International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 169-182
Published online June 25, 2022
https://doi.org/10.5391/IJFIS.2022.22.2.169
© The Korean Institute of Intelligent Systems
Reham Raouf and Saad Elsaieed
Department of Insurance & Actuarial Sciences, Faculty of Commerce, Cairo University, Cairo, Egypt
Correspondence to :
Reham Raouf (rehamraouf@foc.cu.edu.eg)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In recent years, many methods have been proposed to forecast data in different fields based on successful fuzzy time series models (FTS). Egyptian social insurance systems (SISs) need support to optimally define and estimate yearly total benefits (pensions), which helps the actuaries who are responsible for the system make optimal decisions. Given that the total benefits have not been forecasted before by prediction methods, this paper proposes FTS models by Chen, Cheng, Yu, and Song to forecast Egyptian social insurance benefits, proposes Huarng for appropriate partition lengths, and constructs the interval length using the difference in the transformation data method, given that the data has not been stationary in recent years and has increased significantly. The proposed approach is based on experiments implemented using four models with interval length partitions of 5, 10, 50, 100, and Huarng partitions of 465. The results show great progress in the performance of yearly benefit forecasting, especially in the Chen model with a Huarng 465 partition, which has high accuracy prediction with low error when training and testing data.
Keywords: Fuzzy time series, Social insurance, Benefits, Pensions
Forecasting activities play a significant part in our everyday lives; they are critical in a variety of businesses because projections of future occurrences must be factored into decision-making. Given that social insurance is one of the most significant concerns that most nations confront because of the sensitivity of its relationship with public budget impacts, forecasting insurance is an important issue, and better forecasting aids actuaries and the responsibility of making strategic choices in the social insurance system. In this context, this study proposes a prediction of the total benefits in millions (pensions) required by Egyptian insurance systems.
According to their conventional definition, pension schemes and social insurance systems (SISs) are one of the three primary components of social protection, together with social assistance, social safety nets, employment strategies, and labor market interventions [1]. SISs are a mechanism that allows workers to engage in securing their future, and they are partially or entirely supported by worker and employee contributions. Egypt has a broadly stratified SIS operating as a fully funded plan, where employees make payments that are invested and subsequently reimbursed as pensions. The system has gradually evolved with a defined benefit plan. The system is primarily governed by several laws, and the present legal framework, Law 148/2019, was formally established on the first of January 2020. This, in turn, plays a significant role in changing the system and overcoming challenges due to the complexity of the old social insurance regulatory and legal framework. In practice, the system faces serious challenges and is extremely fragmented with several benefit packages available to various parts of the workforce, offering payment types for employees and their families categorized by old age, disability, survivors, illness, work injury, and unemployment [2,3]. In addition, several factors have led to Egypt’s benefits being unsustainable and inefficient. For example, the reserves of pension funds are invested at low, occasionally negative, real interest rates. They provide extremely generous minimal pensions and opportunities for early retirement. Members may also easily alter the size of their pensions, which are based solely on payments made in the few years leading to retirement. Many workers agree with their employers to under-report their income throughout the majority of their working careers to reduce their payments. As a result of these factors, Egypt’s pension funds will soon spend more on pension payouts than is received via contributions from members [4]. Therefore, it should be noted that the rate of total Egypt’s benefits (pensions) in recent years is increasing sharply, forcing us to forecast the total benefits in the next few years. Annual benefits are estimated based on actuarial models and used to provide demographic and financial forecasts for pension systems. They are often generated from models applied to vocational pension schemes that cover groups of employees and are based on demographic and economic factors. Consequently, actuarial methods in connection with predictive analytics must significantly enhance their understanding of predicted behavior or events to support their policies and judgments [5]. In addition, implementing a year-by-year simulation approach to estimate the future costs of benefits is one of the most significant tasks for actuaries [6]. Therefore, benefit prediction and analysis will aid actuaries in their professional responsibility of making strategic choices in the social insurance system. To the best of our knowledge, our proposed approach is the first to predict total benefits regarding Egyptian social insurance using fuzzy time series (FTS) models with a yearly dataset (pensions); this approach leads to sufficient and outperformed results. In this study, we present the use of FTS models to forecast Egyptian social insurance benefits (pensions) rather than traditional assumption methods for better forecasting pension demand.
Fuzzy set theory was developed to address the ambiguity and uncertainty that characterize most real-world problems. The highly popular fuzzy set theory, proposed by Zadeh [7] in 1965, is used to explain linguistic fuzzy information through mathematical modeling and generates various findings. It is still extensively utilized in a wide range of applications. Subsequently, a number of academics proposed several models, the first of which was a FTS forecasting model. Song and Chissom [8] developed books explaining the concepts of the fuzzy set theory to overcome the challenges of classical time series. However, this model incorporates the max-min composition method, which makes the computing process substantially more difficult [9]. The Chen model also contains a series of studies on the creation and execution of experimental University of Alabama enrollment data forecasting. Chen [10] proposed a simpler calculation technique to address this flaw with the benefit of reducing computing time and making the procedure more understandable. However, for fuzzy logical relationships (FLRs), this model lacks an appropriate weight mechanism. The model has now been widely modified, with academics attempting to increase the prediction accuracy by modifying the weighting technique or increasing the duration of linguistic intervals. Hwang et al. [11] proposed a forecasting approach based on a FTS. Chen and Hwang [12] proposed a FTS-based temperature forecasting algorithm. Chen [10] improved forecasting by using a high-order FTS model. The expanded Chen’s model was used in [1], and Huarng in [13] predicted enrollments using heuristic principles, which reduced computations. Huang [14] calculated interval lengths using both average-based lengths and distribution bases. Fuzzy theory was further refined and suited by Yu and his colleague [15,16] to address weighing and recurring issues using typical FTS models, including stock price predictions. To improve forecasting accuracy, Yu [15] used multiple weighted technologies on FTS models. Then, the ratio-based interval length was added to the FTS model proposed by Huarng and Yu [17]. Cheng [18] used a FTS model combined with a trend-weighting method to forecast real stock price trading data and university enrolment.
This paper is organized as follows. In Section 2, we review related studies that implemented FTS in different domains with various datasets to extract model features. Section 3 presents the steps of the FTS models, and Section 4 illustrates the implementation of the proposed FTS model and the forming benefits dataset. In Section 5, the results and discussion of the model’s approach and accuracy in existing work are presented. Finally, Section 6 summarizes and concludes the study.
Machine-learning models have contribute to SIS. The author in [19] proposed the use of three algorithms, namely the decision tree, native Bayes, and CN2 rule induction, to predict classification based on some of the social insurance features for people. The three algorithms obtained high accuracy results when classifying. The author in [20] proposed a novel deep learning-based framework based on the recurrent neural network (RNN) architecture for forecasting an individual’s payment status using a real dataset gathered by Taiwan’s Ministry of Health and Welfare. The model is being developed to effectively abstract people’s payment behavior and accurately forecast their future payment behavior over a lengthy period. Compared with state-of-the-art approaches, such as support vector machines (SVM) and hidden Markov models (HMM), the model outperforms even under varied settings. However, these studies are related to the prediction of individuals’ personal payments, which is an issue that differs from our target.
