J Syst Sci Complex (2014) 27: 117–129
FORECASTING TIME SERIES WITH GENETIC PROGRAMMING BASED ON LEAST SQUARE METHOD∗ YANG Fengmei · LI Meng · HUANG Anqiang · LI Jian
DOI: 10.1007/s11424-014-3295-2 Received: 10 April 2012 / Revised: 22 April 2013 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2014 Abstract Although time series are frequently nonlinear in reality, people tend to use linear models to fit them under some assumptions unnecessarily in accordance with the truth, which unsurprisingly leads to unsatisfactory performance. This paper proposes a forecast method: Genetic programming based on least square method (GP-LSM). Inheriting the advantages of genetic algorithm (GA), without relying on the particular distribution of the data, this method can improve the prediction accuracy because of its ability of fitting nonlinear models, and raise the convergence speed benefitting from the least square method (LSM). In order to verify the validity of this method, the authors compare this method with seasonal auto regression integrated moving average (SARIMA) and back propagation artificial neural networks (BP-ANN). The results of empirical analysis show that forecast accuracy and direction prediction accuracy of GP-LSM are obviously better than those of the others. Keywords
1
Forecast, genetic programming, least square method, time series.
Introduction
Economic forecast can predict the future by the rules extracted from historical information and bring the information advantage to decision makers. Therefore, economic forecast has already gotten more and more attention. YANG Fengmei · LI Meng Faculty of Science, Beijing University of Chemical Technology, Beijing 100029, China. Email :
[email protected];
[email protected]. HUANG Anqiang School of Economics and Management, Beihang University, Beijing 100083, China. Email :
[email protected]. LI Jian (Corresponding author) School of Economics and Management, Beijing University of Chemical Technology, Beijing 100029, China. Email:
[email protected]. ∗ This research was supported by the National Natural Science Foundation of China under Grant Nos. 71171011 and 91224001, Program for New Century Excellent Talents in University (NCET-12-0756). This paper was recommended for publication by Editor WANG Shouyang.
118
YANG FENGMEI, et al.
In recent years, many forecast methods have been presented that can be categorized into linear models[1−6] and nonlinear models[7−11] , and most of the latter are intelligent models. One of the notable nonlinear models with thrilling performance is symbolic regression from genetic programming (GP) proposed by Koza[12] . GP has been successfully applied in an increasingly number of areas including optimal control, symbolic regression, programming, solving differential equations, searching game strategy, evolution of spontaneous behavior, and so forth[13−18] , owing to its high effectiveness of analyzing nonlinear problems. Different forecast models have their own advantages and disadvantages. Linear forecast models are simpler and thus can be implemented more easily, but they can’t capture the nonlinear trend. Nonlinear models, especially intelligent models such as artificial neural networks (ANN) and support vector machine (SVM), can effectively fit the nonlinear trend. However, unfortunately their mechanisms are so-called black boxes because of their low understandability. Genetic programming based on least square method (GP-LSM) proposed by this paper may be one of good candidates for solutions to the above dilemma. Through combining GP and least square method (LSM), GP-LSM has four merits: (a) It doesn’t depend on the distribution of data; (b) theoretically, it can find the global optimum through multi-step iterations; (c) it can provide the explicit optimal forecast equation; (d) GP-LSM converges fast and is able to generate the optimal variable coefficients of the corresponding forecast equations, owing to the utilization of LSM. The remainder of this paper is outlined as follows. Section 2 introduces GP-LSM in detail. Section 3 presents a brief introduction of SARIMA and BP-ANN method. Empirical studies are discussed in Section 4. Some concluding remarks and suggestions for future research directions are provided in Section 5.
