Quartile-Based Defuzzify Scheme Integrated with PSO for Revenue Change Forecasting. Abstract

Transcription

1 Quartile-Based Defuzzify Scheme Integrated with PSO for Revenue Change Forecasting Yao-Lin Huang 1 I-Hong Kuo 2 Shu-Mei Pai 3 1 Department of Computer and Communication Engineering, De Lin Institute of Technology 2 Information and Communications Research Laboratories (ICL), Industrial Technology Research Institute (ITRI) 3 Department of Nursing, Hsin Sheng Junior College of Medical Care and Management yaolinhuang@gmail.com yihonguo@gmail.com dutypai@gmail.com Abstract In this paper, we propose an improved forecasting model which integrates fuzzy time series approach and particle swarm optimization (PSO) for revenue change. The proposed model uses Quartile-Based Defuzzify (QBD) scheme to calculate the fuzzy center of each interval for the defuzzification process. The empirical data of revenue changes in e-reader are used to assess the model s effectiveness. The experimental results prove that the proposed QBD scheme is applicable and performs better than other listing methods. Keywords: Particle swarm optimization, Time series forecasting, Quartile-based defuzzify 1. Introduction Accuracy prediction of revenue changes can help the companies of e-reader make investment policies. In recent years, the forecasting approaches of time series have great progress in various applications. However, some problems encounter limited historical data and large fluctuations, and thus it is difficult to make accurate prediction for decision-maker. Traditional forecasting methods can only deal with numerical data and fail to solve the problems in which the historical data are linguistic values. Fuzzy set theory was firstly introduced by Zadeh [1-3] to deal with linguistic values. Song and Chissom [4] successfully applied the concept of fuzzy sets in the time-invariant and time-variant fuzzy time series models [5-7] to forecast enrollments. To reduce the complexity of the forecasting procedure and improve forecasting accuracy, Chen [8] introduced the first-order fuzzy logical relationships (FLRs) and IF-THEN rules based on fuzzy theory. Huarng [9, 10] presented computational and heuristic schemes by properly partitioning the lengths of intervals. Wong et al. [11] presented automatically adaptive method which can analyze the window size of fuzzy time series according to the prediction accuracy in the training phase and heuristic rules to yield forecasted values in the testing phase. Yolcu et al. Yu [12] presented the refined approach in the formulation of FLRs to apply the FLRs more effectively. Hwang et al. [13] presented the time-variant fuzzy model using the variations in fuzzy relationship matrix. [14] applied the constrained optimization which used a single-variable to calculate the ratio of the interval lengths. Kuo et al. [15, 16] and Huang et al. [17, 18] presented the hybr forecasting models which integrated fuzzy methods and particle swarm optimization 390

