.95 473 471 501 GTS(M,2) 142.06 138.02 138.62 372 375 390 RMSE GTS(M,1) 185.9 184.1 209.7 625 632 643 GTS(M,2) 186.4 180.1 165.0 447 449 460 MAPE GTS(M,1) 0.00937 0.00903 0.01039 0.0293 0.0291 0.0307 GTS(M,2) 0.00882 0.00846 0.00844 0.00227 0.00227 0.0023 4. The Comparison Study Forecasting of the University of Alabama enrollment data has been done in a fuzzy time series. The existing methods aim to achieve the smallest evaluation values (MAE, RMSE, MAPE). Based on Table 11, it can be seen that the proposed method produces evaluation values of all orders of the smallest compared to the evaluation values [49]. The lower the RMSE value, the more variation in the value generated by the forecasting model is closer to the interpretation. The proposed method obtains an average MAPE value of less than 10%, so the model’s forecasting performance is very good. Moreover, the proposed method can handle high forecasting errors for each data series by using adaptive defuzzification with Eq. (4). The proposed method can obtain small errors in each data series forecast. The comparison results of forecasting errors for each enrollment data series is shown in Figure 2. The figure shows that the proposed model, namely GTS (1.1), GTS (1.2), GTS (2.1), GTS (2.2), GTS(3,1) 1319 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1309-1323, 2024 DOI: 10.55214/25768484.v8i4.1508 © 2024 by the authors; licensee Learning Gate and GTS(3,2), produces an error of less than 500 in each data series and GTS(1.2) is the model that has the smallest error compared to other models. Correspondingly, Qiu’s method obtained GTS (1,2) as the best model with the smallest MSE [37] We compare the proposed method with the existing methods methods [6], [8], [12], [15], [16], [49], [52], [53] for enrollment data forecasting. Figure 2. The comparison results of forecasting errors for the different of GTS(m,n) Figure 3. The comparison results of forecasting errors for each enrollment data series. A comparison of the Mean Square Error (MSE) of the proposed method with existing methods is shown in Table 12. The MSE of the proposed method is the smallest compared to existing methods. We also compare the forecasting error values for each data series provided in Figure 3. Similarly, the MSE value, the error value of the proposed method, is smaller than the existing methods. Meanwhile, the comparison of all evaluations on data enrollment forecasting is shown in Figure 4. The proposed method is superior compared to existing methods from all the evaluations provided (MAE, RMSE, MAPE). The comparative study explained above clearly shows that the proposed method can overcome the problem of forecasting error for each time series data. Based on the evaluation, the proposed method is superior to existing methods. Therefore, the proposed method has a good chance of being developed in forecasting with different data. 1320 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1309-1323, 2024 DOI: 10.55214/25768484.v8i4.1508 © 2024 by the authors; licensee Learning Gate Figure 4. The comparison of all evaluations on data enrollment forecasting. 5. Conclusion This paper proposes a new defuzzification method with adaptive rules by considering the smallest distance between the actual value and the defuzzification result as the selected forecasting value. The proposed method can minimize errors for each data series. The proposed method is also tested for the efficiency of forecasting fuzzy time series enrollment data at the University of Alabama and provides a comparative study with existing methods [6], [8], [12], [15], [16], [49], [52], [53]. From Table 11, we see that in forecasting the enrollments at the University of Alabama, our proposed method outperforms the method proposed by the existing methods. Although this study made great improvements in dealing with the high error problem for each data series forecaster, there is a limitation 1321 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1309-1323, 2024 DOI: 10.55214/25768484.v8i4.1508 © 2024 by the authors; licensee Learning Gate of the proposed method: adaptive defuzzification is applied to partitioned time series data with universes of discourse into equal-length sub-intervals. If the interval lengths differ and the partitions increase, the defuzzification calculation process becomes more complicated. Table 11. A comparison of the Mean Square Error (MSE) of the proposed method with existing methods. Year Actual (x) The existing method [52] [53] [8] [49] [12] [15] [6] [16] Propose 1971 13055 - - - - - - - - - 1972 13563 14230 14242 13512 13902 13680.75 14049 15049 13637 13583 1973 13867 14230 14242 13998 13902 13844.43 14349 15149 14120 13750 1974 14696 14230 14242 14658 13902 14951.36 14549 15149 14408 14250 1975 15460 15541 15474.3 15341 15576 15532.34 15049 15349 15195 15250 1976 15311 15541 15474.3 15501 15576 15533.19 15549 15549 15712 15283 1977 15603 15541 15474.3 15501 15576 15533.19 15449 15549 15635 15750 1978 15861 16196 15474.3 15501 16246 15533.19 15649 15649 15786 15750 1979 16807 16196 16146.3 17065 16246 16298.77 15749 15649 15918 16750 1980 16919 16196 16988.3 17159 17251 17113.79 16349 15849 16406 17010 1981 16388 17507 16988.3 17159 17251 17113.79 16449 15949 16466 16250 1982 15433 16196 16146.3 15341 15576 16298.77 16049 15749 16190 15523 1983 15497 15541 15474.3 15501 15576 15533.19 15549 15549 15698 15250 1984 15145 15541 15474.3 15501 15576 15533.19 15549 15549 15731 15167 1985 15163 15541 15474.3 15501 15576 15532.34 15349 15549 15550 15250 1986 15984 15541 15474.3 15501 15576 15532.34 15349 15549 15559 16212 1987 16859 16196 16146.3 17065 16246 16298.77 15849 15649 15982 16750 1988 18150 17507 16988.3 17159 17251 17113.79 16349 15849 16433 17750 1989 18970 18872 19144 18832 18591 18741.35 17149 16149 17366 18890 1990 19328 18872 19144 19333 19596 19190.44 17649 16249 17967 19250 1991 19377 18872 19144 19083 19094 18972.15 17849 16349 18230 19250 1992 18876 18872 19144 19083 19094 18972.15 17849 16349 18236 18890 MSE 251055 219031 117532 192067 179247 772901 2145801 572931 34759 Acknowledgment: This works was financially supported by research grant of Faculty of Science and Mathematics Universitas Diponegoro. 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