Social insurance benefits (pensions) still make traditional assumptions. In this section, we focus on related studies that implemented FTS in different domains using various datasets. There are studies related to FTS models in different industries and domains that predict enrollment, stock marketing, tourism, shipping, and transportation. We led the forced implementation of FTS to predict Egypt’s social insurance in a series of years, including the total amounts of pensions.
Although the dataset related to our proposal is represented in a series of years, the critical data must be predicted using the FTS model. The available FTS models implemented for prediction in the studies are as follows. The author in [21] proposed two models to predict the demand for tourism arrival in Taiwan as well as arrivals during the period of 1989–2000 from Hong Kong, Germany, and the United States. The FTS is more appropriate for predicting tourism demand, especially from Hong Kong arrival to Taiwan with an error of 1.93, when compared to the greedy algorithm. The author in [22] measured the prediction performance for tourist arrivals in Indonesia using a FTS compared with classical methods such as seasonal autoregressive integrated moving average (SARIMA), Box–Jenkins methods, time series regression, and Holt-Winters. The dataset was divided into training datasets from January 1989 to December 1996. The testing data were collected from January 1997 to December 1997. The root mean square error (RMSE), mean square error (MSE), mean absolute deviation (MAD), and mean absolute percentage error (MAPE) were measured. The accuracy of the resultant Chen’s FTS was outperformed by classical methods after used data transformation. Although FTS methods are simple, they outperformed all the statistical classical methods, which were used previously to predict social insurance; this prompted our proposal for using FTS methods for predictions. In [23], power companies used the proposed FTS model for electrical power to predict electrical load to make decisions. The dataset of the regional electric load in Taiwan, represented yearly, was used in this study and a comparison between FTS and previously implemented models’ linear and nonlinear regression was conducted. The accuracy of FTS outperforming in the MAPE is smaller than in other implemented models, especially when given appropriate intervals using an efficient length of discourse universe. This study adds attention to interval length, which is very important to the experimental results in the current proposal as follows. In this study [24], FTS models were analyzed for prediction as the author’s objective, using monthly data of three agro-products that represent available time-series data. Chen, Huarng, and Singh critically tested the FTS model’s production time-series data for 22 years, from 1988 to 2010. the MSE in predicting sugar production the (Chen) had the best acceptable accuracy of 63.45, whereas in the prediction of Lahi and Rice production (Singh), the best accuracies were 2605 and 913.62, respectively. The possible advantages of this study are that the varied FTS model’s predictions probably differ according to the value of the dataset, which leads to various FTS model experiments in our approach. The author in [25] enhanced FTS by reducing the prediction error and proposed a new prediction model set on the fuzzy transform (F-transform), which is established based on partitioning the universe, and the fuzzy logical relationships are utilized for prediction. This study showed that it can improve prediction accuracy by implementing the F-transform with applicable models in enrollments of the University of Alabama and many patents awarded in Taiwan, which could be the starting point in our proposed model taking dataset transformation into consideration.
FTS has multiple models implemented in different domains and varies discrimination according to the model, data set values, partition method, accuracy measure tools, and so on. This forced us to implement various experiences in our proposed model to make predictions that outperform by observing the results.
This paper’s proposed FTS models, Chen, Cheng, Yu, and Song, forecast Egyptian social insurance benefits, construct the interval length using methods of difference transformation data, use Huarng for appropriate partition lengths, and conduct experiments of interval length partitions (5, 10, 50, and 100). In light of the proposed study, the definition of FTS proposed by Song and Chissom [8,9,26] is based on fuzzy sets [7]. Chen [10] developed a novel approach that is more efficient than Song and Chissom’s proposed method because it employs simpler arithmetic operations rather than the difficult max-min composition operation. The steps of the Chen method are briefly reviewed in the following steps and depicted as a flowchart in Figure 1.
This step collects historical dataset and processing
(1) Defining and setting the universe of discourse (U) according to the available historical data time series using the following formula:
where
(2) Dividing the universe of discourse U into multiple partitions of equal interval lengths. To determine the suitable length, Huarng [28] proposed an applied computed length of interval
a) For the first differences, compute all the absolute differences between the values
b) Assume that the length is half the average.
c) Using Table 1, we identified the length range and calculated the base length.
d) The length of the required
Then, U can be divided into equal-length intervals U = [
Define the fuzzy set as
where
Fuzzy variances depend on the value of the membership degree. During this step, a fuzzy set was created for each set of data. The degree to which each historical data point correlates to each Ai may be estimated if the fluctuation is within ui. In this phase, the set of fuzzy logical relationships is between the
The defuzzification principles for the fuzzified forecasted variants in Step 4 are as follows:
(1) If the membership of the output has just one maximum of (
(2) If the membership of the output consists of one or more sequential maxima, use the midpoint of the relevant conjunct period as the prediction.
(3) If the membership of the output is 0, there is no limitation, and no change is anticipated compared to 0.
Adding the actual value of the change from the previous year to the expected degree of change yields the predicted value for the current year. The robustness of a model can be measured using statistical tools.
The dataset was collected from the National Fund for Social Security and the Ministry of Insurance and Social Affairs (MOISA). All insurance data cover 43 years from 1976 to 2019; each year refers to the total benefits contained (government sector, public, and private sector) that are required from the insurance system to cover. The data are divided into training and testing datasets to evaluate the performance and compare it to other models to validate the proposed FTS model. Figure 2 shows plots of the total benefits (in millions), training data, and testing data. The dataset was divided into 80% training data from (1976 to 2009) and 20% testing data from (2010 to 2019). Table 2 shows the total benefits in one million datasets in detail.
Given that the benefit values have been successively increasing over the years, the transformation method is the most suitable method for use with the present data and it is advantageous for making data stationary [27] where the partitions are based on the differences and not on the original values. Figure 3 shows the yearly differences during the 44 years, and Table 2 shows the different column differences. Then, we built the partition set based on these values.
The suggested (Chen, Cheng, Song, Yu) model technique for utilizing insurance pension data was implemented with different partitions using the PYFTS library [28]. Figure 4 shows the group of model training and testing prediction data in the curve for all experimental models and partitions. Figure 4 contains the four models, representing Chen, Cheng, Song, and Yu, respectively.
Given the multitude of experimentally implemented models and partitions, we briefly explain the model’s implementation with different forms in Python FTS libraries as follows. However, we did not mention the implementation steps in all experiments to avoid redundancy in the explanation. We have already displayed all the results of the experiments of the different models with partitions and analysis in the next section. To avoid repetition, we displayed the implementation steps for one instance of the experimental results of the Chen model with partition 5 in detail.
