Application of High Order Intuitionistic Fuzzy Time Series

Article Study Case APPLICATION OF HIGH ORDER INTUITIONISTIC FUZZY TIME SERIES FOR JOINT STOCK PRICE INDEX PREDICTION Abstract The Jakarta Composite Index (JCI) is an index that measures the price performance of all stocks listed on the Indonesia Stock Exchange (IDX). The movement of the JCI is an important reference for investors in making financial decisions related to investment gains and losses. Therefore, accurate JCI forecasting information is very important for investors. This research discusses the application of the Deep Learning Long Short-Term Memory (LSTM) method in forecasting the JCI on the IDX. The LSTM method is used because of its ability to handle time series data with complex and non-linear patterns. Important stages in LSTM modeling include data preprocessing, network architecture formation, model training, and performance evaluation. This study compares the LSTM method with the traditional Autoregressive Integrated Moving Average (ARIMA) model to see how successful the LSTM method is in forecasting daily JCI data. Comparison of the RMSE (root mean square error) and MAPE (mean absolute percentage error) values generated by the LSTM model and the ARIMA model shows that the LSTM method has a smaller error value, namely the RMSE value of 25.314 and MAPE of 0.745% on training data, while on testing data the RMSE value is 20.786 and MAPE of 0.589%. Thus, the LSTM method is more recommended in forecasting JCI than the ARIMA method discussed in this study. Keywords: application, high order, intuitionistic, fuzzy, time series, stock price, index prediction INTRODUCTION The Indonesia Stock Exchange (IDX) is an organized system in Jakarta that brings together sellers and buyers of securities. As a means of stock investment for domestic and foreign investors, the IDX is able to drive the country's economy. Indicators of stock price movements listed on the IDX are expressed in an index called the composite stock price index (JCI). According to Rosalyn (2018), the movement of the JCI is influenced by the rupiah exchange rate against the dollar and inflation. The rise and fall of the JCI value shows the condition of the capital market and is one of the bases for determining monetary policy. For investors, understanding the movement of the JCI is very important because they are often faced with decisions related to financial risk. In this case, investors must be able to accurately predict the likely future position of the JCI to minimize losses on their decisions. AMELIA ROSE SMITH 1 Amelia Rose Smith Instinctively, the future JCI can be estimated by observing the time series behavior pattern of JCI in the past, i.e. the historical value over time, assuming that the value in the previous time has a relationship or influences the value in the next time. However, the fluctuating and irregular time series pattern of JCI makes it difficult to model and this is a challenge for researchers. Various method approaches have been applied to JCI forecasting modeling, such as autoregressive integrated moving average, singular spectrum analysis, autoregressive distributed lag and fuzzy time series. Each method has its own strengths, for example the autoregressive method views the JCI as having a stochastic linear relationship. However, this method in its application takes into account the assumption of stationarity in the data. If the data is not stationary then it needs special handling before the modeling process, either the transformation or differentiation process so that the data meets stationary properties. Meanwhile, the singular spectrum analysis (SSA) method discussed by Chrisantama et al. (2021) does not require special assumptions as in the autoregressive method. This method is able to decompose the JCI data into signal and noise components. Unlike the autoregressive method, SSA is able to represent deterministic relationships in the data. On the other hand, the fuzzy time series (FTS) method uses a linguistic variable approach. Quantitative values in the form of real numbers are first transformed into linguistic values with predefined membership degrees. The resulting