2 (PSO) to increase forecast accuracy. Adebiyi et al. [19] presented a hybrized approach in which artificial neural networks was used to increase forecasting accuracy in stock index. Sadaei et al. [20] proposed a sophisticated exponentially weighted fuzzy algorithm that is aligned with an enhanced harmony search. Chen et al. [21, 22] presented a new model for fuzzy forecasting based on two-factors second-order fuzzy-trend logical relationship groups and the probabilities of trends of fuzzy-trend logical relationships After inspecting the prediction approaches in literature, the lengths of intervals and forecasting rules are two key issues impacting prediction accuracy. Hence, a Quartile-Based Defuzzify (QBD) scheme is applied in fuzzy time series to improve fuzzy approach. Then, The PSO algorithm is applied to adjust interval lengths in the universe of discourse by minimizing the MSE value. The proposed forecasting method is called the QBDPSO model. The empirical example of revenue changes in e-reader from Prime View International Co., Ltd. [23] is used to assess the effectiveness of the proposed forecasting model. The indicators of mean squared error (MSE) and mean absolute percentage error (MAPE) are used to estimate the forecasting accuracy. In QBDPSO model, Chen s method [8] is applied to construct fuzzy logical relationships (FLRs) and the QBD scheme is used to calculate defuzzify values in the training phase. The experimental results show that the QBDPSO model outperforms other existing models. The remaining content of this paper is organized as follows. Section 2 introduces fuzzy time series and PSO algorithm. Section 3 depicts improved quartile-based defuzzify scheme and the detail procedures of the empirical case of revenue changes in e-reader. Section 4 evaluates the forecasting accuracy of the proposed model by compared with the existing methods. Finally, the paper is concluded in section Related works In this section, some basic definitions and principles of fuzzy time series are introduced to cope with the ambiguity and chaotic of prediction problems. The algorithm of Particle swarm optimization (PSO) is also reviewed to integrate with fuzzy approach. 2.1 Fuzzy Time series Let U be the universe of discourse, where U = {u 1, u 2,, u n }. A fuzzy set A in U can be defined as: A ( u ) / u (1) A( u1) / u1 A( u2) / u2 A where A is the membership function of the fuzzy set A, A : U [0, 1] and A (u i ), 1 i n, denotes the grade of membership of u i in U and A (u i ) [0, 1]. In [4-6, 8], some definitions of the fuzzy time series are described as follows. Definition 1: Let Y(t)(t =, 0, 1, 2, 3, 4, ), a subset of real numbers, be the universe of discourse by which fuzzy sets i (t) are defined. If F(t) is a collection of 1 (t), 2 (t),, then F(t) is the fuzzy time series defined as Y(t). Definition 2: When F(t) and F(t 1) are fuzzy sets, if there exists a FLR, R(t 1, t), such that F(t) = F(t 1) R(t, t 1), where the symbol denotes an operation, then F(t) is sa to be caused by F(t 1). If F(t) is caused by F(t 1) only, the first-order FLR is represented by F(t 1) F(t), where F(t 1) is the current state and F(t) is the next state, respectively. n n 391

3 Definition 3: Let F(t n), F(t n + 1),, F(t 1), and F(t) be the fuzzy sets in a time series. If F(t) is caused by F(t n), F(t n + 1), and F(t 1), then the nth-order FLR is defined as follows: F(t n), F(t n + 1),, F(t 1) F(t), where F(t n), F(t n + 1),, F(t 1) denotes to the current state and F(t) denotes to the next state. 2.2 Particle Swam Optimization The PSO algorithm was first introduced by Eberhart and Kennedy [24] in The PSO approach can efficiently compute the near optimal or optimal solution for complicated problems. Based on the PSO algorithms [15, 17, 25, 26], a cooperative swarm of particles searches the space of problem domains to find all possible solutions. In [24] at each optimization step, a flying particle, indexed by, adjusts its candate position and velocity as (2) and (3): t t 1 t 1 t 1 V V c rand () ( P X ) c rand () ( P X ) (2) 1 2 gd X t X t 1 V t (3) where is the inertia weight factor; and denote the ith particle s position in dimension d at time t 1 and t; P is the position where the best fitness value that ith particle has ever obtained since time t; P gd is the position where the highest fitness value that all particles have ever found since time t; c 1 and c 2 are acceleration values; rand() is a random function which yields a random value from 0 to 1; and and represent the current and next velocities of a particle. If exceeds V ma x, then is set to V max, where V max is a constant value. In this paper, the standard PSO algorithm is used and depicted as follows. Algorithm 1. Standard PSO algorithm l: Randomly yield position and velocity for each particle 2: while stopping criteria (maximum iterations or minimum error ) is not reached do 3: for each particle do 4: Compute the fitness 5: Update the local best position and global best position 6: Update the velocity 7: Update the position 8: end for 9: end while 3. QBDPSO forecasting model In QBDPSO forecasting model, an improved defuzzify scheme is applied to improve fuzzy approach. The key issue of the improved defuzzify scheme is based on the quartiles of the intervals. For a defuzzification process, the QBD scheme cuts each interval into four sub-intervals with equal size. The QBD scheme calculates the fuzzy center of an interval according to two rules: (1) Each data located in the first and four quartiles of a interval is replaced with the value of mpoint of the interval; (2) Each data between the second and third quartiles of a interval, the value is unchanged. Based on above two rules, the fuzzy center of each interval can be obtained. 392