In the partition step, we propose using the main method grid partitions proposed in [8–10,29,30] with a varying number of partitions to divide the universe of discourse (U) over four models where the length of each partition is the same.
The experimental implementation used partitions 5, 10, 50, 100, and 465 (based Huarng) partitions. It establishes the best partition number for improving model accuracy [31–33]. Figure 5 shows the group of partition interval lengths of the training datasets representing partitions 5, 10, 50, 100, and 465 Huarng, respectively. For example, the partition 5 triangle grids divide the universe of discourse in detail as follows:
In this step, we classify each value as a benefit to define belonging as fuzzified. Table 3 lists each benefit fuzzified in the fuzzified column.
The Chen model establishes fuzzy logical relationship (FLR) rule groups based on partition 5, the fuzzy logical relationship is as follows: [‘
The derived FLRs are then divided into groups based on the current benefits of FLRs. Consequently, five FLR groups were obtained, as shown in Figure 6. Table 2 illustrates each benefit with its FLR group column as follows:
The five FTS models proposed that the predicted benefits evaluated using statistical measure tools were MAPE, RMSE, SMAPE, and Thiel’s U statistic the measures were chosen as the model error assessment metrics, which can assess the degree of change and accuracy of data while assessing the model’s prediction quality. The tool are defined as the RMSE in
In the training dataset, accuracy was measured using four tools to show the error value in each model with each experimental partition used. Figure 7 shows the training accuracies of the models containing the four models representing the RMSE, MAPE, SMAPE, and Thiel’s coefficient, respectively. The four tools had the same accuracy rates for the models. It should be noted that the three models Chen, Cheng, and Yu start with approximate errors in partition 5 and start decreasing to have the lowest error in partition 465. Conversely, the Song model had low errors in partition 5, and the errors increased when the partitioning was increased. The three models, Chen, Cheng, and Yu, feature stability and have low errors starting at partition 50 to partition 100, and in partition 465 (based Huarng) get the same lowest error among the four tools. Table 4 shows the training accuracy, RMSE error in the range of 17.89 to 18.99, error in MAPE in the range of 0.36 to 0.37, error in SMAPE is 0.18, and error in Thiel’s U is 0. The three models, Chen, Cheng, and Yu, had identical accuracies. In contrast, in the Song model in partition 465 (based Huarng), the error in RMSE is 1540.18, the error in MAPE is 122.32, the error in SMAPE is 23.8, and the error in Theil’s U is 0.06. The Song model had extreme errors compared with the three models. Even in partition 5, the Song model had slightly more errors than the three models Chen, Cheng, and Yu. Overall, the Chen, Cheng, and Yu models have an optimal prediction in the training dataset with slight errors, particularly with partition 465.
Testing accuracy is an important factor that represents the power required to forecast the data. In the testing dataset, the accuracy was measured using four tools to show the error value in each model with each experimental partition used. Figure 8 shows the model value testing accuracies, representing the four tools RMSE, MAPE, SMAPE, and Thiel’s coefficient, respectively. It should be noted that the four tools have the same rates of accuracy for the models, which is similar to the training accuracy. Overall, the four models have approximate error values in partition 5, but the three models Chen, Cheng, and Yu decreased the errors from partition 10 and remained stationary with low errors to partitions of 465. The Song model remained stationary with high errors, even in partitions of 465. Table 5 shows the testing dataset accuracy, where the accuracies of the four models in partition 5 are too close together. The error in RMSE in range 11306 to 11314, the error in MAPE in range 10.95 to 11.98, the error in SMAPE in range 5.89 to 6.03, and the error in Thiel’s Coefficient in range 0.069 to 0.059. Then, the three models, Chen, Cheng, and Yu, have error estimates that are the same as completely after partitions 5. In parallel, the Song model had the highest error for each partition, even with slight errors. However, the ideal result was in partition 465; the three models had higher accuracy with low errors identical and in four measure tools RMSE, MAPE, SMAPE, and Thiel’s U, 11411.13, 8.87, 4.70, and 0.059, respectively. Song were 12927.04, 10.62, 5.68, and 0.067, respectively.
This paper proposed to predict yearly Egyptian total benefits (pensions) using FTS models Chen, Cheng, Song, and Yu, construct the interval length using methods of difference transformation data method, given the data has not been stationary in the last years and increasing significantly, and Huarng method for appropriate partitions length, in addition, experiments on interval length partitions 5, 10, 50, 100 for further validation of the models. Considering that we have many experiments, to avoid repetition, we sufficed to explain the Chen model implemented with partition 5 in detail. The performance of the suggested models was assessed by predicting the yearly benefits using four statistical criteria. According to the data, the proposed techniques significantly enhance the forecast performance and prediction accuracy. In particular, the Chen, Cheng, and Yu models proved to outperform, especially in training accuracy at partition 465 based Huarng, and predict with acceptable accuracy in testing data, whereas the Song model has the highest error in training and testing. The proposed model motivates the implementation of FTS to predict (pensions) side-by-side with recent learning-based algorithms in different SISs in other countries.
No potential conflict of interest relevant to this article was reported.
Partitions of interval lengths (training dataset): (a) partition 5, (b) partition 10, (c) partition 50, (d) partition 100, and (e) Huarng 465.
Training accuracy of the four models: (a) RMSE, (b) MAPE, (c) SMAPE, and (d) Thiel’s coefficient.
Testing accuracy of the four models: (a) RMSE, (b) MAPE, (c) SMAPE, and (d) Thiel’s coefficient.
Table 2. Total benefits dataset.
Year | Government sector | Public & private sector | Total benefits in millions | Difference | Year | Government sector | Public & private sector | Total benefits in millions | Difference |
---|---|---|---|---|---|---|---|---|---|
121 | 57 | 179 | 0 | 145 | 40 | 185 | 6 | ||
162 | 51 | 214 | 29 | 195 | 68 | 264 | 50 | ||
243 | 105 | 348 | 84 | 207 | 121 | 328 | −20 | ||
401 | 351 | 753 | 425 | 436 | 424 | 861 | 108 | ||
557 | 485 | 1042 | 181 | 641 | 553 | 1194 | 152 | ||
779 | 631 | 1410 | 216 | 841 | 688 | 1529 | 119 | ||
1013 | 823 | 1837 | 308 | 1275 | 969 | 2244 | 407 | ||
1417 | 1133 | 2551 | 307 | 1714 | 1386 | 3100 | 549 | ||
2037 | 1705 | 3743 | 643 | 2353 | 2084 | 4438 | 695 | ||
3019 | 2562 | 5582 | 1144 | 3587 | 3000 | 6587 | 1005 | ||
4107 | 3469 | 7577 | 990 | 4810 | 4090 | 8901 | 1324 | ||
5252 | 5003 | 10255 | 1354 | 6034 | 5845 | 11880 | 1625 | ||
6689 | 6797 | 13487 | 1607 | 7742 | 7563 | 15305 | 1818 | ||
9590 | 8314 | 17904 | 2599 | 10791 | 8996 | 19787 | 1818 | ||
12240 | 9756 | 21996 | 1818 | 13988 | 10590 | 24578 | 2599 | ||
15441 | 12865 | 28306 | 3728 | 16867 | 13398 | 30265 | 1959 | ||
19311 | 15170 | 34481 | 4216 | 19488 | 18139 | 37627 | 3146 | ||
22660 | 18456 | 41116 | 3489 | 28724 | 22124 | 50848 | 9732 | ||
35568 | 28164 | 63732 | 12884 | 33640 | 35777 | 69417 | 5685 | ||
40558 | 43175 | 83733 | 14316 | 48554 | 51816 | 100370 | 16637 | ||
55495 | 59169 | 114664 | 14294 | 62300 | 67600 | 129900 | 15236 | ||
72800 | 77600 | 150400 | 20500 | 85700 | 92800 | 178500 | 28100 |
Table 3. Dataset training fuzzyfied and FLR group rules.