model is a relationship between the linguistic value of data in the past time and the time after that. This model is considered capable of representing nonlinear relationships in JCI data. The FTS method has been popular since Chen (1996) simplified the computation of FTS introduced by Song & Chissom (1994) on the application of forecasting the number of applicants of the University of Alabama. Starting from Chen's (1996) research, many researchers began to be interested in developing FTS methods on the same data, including Abhishek et al. (2018). Abhishekh et al. (2018, 2020) discussed the high order intuitionistic fuzzy time series (HOIFTS) method, which involves intuitionistic fuzzification that takes into account both membership and non-membership degrees. Furthermore, it has demonstrated the success of the HOIFTS method on stock price forecasting. The HOIFTS modeling procedure is generally the same as that discussed by Abhishek et al. (2018, 2020), differing only in the procedure for determining the fuzzy interval and class length. Yolcu et al. (2022) developed HOIFTS by considering the multivariate time series relationship while Ashraf et al. (2024) AMELIA ROSE SMITH 2 Article Study Case developed HOIFTS by calculating the radius of membership and non-membership degrees. Both studies show that the intuitionistic fuzzification process is able to handle uncertainty in the movement of stock data. This research focuses on the application of the HOIFTS method to JCI forecasting. The method used refers to the research of Abhishekh et al. (2018) which is considered simpler than the HOIFTS method introduced by Yolcu (2022) and Ashraf et al. (2024). Although simpler, the HOIFTS method is considered capable of handling uncertainty and representing nonlinear relationships in JCI data. To show how successful HOIFTS is in forecasting JCI, this study uses the first order Chen method (Chen, 1996) and the higher order Chen method (Chen, 2002) as benchmark methods. The error evaluation of the three methods is measured based on the root mean square error (RMSE) and mean absolute percentage error (MAPE) values. For convenience, the first order Chen method is hereafter referred to as the Chen method and the high order Chen method is hereafter referred to as the HOC (high order Chen) method STUDY OF LITERATURE The data discussed in this study are monthly JCI data on the IDX obtained from the investing.com website. The data consists of 48 observations, with data from January 2018 April 2021 as training data and data from May - December 2021 as data. Furthermore, the research steps are divided into three main parts as presented in Figure 1. AMELIA ROSE SMITH 3 Amelia Rose Smith Part A Creating intuitive fuzzy sets (IFS) Part B Calculating forecasting values Part C Error evaluation Step 1 Defines the universal set U Step 1 Calculating the score function Step 1 Calculating RMSE Step 2 Step 2 IFS fuzzification Step 2 Calculate the MAPE value Determines the length and number of intervals Step 3 Defining fuzzy sets Step 4 Calculating degrees of membership and nonmembership Step 3 Formed IFLR and IFLRG Step 4 Defuzzification Step 5 Defines fuzzy sets to IFS Figure 1. Research Flow A. Creating Intuitionistic Fuzzy Sets (IFS) Step 1: define the universe set U which can be expressed in equation (1) where 𝐷𝑚𝑖𝑛 is the minimum data, 𝐷𝑚𝑎𝑥 is the maximum data, while 𝐷1 and 𝐷2 are arbitrary positive numbers determined by the researcher. Step 2: Determining the length and number of intervals. The length and number of intervals are determined using the average-based method with the following steps. 1) Calculate the absolute value of the difference of all consecutive data 2) Determine the average value of the data obtained from step 1 and then divide the average value by 2. This value is then used as the basis for determining the length of the interval. AMELIA ROSE SMITH 4 Article Study Case 3) Determine the basis of the interval length based on half the average value obtained from stage 2 by referring to Table 1. 