4 Table 1. Revenue change, fuzzy set, mpoint and fuzzy center of QBD scheme Year/Month Revenue change Fuzzy set Mpoint Fuzzy center 2011/ A / A / A / A / A / A / A The PSO algorithm is used to adjust interval lengths in the universe of discourse by minimizing the MSE value. In QBDPSO model, the trained fuzzy rules and forecasted values are used to predict unknown values. The empirical example of revenue changes for QBDPSO model is presented as follows. Step 1: Define the universe of discourse U as U = [200, 4000]. Step 2: Dive U into ten intervals as U = { u 1, u 2, u 3, u 4, u 5, u 6, u 7, u 8, u 9, u 10 }. (4) Step 3: Define the fuzzy sets based on the intervals Step 4: Fuzzify historical data as showed in Table 1 in which Columns 2 lists actual revenue and Column 3 shows fuzzy sets. Step 5: Generate all fuzzy logical relationships. According to Definitions 2 and 3, Table 2 shows all 3rd-order FLRs. Table 2. The 3rd-order fuzzy logical relationships on revenue changes The 3rd-order Fuzzy Logical Relationships A 6, A 3, A 6 A 5 A 1, A 0, A 1 A 1 A 3, A 6, A 5 A 4 A 0, A 1, A 1 A 1 A 6, A 5, A 4 A 5 A 1, A 1, A 1 A 3 A 9, A 6, A 0 A 0 A 7, A 7, A 3 A 3 A 6, A 0, A 0 A 0 A 7, A 3, A 3 A 2 A 0, A 0, A 1 A 0 A 3, A 2, A 2 A 2 A 0, A 1, A 0 A 1 A 2, A 2, A 2 A 2 393

5 Table 3. The 3rd-order fuzzy logical relationship groups on revenue changes The 3rd-order Fuzzy Logical Relationship Groups (A 0, A 0, A 1 ), (A 6, A 0, A 0 ), (A 9, A 6, A 0 ), (A 9, A 9, A 6 ) A 0 (A 0, A 0, A 0 ), (A 0, A 1, A 0 ), (A 0, A 1, A 1 ), (A 1, A 0, A 1 ) A 1 (A 2, A 2, A 2 ), (A 3, A 2, A 2 ), (A 3, A 3, A 2 ), (A 7, A 3, A 3 ) A 2 (A 3, A 3, A 5 ), (A 3, A 5, A 7 ), (A 4, A 5, A 6 ), (A 5, A 7, A 7 ) A 7 (A 5, A 6, A 7 ), (A 6, A 7, A 9 ) A 9 Step 6: Create FLR groups. All FLRs with the same current state are taken together to yield a FLR group. According to Table 2, Table 3 shows FLR groups Step 7: Forecast and defuzzify. The QBD scheme is used to forecast in each forecasting. Table 4 shows the comparison of revenue forecasting from January 2011 to July Step 8: Compute forecasting accuracy. The measure indicators of MSE and MAPE are used to estimate the forecasting accuracy. Let N be the number of forecasted data. The estimated indicators of MSE and MAPE are defined as follows: N 2 ( F i Y i i v ( ) ( )) 1 MSE (5) N 1 MAPE N N i 1 Fv ( i) Y( i) 100% (6) Y( i) Step 9: PSO optimization. According to Algorithm 1, we use the PSO algorithm to adjust the interval lengths of U and repeat Steps 2-8 to find out proper intervals with minimum MSE value when maximum-iterations criteria is not satisfied. Table 4. Comparison of revenue forecasting from January 2011 to July 2013 Year/Month Actual Revenue MV HPSO[15] QBDPSO 2011/ / / / / / / / / / / / / MSE MAPE 38.69% 5.22% 4.95% 394