Year | Benefits | Difference | Fuzzyfied | FLR Group | Defuzzification | Forecasting value |
---|---|---|---|---|---|---|
1976 | 179 | 0 | → | —— | 0 | |
1977 | 185 | 6 | → | [(−22.0) + (909.9)]/2 + 179 | 622.96 | |
1978 | 214 | 29 | → | [(−22.0) + (909.9)]/2 + 185 | 628.96 | |
1979 | 264 | 50 | → | [(−22.0) + (909.9)]/2 + 214 | 657.96 | |
1980 | 348 | 84 | → | [(−22.0) + (909.9)]/2 + 264 | 707.96 | |
1981 | 328 | −20 | → | [(−22.0) + (909.9)]/2 + 348 | 791.96 | |
1982 | 753 | 425 | → | [(−22.0) + (909.9)]/2 + 328 | 771.96 | |
1983 | 861 | 108 | → | [(−22.0) + (909.9)]/2 + 753 | 1196.96 | |
1984 | 1042 | 181 | → | [(−22.0) + (909.9)]/2 + 861 | 1304.96 | |
1985 | 1194 | 152 | → | [(−22.0) + (909.9)]/2 + 1042 | 1485.96 | |
1986 | 1410 | 216 | → | [(−22.0) + (909.9)]/2 + 1194 | 1637.96 | |
1987 | 1529 | 119 | → | [(−22.0) + (909.9)]/2 + 1410 | 1853.96 | |
1988 | 1837 | 308 | → | [(−22.0) + (909.9)]/2 + 1529 | 1972.96 | |
1989 | 2244 | 407 | → | [(−22.0) + (909.9)]/2 + 1837 | 2280.96 | |
1990 | 2551 | 307 | → | [(−22.0) + (909.9)]/2 + 2244 | 2687.96 | |
1991 | 3100 | 549 | → | [(−22.0) + (909.9)]/2 + 2551 | 2994.96 | |
1992 | 3743 | 643 | → | [(909.9) + (1841.8)]/2 + 3100 | 4475.88 | |
1993 | 4438 | 695 | → | [(909.9) + (1841.8)]/2 + 3743 | 5118.88 | |
1994 | 5582 | 1144 | → | [(909.9) + (1841.8)]/2 + 4438 | 5813.88 | |
1995 | 6587 | 1005 | → | [(909.9) + (1841.8)]/2 + 5582 | 6957.88 | |
1996 | 7577 | 990 | → | [(909.9) + (1841.8)]/2 + 6587 | 7962.88 | |
1997 | 8901 | 1324 | → | [(909.9) + (1841.8)]/2 + 7577 | 8952.88 | |
1998 | 10255 | 1354 | → | [(909.9) + (1841.8)]/2 + 8901 | 10276.88 | |
1999 | 11880 | 1625 | → | [(909.9) + (1841.8)]/2 + 10255 | 11630.88 | |
2000 | 13487 | 1607 | → | [(1841.8) + (2773.7) + (3705.6)]/3 + 11880 | 14653.76 | |
2001 | 15305 | 1818 | → | [(1841.8) + (2773.7) + (3705.6)]/3 + 13487 | 16260.76 | |
2002 | 17904 | 2599 | → | [(1841.8) + (2773.7) + (3705.6)]/3 + 15305 | 18078.76 | |
2001 | 15305 | 1818 | → | [(1841.8) + (3705.6)]/2 + 17904 | 20677.76 | |
2001 | 15305 | 1818 | → | [(1841.8) + (2773.7) + (3705.6)]/3 + 15305 | 22560.76 | |
2002 | 17904 | 2599 | → | [(1841.8) + (2773.7) + (3705.6)]/3 + 15305 | 24769.76 | |
2006 | 28306 | 3728 | → | [(1841.8) + (3705.6)]/2 + 17904 | 27351.76 | |
2007 | 30265 | 1959 | → | 1841.8 + 28306 | 30147.84 | |
2008 | 34481 | 4216 | → | [(1841.8) + (2773.7) + (3705.6)]/3 + 30265 | 33038.76 | |
2009 | 37627 | 3146 | → | 1841.8 + 34481 | 36322.84 | |
2010 | 41116 | 3489 | → | 1841.8 + 37627 | 40400.76 | |
2011 | 50848 | 9732 | → | 1841.8 + 41116 | 42957.84 | |
2012 | 63732 | 12884 | → | 1841.8 + 50848 | 52689.84 | |
2013 | 69417 | 5685 | → | 1841.8 + 63732 | 65573.84 | |
2014 | 83733 | 14316 | → | 1841.8 + 69417 | 71258.84 | |
2015 | 100370 | 16637 | → | 1841.8 + 83733 | 85574.84 | |
2016 | 114664 | 14294 | → | 1841.8 + 100370 | 102211.84 | |
2017 | 129900 | 15236 | → | 1841.8 + 114664 | 116505.84 | |
2018 | 150400 | 20500 | → | 1841.8 + 129900 | 131741.84 | |
2019 | 178500 | 28100 | → | 1841.8 + 150400 | 152241.84 | |
1841.8 + 178500 | 180341.84 |
E-mail: rehamraouf@foc.cu.edu.eg
E-mail: saad.elsaieed@hotmail.com
International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 169-182
Published online June 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.2.169
Copyright © The Korean Institute of Intelligent Systems.
