Grey forecasting model with polynomial term and its optimization

Transcription

1 Grey forecasting model with polynomial term and its optimization Luo Dang, Wei Baolei School of Mathematics and Statistics, orth China University of Water Resources and Electric Power, Zhengzhou 4546, PR China Abstract A novel grey forecasting model named GMP(1,1,) model aimed to enlarge the suitable ranges is proposed in this paper GM(1,1) model, GM(1,1,k) model and GM(1,1, t ) model prove to be special cases of GMP(1,1,) model with different polynomial orders Moreover, a criterion for determining polynomial order is given out based on the stepwise ratio sequences for practical application In order to further improve forecasting performance, the initial condition is optimized using least square method Finally, an empirical example about aiwan s integrated circuit industry output forecasting is used to illustrate the accuracy As comparing with exponential smoothing method, DGM(1,1,) model and GABGM(1,1) model, the results indicate that the proposed model is better than other three models Key words: Grey system theory; GMP(1,1,) model; Least square method; Forecasting accuracy; Initial condition optimization 1 Introduction Since grey system theory was pioneered by professor Deng [1], grey forecasting method has been employed in hydrology, transportation, energy, signal processing, industrial production, supply chain management, etc [2-7] As the elements of grey forecasting methodology, GM(1,1) model has been the research focus For example, Ref [8-12] optimized the grey derivative, background coefficient and initial condition, respectively; Ref [13-16] studied its properties of unbiasedness, morbidity and chaos Moreover, Ref [17] researched the sufficient and necessary condition of GM(1,1) modeling; using matrix perturbation theory, Wu et al gave a reasonable explanation for the reason why GM(1,1) model is not used for large sample [18] and then put forward grey models with the fractional order accumulation [19] ; Xiao et al studied the mechanism of GM(1,1) model with the generalized accumulation; Liu et al derived the relative error up bounds of GM(1,1,W) model and GM(1,1,C) model with the developing coefficient located in applicable range; Liu et al studied the properties and suitable rages of four basic Corresponding Author: Baolei Wei, School of Mathematics and Statistics, orth China University of Water Resources and Electric Power, Zhengzhou 4546, PR China; weibaolei_214@163com Luo Dang, Wei Baolei Grey forecasting model with polynomial term and its optimization 217, 29(3), 58-69

2 models of GM(1,1) model [22] hrough analyzing the above results, we find that the sequence applicable for GM(1,1) modeling is with the characteristics of homogenous index law or the approximate homogenous index law herefore, in order to enlarge suitable ranges, scholars extended GM(1,1) model from the following two perspectives: Extension of whitenization differential equation Ref [23] created GM(1,1,k) model for sequence with approximate non-homogenous index law; Ref [24] developed GM(1,1, t ) model with time power term Actually, it degraded into GM(1,1,k) model when 1 But there is still not a method or criterion to select the suitable parameter (2) Construction of discrete grey model Xie et at put forward DGM(1,1) model for approximate non-homogenous index sequences on the basis of the angle from discrete to discrete [25] ; Cui et al studied the morbid property of GM(1,1, k ) model using matrix condition number [26] ; Ref [27] constructed DGM(1,1,) model corresponding with polynomial time-vary sequences In fact, DGM(1,1,) model is the unification of the above two discrete grey models with different values However, how to determine the polynomial order is still not solved until now All above researches are significant for the development and perfection of grey system Based on theoretical analysis and numerical experiments, Ref [28] proved that the conclusion that DGM(1,1) model is the precise form of GM(1,1) model is not correct In other words, the essence of classical models and discrete models are different herefore, GMP(1,1,) model whose discrete form is DGM(1,1,) model is proposed in this paper, and then a criterion for determining the polynomial order is also researched he remaining paper is organized as follows GMP(1,1,) model is constructed and the relationship between GMP(1,1,) model and other models is also discussed in section 2 hen, a criterion for determining the polynomial order is given out in section 3 he initial condition optimization and initial pint optimization of GMP(1,1,) model prove to be equivalent in section 4 An illustrate example about integrated circuit industry output forecasting in aiwan is given out and then the forecasting accuracy are compared with exponential smoothing (ES) method, DGM(1,1,) model and GABGM(1,1) model in section 5 Finally, conclusions and future works are presented in section 6 2 Grey GMP(1,1,) model Definition 1 [1] Assume the nonnegative sequence () (), () (2),, () ( ), (2),, ( ) X x x x n, X x x x n is called first order accumulative generating sequence, () where x ( k) k x ( ) 1 Definition 2 he basic form of GMP(1,1,) model is defined as 1 1 () x ( k 1) x ( k) 2k 1 k ( k 1) x ( k) 1, and the whitenization equation of GMP(1,1,) model is expressed as d x ( t) x () t 1t t (2) dt Actually, Eq is a discrete approximate representation of Eq (2) to obtain the estimate of parameters Integrating Eq (2) in the interval [k-1, k], the right side becomes