4) Determine the length of the interval according to the base obtained in step 3. Table 1. Basis of Intervals Range Next, U is divided into several intervals with the same interval length. Step 3: Defining fuzzy sets. Fuzzy sets are defined in equation (2) The notation 𝜇𝐴𝑖(𝑢𝑖) indicates the value of the degree of membership of the data in the interval 𝑢𝑖 which in this study can be determined using the triangular function in equation (3). Where x is the data to be determined and a, b, c, are the starting, middle, and ending points of the interval as presented in Figure 2, respectively. Figure 2: Triangular Membership Function Step 4: Calculating the degree of membership and non-membership. Membership and non-membership degrees are calculated using In equation (4), 𝜋(𝑢𝑖) and 𝛿(𝑢𝑖) are formulated in equations (5) and (6) respectively. AMELIA ROSE SMITH 5 Amelia Rose Smith The degree of membership and non-membership for each 𝑢𝑖 is calculated as follows. In equation (7), 𝑥 is the actual data, 𝜋 is the minimum value of membership degree in fuzzy sets, 𝛿 is the maximum value of membership degree in fuzzy sets. Step 5: Define fuzzy sets to IFS. Suppose 𝑋 is a non-empty set. IFS I in 𝑋 is a set of ordered triples expressed in equation (8). with functions 𝜇𝐼(𝑥), 𝑣𝐼(𝑥): 𝑋 ⟶ [0,1], 𝜇𝐼(𝑥) is the degree of membership of 𝑥 in 𝐼 and 𝑣𝐼(𝑥) is the degree of non-membership of 𝑥 in the set 𝐼. As a condition 0 ≤ 𝜇𝐼(𝑥) + 𝑣𝐼(𝑥) ≤ 1 must be met. B. Calculating Forecast Value Step 1: Calculating the score function. For each IFS 𝐼 = (𝜇𝐼, 𝑣𝐼), the score of 𝐼 can be evaluated by a score function (Xu and Yager, 2006) denoted by 𝑆 and expressed in equation (9). where 𝜇𝐼 is the IFS membership degree and 𝑣𝐼 is the IFS non-membership degree. Step 2: Perform IFS fuzzification based on the score function values obtained in Step 6. Step 3: Form Intuitionistic fuzzy logical relationship (IFLR) and Intuitionistic fuzzy logical relationship groups (IFLRG). IFLR is formed based on the relationship of two IFSs related to the fuzzification results of two consecutive historical data, for example Ii is the IFS result of data at time t and Ij is the IFS result of data at time (t+1) then IFLR can be expressed in equation (10). Furthermore, IFLG can be determined based on the formed IFLR. AMELIA ROSE SMITH 6 Article Study Case Step 4: Defuzzification. The IFS defuzzification process is based on the IFLRG result with the following rules. 1) If the IFS with order n for month i is 𝐼𝑖𝑛,𝐼𝑖(𝑛-1), . . . ,𝐼𝑖𝑘 where 𝑛 ≥ 2 and there is an IFLR of order n with IFLRG shown as follows. The forecast value for month i is 𝑚𝑗 which is the midpoint of the linguistic interval 𝑢𝑗 corresponding to IFS 𝐼𝑗. 2) If the IFS of order n for month i is 𝐼𝑖𝑛,𝐼𝑖(𝑛-1), . . . ,𝐼𝑖1 where 𝑛 ≥ 2 and there is an IFLR of order n with IFLRG shown as follows Here it can be seen that there is an ambiguity that the forecast for month i may be 𝐼𝑗1, or 𝐼𝑗2, ..., or 𝐼𝑗𝑝. Thus a further step is needed to find a high-order intuitionistic fuzzification for month i. Assume that there exists an integer m ≥ n that can resolve this ambiguity for month i and there exists an mth order IFLR with IFLRG shown as follows. In this case there exists an integer m ≥ n such that there is no ambiguity for month i and the forecast value for month i is 𝑚𝑗, i.e. the midpoint of the linguistic interval 𝑢𝑗 corresponding to IFS 𝐼𝑗. 3) If the IFSs of order n for month i are 𝐼𝑖𝑛,𝐼𝑖(𝑛-1), . . . ,𝐼𝑖1 where 𝑛 ≥ 2 and there is an nth order IFLR with IFLRG shown as follows. The forecasting value for month I, denoted by 𝐹𝑖 can be calculated by equation (11). with 𝑚𝑖𝑛, 𝑚𝑖(𝑛-1), . . . , 𝑚𝑖1 is the midpoint value of the linguistic interval 𝑢𝑖𝑛, 1), . . . , 𝑢𝑖1 according to IFS 𝐼𝑖𝑛, 𝐼𝑖(𝑛-1), . . . ,𝐼𝑖1. AMELIA ROSE SMITH 7 𝑢𝑖(𝑛- Amelia Rose Smith C. Error Evaluation The error rate of the forecasting results is evaluated using RMSE and MAPE which are calculated using the following formula. and the MAPE value is calculated using the equation with In equations (12) and (13), Yt is the actual data at time 𝑡, Ft is the forecasting value at time 𝑡, 𝑁 is the amount of data predicted. The RMSE and MAPE forecasting error values generated from the HOIFTS model are then compared with the RMSE and MAPE values of the Chen and HOC models. The Chen modeling process follows the procedure in Chen (1996) and the HOC modeling