Reham Raouf and Saad Elsaieed
Department of Insurance & Actuarial Sciences, Faculty of Commerce, Cairo University, Cairo, Egypt
Correspondence to:Reham Raouf (rehamraouf@foc.cu.edu.eg)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In recent years, many methods have been proposed to forecast data in different fields based on successful fuzzy time series models (FTS). Egyptian social insurance systems (SISs) need support to optimally define and estimate yearly total benefits (pensions), which helps the actuaries who are responsible for the system make optimal decisions. Given that the total benefits have not been forecasted before by prediction methods, this paper proposes FTS models by Chen, Cheng, Yu, and Song to forecast Egyptian social insurance benefits, proposes Huarng for appropriate partition lengths, and constructs the interval length using the difference in the transformation data method, given that the data has not been stationary in recent years and has increased significantly. The proposed approach is based on experiments implemented using four models with interval length partitions of 5, 10, 50, 100, and Huarng partitions of 465. The results show great progress in the performance of yearly benefit forecasting, especially in the Chen model with a Huarng 465 partition, which has high accuracy prediction with low error when training and testing data.
Keywords: Fuzzy time series, Social insurance, Benefits, Pensions
Forecasting activities play a significant part in our everyday lives; they are critical in a variety of businesses because projections of future occurrences must be factored into decision-making. Given that social insurance is one of the most significant concerns that most nations confront because of the sensitivity of its relationship with public budget impacts, forecasting insurance is an important issue, and better forecasting aids actuaries and the responsibility of making strategic choices in the social insurance system. In this context, this study proposes a prediction of the total benefits in millions (pensions) required by Egyptian insurance systems.
According to their conventional definition, pension schemes and social insurance systems (SISs) are one of the three primary components of social protection, together with social assistance, social safety nets, employment strategies, and labor market interventions [1]. SISs are a mechanism that allows workers to engage in securing their future, and they are partially or entirely supported by worker and employee contributions. Egypt has a broadly stratified SIS operating as a fully funded plan, where employees make payments that are invested and subsequently reimbursed as pensions. The system has gradually evolved with a defined benefit plan. The system is primarily governed by several laws, and the present legal framework, Law 148/2019, was formally established on the first of January 2020. This, in turn, plays a significant role in changing the system and overcoming challenges due to the complexity of the old social insurance regulatory and legal framework. In practice, the system faces serious challenges and is extremely fragmented with several benefit packages available to various parts of the workforce, offering payment types for employees and their families categorized by old age, disability, survivors, illness, work injury, and unemployment [2,3]. In addition, several factors have led to Egypt’s benefits being unsustainable and inefficient. For example, the reserves of pension funds are invested at low, occasionally negative, real interest rates. They provide extremely generous minimal pensions and opportunities for early retirement. Members may also easily alter the size of their pensions, which are based solely on payments made in the few years leading to retirement. Many workers agree with their employers to under-report their income throughout the majority of their working careers to reduce their payments. As a result of these factors, Egypt’s pension funds will soon spend more on pension payouts than is received via contributions from members [4]. Therefore, it should be noted that the rate of total Egypt’s benefits (pensions) in recent years is increasing sharply, forcing us to forecast the total benefits in the next few years. Annual benefits are estimated based on actuarial models and used to provide demographic and financial forecasts for pension systems. They are often generated from models applied to vocational pension schemes that cover groups of employees and are based on demographic and economic factors. Consequently, actuarial methods in connection with predictive analytics must significantly enhance their understanding of predicted behavior or events to support their policies and judgments [5]. In addition, implementing a year-by-year simulation approach to estimate the future costs of benefits is one of the most significant tasks for actuaries [6]. Therefore, benefit prediction and analysis will aid actuaries in their professional responsibility of making strategic choices in the social insurance system. To the best of our knowledge, our proposed approach is the first to predict total benefits regarding Egyptian social insurance using fuzzy time series (FTS) models with a yearly dataset (pensions); this approach leads to sufficient and outperformed results. In this study, we present the use of FTS models to forecast Egyptian social insurance benefits (pensions) rather than traditional assumption methods for better forecasting pension demand.
Fuzzy set theory was developed to address the ambiguity and uncertainty that characterize most real-world problems. The highly popular fuzzy set theory, proposed by Zadeh [7] in 1965, is used to explain linguistic fuzzy information through mathematical modeling and generates various findings. It is still extensively utilized in a wide range of applications. Subsequently, a number of academics proposed several models, the first of which was a FTS forecasting model. Song and Chissom [8] developed books explaining the concepts of the fuzzy set theory to overcome the challenges of classical time series. However, this model incorporates the max-min composition method, which makes the computing process substantially more difficult [9]. The Chen model also contains a series of studies on the creation and execution of experimental University of Alabama enrollment data forecasting. Chen [10] proposed a simpler calculation technique to address this flaw with the benefit of reducing computing time and making the procedure more understandable. However, for fuzzy logical relationships (FLRs), this model lacks an appropriate weight mechanism. The model has now been widely modified, with academics attempting to increase the prediction accuracy by modifying the weighting technique or increasing the duration of linguistic intervals. Hwang et al. [11] proposed a forecasting approach based on a FTS. Chen and Hwang [12] proposed a FTS-based temperature forecasting algorithm. Chen [10] improved forecasting by using a high-order FTS model. The expanded Chen’s model was used in [1], and Huarng in [13] predicted enrollments using heuristic principles, which reduced computations. Huang [14] calculated interval lengths using both average-based lengths and distribution bases. Fuzzy theory was further refined and suited by Yu and his colleague [15,16] to address weighing and recurring issues using typical FTS models, including stock price predictions. To improve forecasting accuracy, Yu [15] used multiple weighted technologies on FTS models. Then, the ratio-based interval length was added to the FTS model proposed by Huarng and Yu [17]. Cheng [18] used a FTS model combined with a trend-weighting method to forecast real stock price trading data and university enrolment.
This paper is organized as follows. In Section 2, we review related studies that implemented FTS in different domains with various datasets to extract model features. Section 3 presents the steps of the FTS models, and Section 4 illustrates the implementation of the proposed FTS model and the forming benefits dataset. In Section 5, the results and discussion of the model’s approach and accuracy in existing work are presented. Finally, Section 6 summarizes and concludes the study.
Machine-learning models have contribute to SIS. The author in [19] proposed the use of three algorithms, namely the decision tree, native Bayes, and CN2 rule induction, to predict classification based on some of the social insurance features for people. The three algorithms obtained high accuracy results when classifying. The author in [20] proposed a novel deep learning-based framework based on the recurrent neural network (RNN) architecture for forecasting an individual’s payment status using a real dataset gathered by Taiwan’s Ministry of Health and Welfare. The model is being developed to effectively abstract people’s payment behavior and accurately forecast their future payment behavior over a lengthy period. Compared with state-of-the-art approaches, such as support vector machines (SVM) and hidden Markov models (HMM), the model outperforms even under varied settings. However, these studies are related to the prediction of individuals’ personal payments, which is an issue that differs from our target.
Social insurance benefits (pensions) still make traditional assumptions. In this section, we focus on related studies that implemented FTS in different domains using various datasets. There are studies related to FTS models in different industries and domains that predict enrollment, stock marketing, tourism, shipping, and transportation. We led the forced implementation of FTS to predict Egypt’s social insurance in a series of years, including the total amounts of pensions.