6 Jan-11 Feb-11 Mar-11 Apr-11 May-11 Jun-11 Jul-11 Aug-11 Sep-11 Oct-11 Nov-11 Dec-11 Jan-12 Feb-12 Mar-12 Apr-12 May-12 Jun-12 Jul-12 Aug-12 Sep-12 Oct-12 Nov-12 Dec-12 Jan-13 Feb-13 Mar-13 Apr-13 May-13 Jun-13 Jul-13 Revenue changes KC2015知 識 社 群 國 際 研 討 會 4. Experimental Results The actual data of revenue changes in e-reader from Prime View International Co., Ltd. [23] are listed in Table 1 for model valation. The historical data are separated into two parts: in-sample data from January 2011 to July 2013 for training and out-of-sample data from May 2013 to July 2013 for testing. The standard PSO is used to adjust interval lengths for improving forecast accuracy and the parameters are set as follows. The number of particles is 30, the maximal steps of movement for each particle is 100, the value of inertial weight is decreased linearly from 1.4 to 0.4 during each generation, and the self confence c1 and the social confence c2 are set to 2. The velocity of each particle V is limited to [ 100, 100]. In our experiment, the QBDSO model executes 100 runs, and the best result of 100 runs is chosen to be the final result Actual Revenue MV HPSO[15] QBDPSO Month Figure 1. A Comparison of revenue forecasting in the training phase To compare fitted accuracy, the monthly data from January 2011 to July 2013 of revenue changes in e-reader are used to verify the forecasting accuracy in the training phase. The QBDPSO model of 3nd-order is compared with two existing forecasting models, Moving Average (MV) and HPSO (3rd-order) [15]. A comparison of the forecasted results with ten intervals is listed in Table 4. The proposed model of 3rd-order yields the lowest error rates, i.e., MSE = 4051 and MAPE = 4.95%. The graphic comparison of actual data and the forecasted results in e-reader is shown in Figure 1.0 Table 5. Comparison of revenue change forecasting from May 2013 to July 2013 Date Actual 1st-order 2nd-order 3rd-order Data HPSO QBPSO HPSO QBPSO HPSO QBPSO 2013/ / / MSE MAPE 48.04% 32.60% 35.11% 21.06% 20.49% 18.37% 395

7 Revenue changes Revenue changes KC2015知 識 社 群 國 際 研 討 會 In the testing phase, monthly data of revenue changes from May 2013 to July 2013 are used to verify the forecasting accuracy. A comparison of the forecasted revenue changes between HPSO model [15] and QBDPSO model is shown in Table 5. For QBDPSO model, the MSE values are , and and the MAPE values are 32.6%, 21.06% and 18.37% for 1st- to 3rd-order FLRs, respectively. From the trend of time series plot revealed in Figure 2. (a)-(c), we conclude that the proposed model can prove valuable information for future investment in e-book market. Revenue changes Actual HPSO QBDPSO 2013/ / /07 Month (a) The 1st-order fuzzy logical relationships Actual HPSO QBDPSO 2013/ / /07 Month (b) The 2nd-order fuzzy logical relationships Actual 400 HPSO 200 QBDPSO / / /07 Month (c) The 3rd-order fuzzy logical relationships Figure 2. Comparison of revenue forecasting in the testing phase 396