follows the procedure in Chen (2002). AMELIA ROSE SMITH 8 Article Study Case RESULTS AND DISCUSSION A. Building Intuitionistic Fuzzy Sets (IFS) The data used in this study are monthly JCI data for the period January 2018 to December 2021 which can be accessed through the investing.com website. Data from January 2018 - April 2021 is used as training data (presented in Figure 3) and the rest is COMBINED STOCK PRICE INDEX COMBINED STOCK PRICE INDEX used as testing data. MONTH Figure 3. JCI Monthly Data Plot January 2018 - April 2021 Next, the forecasting process follows the following steps. Step 1: Determine the universe set U. Based on JCI data, it is known that the minimum value (𝐷𝑚𝑖𝑛) of monthly JCI data is 4538, while the maximum value (𝐷𝑚𝑎𝑥) is 6605. Furthermore, the values of 𝐷1 = 38 and 𝐷2 = 95 are selected so that the universe set U = [4500, 6700] is obtained. Step 2: Determining the length and number of intervals. Based on the average-based method, the interval length is 100 and the number of intervals is 22. Thus 𝑢1 = [4500, 4600], 𝑢2 = [4600, 4700], 𝑢3 = [4700, 4800], ..., 𝑢22 = [6600, 6700]. Step 3: Defining fuzzy sets 𝐴𝑖 from the universe set U using equation (2) so as to obtain 𝐴1 = [4500, 4600, 4700], 𝐴2 = [4600, 4700, 4800], 𝐴3 = [4700, 4800, 4900], ...,𝐴22 = [6600, 6700, 6700]. Step 4: Calculating membership and non-membership degrees. As an illustration, the membership degree of the actual data 4716 which is in the interval 𝐴2 and 𝐴3 can be determined using formula (3), obtained. AMELIA ROSE SMITH 9 Amelia Rose Smith Thus the degree of membership of the actual data 4716 in the interval 𝐴2 is 0.84 and in the interval 𝐴3 is 0.16. The calculation of the degree of membership is also carried out on all members of the fuzzy sets 𝐴𝑖 so that it is obtained. View fuzzy sets 𝐴3, using equation (4) can be determined as follows : 𝐴𝐹 = {(4716; 0.16), (4753; 0.53), (4870; 0.3)}. Furthermore, it calculates the max-min value using equations (5) and (6) as follows. In this case, the degree of membership and non-membership in the actual data 4716 can be determined using equation (7) and obtained. Based on the results of the above calculations, the actual data 4716 has a membership degree value of 0.146 and a non-membership degree of 0.768. Step 5: Define the fuzzy sets 𝐴𝑖 into IFS form as specified in equation (8) so as to obtain I1 = {(4538; 0.325; 0.530)}, I2 = {(4716; 0.508; 0.096), (4753; 0.284; 0.320)}, I3 = {(4716; 0.146; 0.768), (4753; 0.485; 0.430), (4870; 0.274; 0.640)}, ... , I22 = {(6605; 0,049; 0,947)}. AMELIA ROSE SMITH 10 Article Study Case B. Calculating Forecast Value Step 1: Calculate the score functions 𝑆(𝐼1), 𝑆(𝐼2), 𝑆(𝐼3), . . . , 𝑆(𝐼22). For example, calculating the score function on data 4716 which is a member of the IFS I2 set has a membership degree value of 0.508 and a non-membership degree of 0.096. Furthermore, calculating using equation (9) is obtained as follows Based on the above calculation, the score value obtained on data 4716 is 0.412. The calculation of the score function is also carried out on all members of the IFS set so as to obtain 𝑆(𝐼1)= {(4538; -0.205)}, 𝑆(𝐼2)= {(4716; 0.412), (4753; -0.036)}, 𝑆(𝐼3)= {(4716; -0.622), (4753; 0.054), (4870; -0.366)},... , 𝑆(𝐼22)= {(6605; -0.897)}. Step 2: Perform the IFS fuzzification process based on the maximum score function of the JCI data for each month. For example, the JCI data in April 2020 is 4716 which is a member of the triangular fuzzy sets A2 and A3 which have score function values 𝑆𝐼2 = (4716; 0.412) and 𝑆𝐼3 = (4716; -0.622) so that the score function value 𝑆𝐼2 = 0.412 > 𝑆𝐼3 = -0.622 then the IFS fuzzification for April 2020 data is I2. The results of IFS fuzzification for JCI training data can be seen in Table 2. Step 3: Form IFLR based on the IFS fuzzification results. In March 2020 and April 2020, the IFS fuzzification results are 𝐼1 and 𝐼2, so the situation can be denoted as I1 → I2. The results of the IFLR formation of JCI training data can be seen in Table 3 Table 2. IFS Fuzzification Results of Training Data MONTH January 2018 February 2018 March 2018 February 2020 March 2020 April 2020 AMELIA ROSE SMITH 11 Amelia Rose Smith Table 3. IFLR