3 1 1 k 2k 1 k ( k 1) ( 1 1t )d t t 1 k ; 2 1 the left side is k k () x ( k) x ( k 1) x ( t)d t x ( k) x ( t)dt k k1 k1 ote that x ( t)dt is the area between the curve and x-axis Replacing AB k 1 with AB in Fig 1, the following equation holds true roughly 1 k x ( t)d t x ( k 1) x ( k ) (3) k 1 2 y x ( k) B y x () t x ( k1) A C k 1 D k t as Figure 1 Geometrical schematic diagram of the integral term heorem 1 he estimate of parameter 1 κ [ ] is expressed 1 ˆ ˆ ˆ κˆ [ ˆ ] ( B B) B Y where ( BB ) is the Moore-Penrose inverse matrix of matrix BB, the number of samples and the polynomial order satisfies n 4, 1 x x (2) () x (2) () x (2) x (3) x (3) Y 1, B () x ( n) 1 1 x ( n 1) x ( n) 2n 1 n ( n 1) Proof: Substituting data into Eq gives a linear equation system 1 1 () x x (2) x (2) () x (2) x (3) x (3) () x ( n 1) x ( n) 2n 1 n ( n 1) x ( n) hat is Bκ Y (5) Since the rows of matrix B are greater than the columns ( n 4 ), Eq (5) is an overdetermined equation system without precise solution Let Y ( Y Bκ ) be (4)

4 the error sequence, then 2 κˆ arg min L( κ) Y ( Y Bκ) ( Y Bκ ) (6) κ he k minimizing the convex quadratic loss function L(k) satisfies d L( κ) 2BBκ2BY dκ hus, Eq (4) is proved Especially, the square matrix BB is nonsingular and 1 ( B B) ( B B) when the columns of matrix B are independent So 1 κˆ [ ˆ ˆ ˆ ˆ 1 ] ( ) B B B Y (7) It is proved heorem 2 Let the initial condition be xˆ x, then the time response sequence is given by ˆ ( 1) ˆ ( ) k x k x e 1k k (8) () xˆ ( k) xˆ ( k) xˆ ( k 1), k 2,3,, n p where p is the number of samples to be forecasted, 2 1 γ M β (9) 1 ˆ ˆ 1 ˆ ˆ 1 β, M ˆ ˆ ˆ Proof: Substituting the estimated parameter into Eq (2) gives that d xˆ ( t) ˆ ˆ () ˆ ˆ ˆ x t 1t t (1) dt he general solution to Eq (1) is xˆ ( t) x ( t) x ( t ) where p ˆ ( ) e t xh t c is the solution to homogeneous equation x () t t t is the particular solution satisfying 1 d xp ( t) h p ˆ xˆ ( t) ; t d xˆ ( t) d ˆ ˆ ˆ ˆ xp () t 1t t (11) dt Eq (11) leads to a linear equation system Mγ β Because ˆ, the matrix 1 M is nonsingular and γ M β hus, the general solution is ˆ xˆ ( t) ce t t t (12) Substituting the initial condition c 1 ˆ x x into the Eq (12) gives that ˆ e x (13) So the time response sequence is ( 1) ˆ ( ) e k x k x 1k k (14) ˆ