8 5. Conclusions In this paper, we propose an improved defuzzify scheme for revenue changes in e-reader based on fuzzy time series approach and PSO algorithm. For each forecasting, the QBD scheme is used calculate appropriate defuzzify values. The PSO algorithm generates optimal forecasted values by adjusting the lengths of intervals in U. Hence, the QBDPSO model can predict the revenue changes very well. As shown in Table 5, the proposed model outperforms the work in HPSO model [15] for the testing phase. This paper shows the superior forecasting capability compared to existing models in the training and testing phases. In the future, we suggest applying QBDPSO model to forecast more real-world problems such as temperature, crop production, stock indices, etc. References [1] Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, [2] Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning--i. Information Sciences, 8(3), [3] Zadeh, L. A. (1975).The concept of a linguistic variable and its application to approximate reasoning--ii. Information Sciences, 8(4), [4] Song, Q. & Chissom, B. S. (1993). Fuzzy time series and its models. Fuzzy Sets and Systems, 54(3), [5] Song, Q. & Chissom, B. S. (1993). Forecasting enrollments with fuzzy time series--part I. Fuzzy Sets and Systems, 54(1), 1-9. [6] Song, Q. & Chissom, B. S. (1994). Forecasting enrollments with fuzzy time series--part II. Fuzzy Sets and Systems, 62(1), 1-8. [7] Song, Q., Leland, R. P., & Chissom, B. S. (1995). A new fuzzy time-series model of fuzzy number observations. Fuzzy Sets and Systems, 73(3), [8] Chen, S. M. (1996). Forecasting enrollments based on fuzzy time series, Fuzzy Sets and Systems, 81(3), [9] Huarng, K. (2001). Effective lengths of intervals to improve forecasting in fuzzy time series. Fuzzy Sets and Systems, 123(3), [10] Huarng, K. (2001). Heuristic models of fuzzy time series for forecasting. Fuzzy Sets and Systems, 123, [11] Wong, W. K., Enjian, B., & Chu, A. W. (2010). Adaptive Time-Variant Models for Fuzzy-Time-Series Forecasting. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 40(6), [12] Yu, H. K. (2005). A refined fuzzy time-series model for forecasting. Physica A: Statistical Mechanics and its Applications, 346 (3-4), [13] Hwang, J. R., Chen, S. M., & Lee C. H. (1998). Handling forecasting problems using fuzzy time series. Fuzzy Sets and Systems, 100(1-3), [14] Yolcu, U., Egrioglu, E., Uslu, V. R., Basaran, M. A., & Aladag, C. H. (2009). A new approach for determining the length of intervals for fuzzy time series. Applied Soft Computing, 9(2), [15] Kuo, I. H., Horng, S. J., Kao, T. W., Lin, T. L., Lee, C. L., & Pan, Y. (2009). An improved method for forecasting enrollments based on fuzzy time series and particle swarm optimization. Expert Systems with Applications, 36(3), [16] Kuo, I. H., Horng, S. J., Chen, Y. H., Run, R. S., Kao, T. W., Chen, R. J., et al. (2010). Forecasting TAIFEX based on fuzzy time series and particle swarm optimization. Expert Systems with Applications, 37(2), [17] Huang, Y. L., Horng, S. J., He, M., Fan, P., Kao, T. W., Khan, M. K., et al. (2011). A hybr forecasting model for enrollments based on aggregated fuzzy time series and particle swarm optimization. Expert Systems with Applications, 38(7), [18] Huang, Y. L., Horng, S. J., Kao, T. W., Run, R. S., Lai, J. L., Chen, R. J., et al. (2011). An improved forecasting model based on the weighted fuzzy relationship matrix combined with a PSO adaptation for enrollments. International Journal of Innovative Computing, Information 397

9 and Control, 7(7A), [19] Adebiyi, A. A., Ayo, C. K., Adebiyi, M., & Otokiti, S. O. (2012). An Improved Stock Price Prediction using Hybr Market Indicators. African Journal of Computing & ICT, vol. 5(5), [20] Sadaei, H. J., Enayatifar, R., Abdullah, A. H., & Gani, A. Short-term load forecasting using a hybr model with a refined exponentially weighted fuzzy time series and an improved harmony search. International Journal of Electrical Power & Energy Systems, 62, [21] Chen, S. M. & Chen, S. W. (2014). A new method for forecasting the taiex based on two-factors second-order fuzzy-trend logical relationship groups and the probabilities of trends of fuzzy logical relationships. International Conference on Machine Learning and Cybernetics (ICMLC), [22] Chen, S. M. & Chen, S. W. (2015). Fuzzy Forecasting Based on Two-Factors Second-Order Fuzzy-Trend Logical Relationship Groups and the Probabilities of Trends of Fuzzy Logical Relationships. IEEE Transactions on Cybernetics, 45(3), [23] L. Prime View International Co., [24] Eberhart, R. & Kennedy, J. (1995). A new optimizer using particle swarm theory, Proc. 6th Int. Symposium Science, [25] Kuo, I. H., Horng, S. J., Kao, T. W., Lin, T. L., Lee, C. L., Terano, T., et al. (2009). An efficient flow-shop scheduling algorithm based on a hybr particle swarm optimization model. Expert Systems with Applications, vol. 36(3), [26] Eberhart, R. & Shi, Y. (2001). Particle swarm optimization: developments, applications and resources. Proc. IEEE Int. Conf. on Evolutionary Computation,