Training Data Next, form the IFLRG obtained from the IFLR results. The results of the IFLRG formation of JCI training data order 1 to order 4 can be seen in Table 4 to Table 7. Table 4. IFLRG Training Data Order 1 Table 5. IFLRG Training Data Order AMELIA ROSE SMITH 12 Article Study Case Step 4: Defuzzification is calculating the JCI training data forecasting value. For example, JCI forecasting in April 2020. Based on Table 2, the IFS fuzzification in February 2020 and March 2020 are 𝐼10 and 𝐼1 so that IFLRG order 2 for April 2020 is as follows. Based on Table 5, there is no ambiguity to forecast JCI in April 2020 so the JCI forecasting value in April 2020 is the midpoint of 𝑢2 which is 4650. The defuzzification results of JCI training data using the HOIFTS method can be seen in Table 8. Table 6. IFLRG Training Data Order 3 Table 7. IFLRG Training Data Order 4 AMELIA ROSE SMITH 13 Amelia Rose Smith Table 8. Defuzzification of Training Data May Next, calculate the forecasting value of JCI testing data. The results of defuzzification of JCI testing data using the HOIFTS method can be seen in Table 9. Table 9. Defuzzification of Testing Data May 2021 June 2021 July 2021 August 2021 September 2021 October 2021 November 2021 December 2021 Chen and HOC modeling process is not discussed in this article. The modeling procedure follows Chen (1996) and Chen (2002) with universe of discourse U = [4500, 67000], class length 100 and defined 22 interval classes as in HOIFTS. C. Evaluation of Forecasting Errors The results of the IFLR for JCI training data can be seen in Table 3. The results of RMSE and MAPE calculations on training and testing data for HOIFTS, Chen, and HOC methods are presented in Table 10. Based on the values in the table, HOIFTS consistently produces the smallest RMSE and MAPE values compared to the other two methods. On training data, the HOIFTS method is able to reduce the RMSE value of the Chen method by 64% and the HOC method by 25% while on testing data, the RMSE value produced AMELIA ROSE SMITH 14 Article Study Case by the HOIFTS method is not much different from the other two methods. The HOIFTS method only managed to reduce the MAPE value generated by the Chen method by about 1% and the HOC method by about 3%. Table 10. Comparison of RMSE and MAPE obtained from HOIFTS, Chen and HOC Methods on JCI Data (*) Indicates the smallest value Meanwhile, when viewed from the MAPE value, the HOIFTS method is able to reduce the MAPE of the Chen method to 62% on training data and 7.5% on testing data. When compared to the HOC method, the MAPE value produced by the HOIFTS method is approximately 23% smaller than the HOC method on the training data and about 2% smaller on the testing data. Furthermore, it is necessary to visually see the comparison between the forecasting values of HOIFTS, Chen, and HOC on the testing data. Based on Figure 4, the forecasting results of the HOIFTS method (red line) tend to be closer to the actual values (blue line) than the other two methods, except in November 2021. Figure 4. Comparison Chart of Actual Value and Forecasting Value using HOIFTS, Chen and HOC methods on Testing Data AMELIA ROSE SMITH 15 Amelia Rose Smith Although the HOIFTS method is more complex than the Chen and HOC methods, this method is more recommended for forecasting JCI on the IDX considering that forecasting accuracy in this case plays a major role in decision making related to the risk of financial loss. CONCLUSION This research discusses the application of the HOIFTS method on forecasting JCI monthly data. The results of the forecasting error evaluation show that the RMSE value is 57.042 and MAPE is 0.837% on the training data, while on the testing data, the RMSE value is 38.466 and MAPE is 0.487%. This value is smaller than the RMSE and MAPE values produced by the Chen and HOC models. However, further research related to the development of the HOIFTS model needs to be done considering the results of error evaluation on testing data are almost the same as the Chen method. AMELIA ROSE SMITH 16 Article Study Case REFERENCES AMELIA ROSE SMITH 17 Amelia Rose Smith AMELIA ROSE SMITH 18 Article Study Case AMELIA ROSE SMITH 19