5 Based on first order inverse accumulative generating operator, the fitting and forecasting values corresponding with original sequence is expressed as () xˆ ( k) xˆ ( k) xˆ ( k 1), k 2,3,, n p (15) It is proved Corollary 1 In GMP(1,1,) model When =, GMP(1,1,) model degrades into GM(1,1) model [1] he basic form and whitenization equation are respectively expressed as () x ( k 1) x ( k) d x ( t) x ( k) and x () t 2 dt he estimated parameter is ˆ κˆ [ ˆ ] ( B1 B1 ) B1 Y where x x (2) x (2) x (3) x ( n 1) x ( n) B Be1 e (2) When =1, GMP(1,1,) model degrades into GM(1,1,k) model [23] he basic form and whitenization equation are respectively expressed as () x ( k 1) x ( k) 2k 1 d x ( t) x ( k) 1 and x () t 1t 2 2 dt he estimated parameter is ˆ ˆ κˆ [ ˆ 1] ( B2 B2 ) B2 Y where x x (2) x (2) x (3) x ( n 1) x ( n) B Be1 e2 e n (3) When 1 2 1, GMP(1,1,) model degrades into GM(1,1, t ) model [24] he basic form and whitenization equation are respectively expressed as 1 1 () x ( k 1) x ( k) k ( k 1) d x ( t) x ( k) and x () t t 2 1 dt he estimated parameter is ˆ ˆ κˆ [ ˆ ] ( B3 B3 ) B3 Y where x x (2) x (2) x (3) x ( n 1) x ( n) B Be1 e2 e n ( n1) Corollary 1 indicates that GM(1,1) model, GM(1,1,k) model and GM(1,1, t ) model are all special cases of GMP(1,1,) model with different polynomial orders Property 1 GMP(1,1,) model is suitable for sequence with the characteristic of partly index law and partly time polynomial law, that is to say, the original sequence is approximated to

6 , c and c 1 () t p x () t ac b b1t bpt From Property 1, we know that when p and b, GMP(1,1,) model is suitable for homogeneous index sequences; when p and b, GMP(1,1,1) model is suitable for non-homogeneous index sequences 3 he criterion for determining the polynomial order he polynomial order has a greatly influence on modeling accuracy in practical application herefore, it is important to select a suitable order according to the characteristics of original sequence Definition 3 Assume that the r order difference sequence of original sequence is D d ( r 1), d ( r 2),, d ( n ), then the r order stepwise ratio sequence is defined as ( r 2), ( r 3),, ( n) where d ( k) ( k), 2, 3, d ( k 1), (16) Corollary 2 In the r order stepwise ratio sequence When () () () (2), (3), (), ( n) is the homogeneous stepwise ratio sequence [1], where (2) When 1 r, () () () d k x k ( k) () () ( ) ( ) d ( k 1) x ( k 1), k 2,3,, n (17) r, (3), (4),, ( n) is the non-homogeneous stepwise ratio sequence [1], where () () d ( k) x ( k) x ( k 1) ( k), k 3,4, () () d ( k 1) x ( k 1) x ( k 2), n (18) Definition 4 Assume max ( k) and max r2kn degree of grey index law is defined as min ( k), then the min r2kn max min Especially, the sequence is called r order homogeneous when ( r 1) Property 2 For any non-negative integer r, if, then Proof: From Definition 3, we know that ( r1) ( r1) d ( k) d ( k) d ( k 1) ( k) 1 ( k) ( k1) ( r1) d ( k 1) d ( k 1) d ( k 2) ( k 1) 1 According to hus,, we have max min ( ) ( ) ( ) r ( k) max r ( k) min r ( k) const ( r1) ( k) const, r2kn r3kn r2kn ( 1) ( 1) ( 1) r max r ( k) min r ( k) r3kn It is proved Property 3 he sufficient and necessary condition for the original sequence to be r order homogeneous is

7 where ( r a d ) ( r 1), Proof: Suppose that Hence, k r1 d ( k) ac, 1, 2,, c d ( n) d ( n 1) k r1 d ( k) ac ; we have d ( k) ac d ( k 1) ac k r r n (19) k r1 ( k) c, 2, 3,, k r 2 ( ) ( ) ( ) r max r ( k) min r ( k) r2kn r2kn k r r n (2) Conversely, let, then according to Definition 4 ( ) ( ) ( ) ( ) ( k) r max r ( i) r min r ( i) const max r2in mn i So D d ( r 1), d ( r 2),, d ( n) first term It is proved r2in is a geometric sequence with the ( r d ) ( r 1) and the stepwise ratio d ( n) d ( n 1) Property 3 shows that is a measurement of r order homogeneous index law herefore, taking the volatility characteristics of higher order polynomial and the principle of parsimony [29] into consideration, we have,,1,2,3 min arg min min arg min r max min r (21) Based on Eq (21), we can preliminarily determine the alternative polynomial orders However, the stepwise ratio sequence diagrams and debugging method are also used because of the biasness of GMP(1,1,) model for order homogenous index sequence 4 he initial condition optimization of GMP(1,1,) model GMP(1,1,) model is based on the initial condition xˆ x to obtain c Actually, it is an unreasonable constraint that the first point (1, x () ) lies on time response function curve [1] hus, GMP(1,1,) model is optimized by minimizing the mean square relative error, attributed as OGMP(1,1,) model hat is, c n ˆ arg min L ( c ) x ˆ ( k ) x ( k ) 2 (22) where Let c k 1 ˆ xˆ ( k) ce t 1k k c x, then ˆ 2 ˆ arg min L( ) n x ( k) x ( k) (23) k 1 ˆ where Hence, xˆ ( k) x e k k m t 1 k 2 (24) 2 ( 1) L( ) n x e k x ( k) k 2 It is obvious that L( ) is a convex quadratic function Based on extreme value theory, we have