Although the dataset related to our proposal is represented in a series of years, the critical data must be predicted using the FTS model. The available FTS models implemented for prediction in the studies are as follows. The author in [21] proposed two models to predict the demand for tourism arrival in Taiwan as well as arrivals during the period of 1989–2000 from Hong Kong, Germany, and the United States. The FTS is more appropriate for predicting tourism demand, especially from Hong Kong arrival to Taiwan with an error of 1.93, when compared to the greedy algorithm. The author in [22] measured the prediction performance for tourist arrivals in Indonesia using a FTS compared with classical methods such as seasonal autoregressive integrated moving average (SARIMA), Box–Jenkins methods, time series regression, and Holt-Winters. The dataset was divided into training datasets from January 1989 to December 1996. The testing data were collected from January 1997 to December 1997. The root mean square error (RMSE), mean square error (MSE), mean absolute deviation (MAD), and mean absolute percentage error (MAPE) were measured. The accuracy of the resultant Chen’s FTS was outperformed by classical methods after used data transformation. Although FTS methods are simple, they outperformed all the statistical classical methods, which were used previously to predict social insurance; this prompted our proposal for using FTS methods for predictions. In [23], power companies used the proposed FTS model for electrical power to predict electrical load to make decisions. The dataset of the regional electric load in Taiwan, represented yearly, was used in this study and a comparison between FTS and previously implemented models’ linear and nonlinear regression was conducted. The accuracy of FTS outperforming in the MAPE is smaller than in other implemented models, especially when given appropriate intervals using an efficient length of discourse universe. This study adds attention to interval length, which is very important to the experimental results in the current proposal as follows. In this study [24], FTS models were analyzed for prediction as the author’s objective, using monthly data of three agro-products that represent available time-series data. Chen, Huarng, and Singh critically tested the FTS model’s production time-series data for 22 years, from 1988 to 2010. the MSE in predicting sugar production the (Chen) had the best acceptable accuracy of 63.45, whereas in the prediction of Lahi and Rice production (Singh), the best accuracies were 2605 and 913.62, respectively. The possible advantages of this study are that the varied FTS model’s predictions probably differ according to the value of the dataset, which leads to various FTS model experiments in our approach. The author in [25] enhanced FTS by reducing the prediction error and proposed a new prediction model set on the fuzzy transform (F-transform), which is established based on partitioning the universe, and the fuzzy logical relationships are utilized for prediction. This study showed that it can improve prediction accuracy by implementing the F-transform with applicable models in enrollments of the University of Alabama and many patents awarded in Taiwan, which could be the starting point in our proposed model taking dataset transformation into consideration.
FTS has multiple models implemented in different domains and varies discrimination according to the model, data set values, partition method, accuracy measure tools, and so on. This forced us to implement various experiences in our proposed model to make predictions that outperform by observing the results.
This paper’s proposed FTS models, Chen, Cheng, Yu, and Song, forecast Egyptian social insurance benefits, construct the interval length using methods of difference transformation data, use Huarng for appropriate partition lengths, and conduct experiments of interval length partitions (5, 10, 50, and 100). In light of the proposed study, the definition of FTS proposed by Song and Chissom [8,9,26] is based on fuzzy sets [7]. Chen [10] developed a novel approach that is more efficient than Song and Chissom’s proposed method because it employs simpler arithmetic operations rather than the difficult max-min composition operation. The steps of the Chen method are briefly reviewed in the following steps and depicted as a flowchart in Figure 1.
This step collects historical dataset and processing
(1) Defining and setting the universe of discourse (U) according to the available historical data time series using the following formula:
where
(2) Dividing the universe of discourse U into multiple partitions of equal interval lengths. To determine the suitable length, Huarng [28] proposed an applied computed length of interval
a) For the first differences, compute all the absolute differences between the values
b) Assume that the length is half the average.
c) Using Table 1, we identified the length range and calculated the base length.
d) The length of the required
Then, U can be divided into equal-length intervals U = [
Define the fuzzy set as
where
Fuzzy variances depend on the value of the membership degree. During this step, a fuzzy set was created for each set of data. The degree to which each historical data point correlates to each Ai may be estimated if the fluctuation is within ui. In this phase, the set of fuzzy logical relationships is between the
The defuzzification principles for the fuzzified forecasted variants in Step 4 are as follows:
(1) If the membership of the output has just one maximum of (
(2) If the membership of the output consists of one or more sequential maxima, use the midpoint of the relevant conjunct period as the prediction.
(3) If the membership of the output is 0, there is no limitation, and no change is anticipated compared to 0.
Adding the actual value of the change from the previous year to the expected degree of change yields the predicted value for the current year. The robustness of a model can be measured using statistical tools.
The dataset was collected from the National Fund for Social Security and the Ministry of Insurance and Social Affairs (MOISA). All insurance data cover 43 years from 1976 to 2019; each year refers to the total benefits contained (government sector, public, and private sector) that are required from the insurance system to cover. The data are divided into training and testing datasets to evaluate the performance and compare it to other models to validate the proposed FTS model. Figure 2 shows plots of the total benefits (in millions), training data, and testing data. The dataset was divided into 80% training data from (1976 to 2009) and 20% testing data from (2010 to 2019). Table 2 shows the total benefits in one million datasets in detail.
Given that the benefit values have been successively increasing over the years, the transformation method is the most suitable method for use with the present data and it is advantageous for making data stationary [27] where the partitions are based on the differences and not on the original values. Figure 3 shows the yearly differences during the 44 years, and Table 2 shows the different column differences. Then, we built the partition set based on these values.
The suggested (Chen, Cheng, Song, Yu) model technique for utilizing insurance pension data was implemented with different partitions using the PYFTS library [28]. Figure 4 shows the group of model training and testing prediction data in the curve for all experimental models and partitions. Figure 4 contains the four models, representing Chen, Cheng, Song, and Yu, respectively.
Given the multitude of experimentally implemented models and partitions, we briefly explain the model’s implementation with different forms in Python FTS libraries as follows. However, we did not mention the implementation steps in all experiments to avoid redundancy in the explanation. We have already displayed all the results of the experiments of the different models with partitions and analysis in the next section. To avoid repetition, we displayed the implementation steps for one instance of the experimental results of the Chen model with partition 5 in detail.
In the partition step, we propose using the main method grid partitions proposed in [8–10,29,30] with a varying number of partitions to divide the universe of discourse (U) over four models where the length of each partition is the same.
The experimental implementation used partitions 5, 10, 50, 100, and 465 (based Huarng) partitions. It establishes the best partition number for improving model accuracy [31–33]. Figure 5 shows the group of partition interval lengths of the training datasets representing partitions 5, 10, 50, 100, and 465 Huarng, respectively. For example, the partition 5 triangle grids divide the universe of discourse in detail as follows:
In this step, we classify each value as a benefit to define belonging as fuzzified. Table 3 lists each benefit fuzzified in the fuzzified column.