8 ˆ x k k x n 2 ( k 1) 1 e k 2 n ( k 1) 2 ( k 1) ( ) e e k 2 (25) hus, the restored values based on OGMP(1,1,) model are m ˆ ( k ˆ ( ) e 1) x k x 1k k (26) () xˆ ( k) xˆ ( k) xˆ ( k 1), k 2,3,, n p In order to evaluate the modeling accuracy, the mean absolute percentage error for prior-sample data (MAPEPR) and post-sample data (MAPEPO) and the root mean square percentage error for post-sample period (RMSPE) are defined as 1 () () () MAPEPR n x ( k) xˆ ( k) x ( k) 1%, k 1 n (27) MAPEPO 1 () () () n p ( ) ˆ ( ) ( ) 1%, kn1 p (28) RMSPE 1 p 5 Empirical analysis 2 () () () n p x k xˆ k x k kn1 ( ) ( ) ( ) 1% (29) Hsu forecasted integrated circuit industry output in aiwan using the genetic algorithm based nonlinear grey Bernoulli model [3] o compare modelling results conveniently, the partition of sample data is the same with that in Ref [3] In other words, the data from 199 to 23 are regarded as prior-sample to training models, and 24 to 27 are reserved for ex post testing he training and testing data are listed in able 1 he stepwise ratio sequences with different orders are respectively shown in Figure 2 Figure 2 he stepwise ratio sequences with different orders of integrated circuit industry output sequence from 199 to 23

9 () (3) (2) From Figure 1, we have that 1 and the stepwise ratio sequences with r equal to and 1 both fluctuate heavily herefore, we determine the alternative polynomial orders, = and 1 Moreover, in order to illustrate the effectiveness of the method, we also construct GMP(1,1,) model with =2 he modeling results with different polynomial orders are shown in able 1 able 1 Actual data and fitting and forecasting values for aiwan s integrated circuit industry output by different models (Units: 1 9 dollars) Year Actual value GM(1,1) model OGM(1,1) model GMP(1,1,1) model OGMP(1,1,1) model Model value Error Model value Error Model value Error Model value Error MAPEPR (%) MAPEPO (%) (24-27) RMSPE (%) (24-27) Continued) GMP(1,1,2) model OGMP(1,1,2) model GABGM(1,1) model ES model DGM(1,1,1) model Model value Error Model value Error Model value Error Model value Error Model value Error

10 Absolute percentage error (%) OGM(1,1) model OGMP(1,1,1) model GMP(1,1,2) model GABGM(1,1) model Fitting 3 Forecasting Year Figure 3 he absolute percentage error for prior-sample data and post-sample data of four models From table 1, the MAPEPR of the GMP(1,1,) models are almost same with that of OGMP(1,1,) models with equal to and 1 But the MAPEPO and RMSPE of GMP(1,1,) models are respectively greater than those of OGMP(1,1,) models, which indicates that the optimized models perform better he result of GMP(1,1,2) model are almost equal to that of OGMP(1,1,2) model with ˆ he MAPEPR, MAPEPO and RMSE of ES method are 1581%, 1511% and 1563%, indicating that ES method has higher error than GABGM(1,1) model and OGMP(1,1,) models with equal to and 1 he reason for OGMP(1,1,) model performing worse is under-fitting in which the prior-sample data clearly shows structure not captured; on the contrary, OGMP(1,1,2) model with lower MAPEPR but higher MAPEPO and RMPSE is due to overfitting in which the disturbance or noise of prior-sample data are identified as structure From figure 3, we know that GABGM(1,1) model and OGMP(1,1,1) model are not much different in both prior-sample data and post-sample data However, OGMP(1,1,1) model is superior to GABGM(1,1) model because there only exists numerical solution to GABGM(1,1) model and its complexity is higher In summary, the OGMP(1,1,1) model has a MAPEPR of 1267% in prior-sample data form 199 to 23, MAPRPO of 596% and RMSPE of 84% in post-sample data form 24 to 27, indicating a highly accurate forecasting method 6 Concluding Remarks From the above researches, we conclude that: GMP(1,1,) model is based on the accumulated generating operation either It is no longer only suitable for sequences with homogenous and non-homogenous index law (2) GMP(1,1,) model is not only the unification of GM(1,1) model, GM(1,1,k) model and GM(1,1, t ) model We can construct novel models suitable for practical sequences by selecting polynomial order (3) he initial condition optimization and initial point optimization proves to be equivalent he properties such as unbiasedness and robustness should be focused in future studies (4) Based on the stepwise ration sequence of original sequence, we proposed a