The Chen model establishes fuzzy logical relationship (FLR) rule groups based on partition 5, the fuzzy logical relationship is as follows: [‘
The derived FLRs are then divided into groups based on the current benefits of FLRs. Consequently, five FLR groups were obtained, as shown in Figure 6. Table 2 illustrates each benefit with its FLR group column as follows:
The five FTS models proposed that the predicted benefits evaluated using statistical measure tools were MAPE, RMSE, SMAPE, and Thiel’s U statistic the measures were chosen as the model error assessment metrics, which can assess the degree of change and accuracy of data while assessing the model’s prediction quality. The tool are defined as the RMSE in
In the training dataset, accuracy was measured using four tools to show the error value in each model with each experimental partition used. Figure 7 shows the training accuracies of the models containing the four models representing the RMSE, MAPE, SMAPE, and Thiel’s coefficient, respectively. The four tools had the same accuracy rates for the models. It should be noted that the three models Chen, Cheng, and Yu start with approximate errors in partition 5 and start decreasing to have the lowest error in partition 465. Conversely, the Song model had low errors in partition 5, and the errors increased when the partitioning was increased. The three models, Chen, Cheng, and Yu, feature stability and have low errors starting at partition 50 to partition 100, and in partition 465 (based Huarng) get the same lowest error among the four tools. Table 4 shows the training accuracy, RMSE error in the range of 17.89 to 18.99, error in MAPE in the range of 0.36 to 0.37, error in SMAPE is 0.18, and error in Thiel’s U is 0. The three models, Chen, Cheng, and Yu, had identical accuracies. In contrast, in the Song model in partition 465 (based Huarng), the error in RMSE is 1540.18, the error in MAPE is 122.32, the error in SMAPE is 23.8, and the error in Theil’s U is 0.06. The Song model had extreme errors compared with the three models. Even in partition 5, the Song model had slightly more errors than the three models Chen, Cheng, and Yu. Overall, the Chen, Cheng, and Yu models have an optimal prediction in the training dataset with slight errors, particularly with partition 465.
Testing accuracy is an important factor that represents the power required to forecast the data. In the testing dataset, the accuracy was measured using four tools to show the error value in each model with each experimental partition used. Figure 8 shows the model value testing accuracies, representing the four tools RMSE, MAPE, SMAPE, and Thiel’s coefficient, respectively. It should be noted that the four tools have the same rates of accuracy for the models, which is similar to the training accuracy. Overall, the four models have approximate error values in partition 5, but the three models Chen, Cheng, and Yu decreased the errors from partition 10 and remained stationary with low errors to partitions of 465. The Song model remained stationary with high errors, even in partitions of 465. Table 5 shows the testing dataset accuracy, where the accuracies of the four models in partition 5 are too close together. The error in RMSE in range 11306 to 11314, the error in MAPE in range 10.95 to 11.98, the error in SMAPE in range 5.89 to 6.03, and the error in Thiel’s Coefficient in range 0.069 to 0.059. Then, the three models, Chen, Cheng, and Yu, have error estimates that are the same as completely after partitions 5. In parallel, the Song model had the highest error for each partition, even with slight errors. However, the ideal result was in partition 465; the three models had higher accuracy with low errors identical and in four measure tools RMSE, MAPE, SMAPE, and Thiel’s U, 11411.13, 8.87, 4.70, and 0.059, respectively. Song were 12927.04, 10.62, 5.68, and 0.067, respectively.
This paper proposed to predict yearly Egyptian total benefits (pensions) using FTS models Chen, Cheng, Song, and Yu, construct the interval length using methods of difference transformation data method, given the data has not been stationary in the last years and increasing significantly, and Huarng method for appropriate partitions length, in addition, experiments on interval length partitions 5, 10, 50, 100 for further validation of the models. Considering that we have many experiments, to avoid repetition, we sufficed to explain the Chen model implemented with partition 5 in detail. The performance of the suggested models was assessed by predicting the yearly benefits using four statistical criteria. According to the data, the proposed techniques significantly enhance the forecast performance and prediction accuracy. In particular, the Chen, Cheng, and Yu models proved to outperform, especially in training accuracy at partition 465 based Huarng, and predict with acceptable accuracy in testing data, whereas the Song model has the highest error in training and testing. The proposed model motivates the implementation of FTS to predict (pensions) side-by-side with recent learning-based algorithms in different SISs in other countries.
FTS model steps.
Plots of total benefits (in millions).
Yearly transformation difference.
Training and testing prediction.
Partitions of interval lengths (training dataset): (a) partition 5, (b) partition 10, (c) partition 50, (d) partition 100, and (e) Huarng 465.
Network FLR rules.
Training accuracy of the four models: (a) RMSE, (b) MAPE, (c) SMAPE, and (d) Thiel’s coefficient.
Testing accuracy of the four models: (a) RMSE, (b) MAPE, (c) SMAPE, and (d) Thiel’s coefficient.
Table 1 . Range of interval.
Range | Base |
---|---|
0.1–1.0 | 0.1 |
1.1–10 | 1 |
11–100 | 10 |
101–1000 | 100 |
Table 2 . Total benefits dataset.
Year | Government sector | Public & private sector | Total benefits in millions | Difference | Year | Government sector | Public & private sector | Total benefits in millions | Difference |
---|---|---|---|---|---|---|---|---|---|
121 | 57 | 179 | 0 | 145 | 40 | 185 | 6 | ||
162 | 51 | 214 | 29 | 195 | 68 | 264 | 50 | ||
243 | 105 | 348 | 84 | 207 | 121 | 328 | −20 | ||
401 | 351 | 753 | 425 | 436 | 424 | 861 | 108 | ||
557 | 485 | 1042 | 181 | 641 | 553 | 1194 | 152 | ||
779 | 631 | 1410 | 216 | 841 | 688 | 1529 | 119 | ||
1013 | 823 | 1837 | 308 | 1275 | 969 | 2244 | 407 | ||
1417 | 1133 | 2551 | 307 | 1714 | 1386 | 3100 | 549 | ||
2037 | 1705 | 3743 | 643 | 2353 | 2084 | 4438 | 695 | ||
3019 | 2562 | 5582 | 1144 | 3587 | 3000 | 6587 | 1005 | ||
4107 | 3469 | 7577 | 990 | 4810 | 4090 | 8901 | 1324 | ||
5252 | 5003 | 10255 | 1354 | 6034 | 5845 | 11880 | 1625 | ||
6689 | 6797 | 13487 | 1607 | 7742 | 7563 | 15305 | 1818 | ||
9590 | 8314 | 17904 | 2599 | 10791 | 8996 | 19787 | 1818 | ||
12240 | 9756 | 21996 | 1818 | 13988 | 10590 | 24578 | 2599 | ||
15441 | 12865 | 28306 | 3728 | 16867 | 13398 | 30265 | 1959 | ||
19311 | 15170 | 34481 | 4216 | 19488 | 18139 | 37627 | 3146 | ||
22660 | 18456 | 41116 | 3489 | 28724 | 22124 | 50848 | 9732 | ||
35568 | 28164 | 63732 | 12884 | 33640 | 35777 | 69417 | 5685 | ||
40558 | 43175 | 83733 | 14316 | 48554 | 51816 | 100370 | 16637 | ||
55495 | 59169 | 114664 | 14294 | 62300 | 67600 | 129900 | 15236 | ||
72800 | 77600 | 150400 | 20500 | 85700 | 92800 | 178500 | 28100 |
Table 3 . Dataset training fuzzyfied and FLR group rules.