11 criterion to determine the alternative polynomial orders However, this method is with the characteristic of both qualitative and quantitative analyses herefore, to select the polynomial order automatically based on regularization method is another work in further [31] Acknowledgements his research is supported by atural Science Foundation of China (os and 71538), Humanistic and Social Science Youth Foundation of Ministry of Education of China (o 14YJC63121), Key Scientific and echnological Proect of Henan Province (o ), Key Research Proect Fund Plan of Henan Universities (o 15A635), and Postgraduate Innovative Proect of orth China University of Water Resources and Electric Power (o YK215-2) References [1] Deng J L he elements of grey system, Press of Huazhong University of Science echnology, Wuhan, 22 [2] Luo D Risk evaluation of ice-am disasters using gray systems theory: the case of ingxia-inner Mongolia reaches of the Yellow River atural Hazards, 214, 71(3): [3] Liu C, Shu, Chen S, et al An improved grey neural network model for predicting transportation disruptions Expert Systems with Applications, 215, 45(C): [4] Xie M, Yuan C Q, Yang Y J Forecasting China s energy demand and self-sufficiency rate by grey forecasting model and Markov model International Journal of Electrical Power & Energy Systems, 215, 66:1-8 [5] Wang Z X An optimized ash nonlinear grey Bernoulli model for forecasting the main economic indices of high technology enterprises in China Computers & Industrial Engineering, 213, 64(3): [6] He Z, Shen Y, Wang Q Boundary extension for Hilbert Huang transform inspired by gray prediction model Signal Processing, 212, 92(3): [7] Samvedi A, Jain V A grey approach for forecasting in a supply chain during intermittent disruptions Engineering Applications of Artificial Intelligence, 213, 26(3): [8] Wang Y, Liu K D, Li Y C GM(1,1) modeling method of optimum the whiting values of grey derivative Systems Engineering-heory & Practice, 21, 21(5): [9] Luo D, Liu S F, Dang Y G he optimization of grey model GM(1,1) Chinese Engineering Science, 23, 5(8):5-53 [1] Jin X, ao, Mao, et al Improvement of grey models by least squares Expert Systems with Applications, 211, 38(11): [11] Wang Y H, Liu Q, Wang J R, et al Optimization approach of background value and initial item for improving prediction precision of GM(1,1) model Journal of Systems Engineering & Electronics, 214, 25:77-82 [12] Guo J H, Ying J W Optimizing the initial condition and the initial point of GM(1,1) Systems Engineering-heory & Practice, 215, 35(9): [13] Ji P R, Huang W S, Hu X Y An unbiased Grey forecasting model Systems Engineering & Electronics, 2, 22(6):6-7, 8 [14] Zheng Z, Wu Y Y, Bao H L Morbidity problem in grey model Chinese Journal of Management Science, 21, 9(5):38-44 [15] Dang Y G, Wang Z X, Liu S F Study on morbidity problem in grey model Systems Engineering-heory & Practice, 28, 28: [16] Wang Z X, Dang Y G, Liu S F Analysis of chaotic characteristics of unbiased GM(1,1) Systems Engineering-heory & Practice, 27, 27(11): [17] Chen C I, Huang S J he necessary and sufficient condition for GM(1, 1) grey prediction model Applied Mathematics & Computation, 213, 219(11): [18] Wu L F, Liu S F, Yao L G, et al he effect of sample size on the grey system model Applied Mathematical Modelling, 213, 37(9): [19] Wu L F, Liu S F, Yao L G, et al Grey system model with the fractional order accumulation Communications in onlinear Science & umerical Simulation, 213, 18(7): [2] Xiao X P, Guo H, Mao S H he modeling mechanism, extension and optimization of grey GM(1,1) model Applied Mathematical Modelling, 214, 38(5-6): [21] Liu J, Xiao X, Guo J, et al Error and its upper bound estimation between the solutions of GM(1,1)

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