Year | Benefits | Difference | Fuzzyfied | FLR Group | Defuzzification | Forecasting value |
---|---|---|---|---|---|---|
1976 | 179 | 0 | → | —— | 0 | |
1977 | 185 | 6 | → | [(−22.0) + (909.9)]/2 + 179 | 622.96 | |
1978 | 214 | 29 | → | [(−22.0) + (909.9)]/2 + 185 | 628.96 | |
1979 | 264 | 50 | → | [(−22.0) + (909.9)]/2 + 214 | 657.96 | |
1980 | 348 | 84 | → | [(−22.0) + (909.9)]/2 + 264 | 707.96 | |
1981 | 328 | −20 | → | [(−22.0) + (909.9)]/2 + 348 | 791.96 | |
1982 | 753 | 425 | → | [(−22.0) + (909.9)]/2 + 328 | 771.96 | |
1983 | 861 | 108 | → | [(−22.0) + (909.9)]/2 + 753 | 1196.96 | |
1984 | 1042 | 181 | → | [(−22.0) + (909.9)]/2 + 861 | 1304.96 | |
1985 | 1194 | 152 | → | [(−22.0) + (909.9)]/2 + 1042 | 1485.96 | |
1986 | 1410 | 216 | → | [(−22.0) + (909.9)]/2 + 1194 | 1637.96 | |
1987 | 1529 | 119 | → | [(−22.0) + (909.9)]/2 + 1410 | 1853.96 | |
1988 | 1837 | 308 | → | [(−22.0) + (909.9)]/2 + 1529 | 1972.96 | |
1989 | 2244 | 407 | → | [(−22.0) + (909.9)]/2 + 1837 | 2280.96 | |
1990 | 2551 | 307 | → | [(−22.0) + (909.9)]/2 + 2244 | 2687.96 | |
1991 | 3100 | 549 | → | [(−22.0) + (909.9)]/2 + 2551 | 2994.96 | |
1992 | 3743 | 643 | → | [(909.9) + (1841.8)]/2 + 3100 | 4475.88 | |
1993 | 4438 | 695 | → | [(909.9) + (1841.8)]/2 + 3743 | 5118.88 | |
1994 | 5582 | 1144 | → | [(909.9) + (1841.8)]/2 + 4438 | 5813.88 | |
1995 | 6587 | 1005 | → | [(909.9) + (1841.8)]/2 + 5582 | 6957.88 | |
1996 | 7577 | 990 | → | [(909.9) + (1841.8)]/2 + 6587 | 7962.88 | |
1997 | 8901 | 1324 | → | [(909.9) + (1841.8)]/2 + 7577 | 8952.88 | |
1998 | 10255 | 1354 | → | [(909.9) + (1841.8)]/2 + 8901 | 10276.88 | |
1999 | 11880 | 1625 | → | [(909.9) + (1841.8)]/2 + 10255 | 11630.88 | |
2000 | 13487 | 1607 | → | [(1841.8) + (2773.7) + (3705.6)]/3 + 11880 | 14653.76 | |
2001 | 15305 | 1818 | → | [(1841.8) + (2773.7) + (3705.6)]/3 + 13487 | 16260.76 | |
2002 | 17904 | 2599 | → | [(1841.8) + (2773.7) + (3705.6)]/3 + 15305 | 18078.76 | |
2001 | 15305 | 1818 | → | [(1841.8) + (3705.6)]/2 + 17904 | 20677.76 | |
2001 | 15305 | 1818 | → | [(1841.8) + (2773.7) + (3705.6)]/3 + 15305 | 22560.76 | |
2002 | 17904 | 2599 | → | [(1841.8) + (2773.7) + (3705.6)]/3 + 15305 | 24769.76 | |
2006 | 28306 | 3728 | → | [(1841.8) + (3705.6)]/2 + 17904 | 27351.76 | |
2007 | 30265 | 1959 | → | 1841.8 + 28306 | 30147.84 | |
2008 | 34481 | 4216 | → | [(1841.8) + (2773.7) + (3705.6)]/3 + 30265 | 33038.76 | |
2009 | 37627 | 3146 | → | 1841.8 + 34481 | 36322.84 | |
2010 | 41116 | 3489 | → | 1841.8 + 37627 | 40400.76 | |
2011 | 50848 | 9732 | → | 1841.8 + 41116 | 42957.84 | |
2012 | 63732 | 12884 | → | 1841.8 + 50848 | 52689.84 | |
2013 | 69417 | 5685 | → | 1841.8 + 63732 | 65573.84 | |
2014 | 83733 | 14316 | → | 1841.8 + 69417 | 71258.84 | |
2015 | 100370 | 16637 | → | 1841.8 + 83733 | 85574.84 | |
2016 | 114664 | 14294 | → | 1841.8 + 100370 | 102211.84 | |
2017 | 129900 | 15236 | → | 1841.8 + 114664 | 116505.84 | |
2018 | 150400 | 20500 | → | 1841.8 + 129900 | 131741.84 | |
2019 | 178500 | 28100 | → | 1841.8 + 150400 | 152241.84 | |
1841.8 + 178500 | 180341.84 |
Table 4 . Training accuracy.
Table 5 . Testing accuracy.
FTS model steps.
|@|~(^,^)~|@|Plots of total benefits (in millions).
|@|~(^,^)~|@|Yearly transformation difference.
|@|~(^,^)~|@|Training and testing prediction.
|@|~(^,^)~|@|Partitions of interval lengths (training dataset): (a) partition 5, (b) partition 10, (c) partition 50, (d) partition 100, and (e) Huarng 465.
|@|~(^,^)~|@|Network FLR rules.
|@|~(^,^)~|@|Training accuracy of the four models: (a) RMSE, (b) MAPE, (c) SMAPE, and (d) Thiel’s coefficient.
|@|~(^,^)~|@|Testing accuracy of the four models: (a) RMSE, (b) MAPE, (c) SMAPE, and (d) Thiel’s coefficient.