2
Genetic Programming Based on Least Square Method (GP-LSM)
The basic principle of GP-LSM is described as follows. First, generate forecast equations randomly and each forecast equation called as an individual of initial population is represented by a tree. Then, adopt certain control policy for the genetic operations of initially generated trees until some newly generated forecast equation meets one of termination conditions. Generally speaking, GP-LSM mainly consists of initialization, evaluation, reproduction, crossover, and mutation. Implementation procedure of GP-LSM is presented in Figure 1. All parts of GP-LSM will be stated in the following sections. 2.1
Initialization
Initialization randomly generates forecast equations represented in the form of trees. Given the function set F composed of arithmetic operators, for example F ={+, −, ×, ÷}, and the variable set V , for example V = {x1 , x2 , · · · , xm }, where xi is the i-th variable that appears in the forecast equation, a tree can be generated through the following steps. First, randomly choose an element from the function set F as the root node of the tree. Second, choose two child nodes from the union of F and V . If all the child nodes are from V , the tree is completely generated. If a child node is from F , consider this child node as a new root and choose child
119
FORECASTING TIME SERIES WITH GP-LSM
nodes for it until all leaves of the tree are from V . An example is presented to demonstrate the procedure of generating the tree depicted in Figure 2. Start Generate initial population Estimate the coefficients with LSM Calculate fitness of each individual Yes
Meet termination conditions? No Reproduction Crossover Mutation
Output optimal forecast equation
New population
Stop
Figure 1 Implementation procedure of GP-LSM
+
+ +
+
+
+
/ yt-3
+
/ u
yt-2
yt-8
/ u
yt-3 yt-2
yt-2
yt-8
yt-3
Figure 2 Procedure of generating a tree
2.2
Estimation of Coefficients with LSM
In Section 2.1, several individuals (forecast equations) represented by trees are generated. It should be pointed out that those individuals contain no coefficients of arithmetic elements. Note that a part of equation between the two arithmetic operators (+ or −) is called an arithmetic element. Take the tree of Figure 2 as an example, the forecast equation is yt = yt−3 + yt−2 ∗ yt−3 + yt−2 /yt−8 ,
(1)
120
YANG FENGMEI, et al.
which contains 3 arithmetic elements including yt−3 , yt−2 ∗ yt−3 , and yt−2 /yt−8 . Obviously there are no coefficients before arithmetic elements. We will estimate these coefficients using LSM. Given Equation (1), transform it to yt = β0 + β1 yt−3 + β2 yt−2 ∗ yt−3 + β3 yt−2 /yt−8 ,
(2)
where βi (i = 0, 1, 2, 3) are coefficients to be estimated. Then, compute arithmetic elements values and replace them with new variables. In the above case, we separately compute the value of yt−3 , yt−2 ∗ yt−3 , and yt−2 /yt−8 , and replace them with w1t = yt−3 , w2t = yt−2 ∗ yt−3 , and w3t = yt−2 /yt−8 , consequently, Equation(2) can be expressed as yt = β0 + β1 w1t + β2 w2t + β3 w3t .
(3)
Assume the length of {yt } is m which means {yt } contains m observations and Equation (3) can be written as Y = W β, where Y = [y9 , y10 , · · · , ym ]T , β = [β0 , β1 , β2 , β3 )]T , and ⎡ 1 w19 w29 w39 ⎢ ⎢ .. .. .. .. W =⎢ . . . . ⎣ 1 w1m w2m w3m
(4)
⎤ ⎥ ⎥ ⎥. ⎦
(5)
Then the estimation of β can be computed as β = (W T W )−1 (W T Y ). 2.3
(6)
Evaluation
In this stage, the task is to find a measurement for identifying forecast equations with the satisfactory goodness-of-fit for reproduction. Mean square error (MSE) is a widely-used criterion, considering its very high time complexity in the reproduction process, this paper introduces R2 as the criterion evaluating the fitness of forecast equation, which can be computed as T (yi − yi )2 Ri2 = i=1 , (7) T 2 i=1 (yi − yi ) where yi denotes the i-th observation, T is the number of observations, yi denotes the predicted value corresponding to yi , yi denotes the mean of observed values. A large enough value of R2 signifies a good forecast equation with high goodness-of-fit that we are eager for. Frequently, a threshold of R2 is predefined and the forecast equation whose R2 is larger than the threshold is considered as the desired equation.
FORECASTING TIME SERIES WITH GP-LSM
2.4
121
Reproduction
According to genetic algorithm (GA) theory, individuals with higher fitness should be reproduced for higher occurring probability. Given a group of individuals, the best one will directly be preserved to the next generation to guarantee the monotone non-decreasing fitness, and the others will be chosen by roulette wheel method. The probability of the k-th individual to be reproduced depends on its fitness, which is expressed as R2 Pk = n k j=1
Rj2
,
(8)
where n is the number of individuals. GP-LSM does not reproduce individuals with the constant probability computed as Equation (8), but employs the reproduction method with an original control strategy of reproduction probability interpreted as follows: Step 1 Calculate the fitness of k-th individual (k = 1, 2, · · · , n) using Equation (7) and compute the reproduction probability Pk using Equation (8). Step 2 Preserve one of the best individuals with the maximal reproduction probability Pmax = maxj {Pj } to the next generation and keep it from the crossover and mutation operations, which will ensure the fitness of individuals in new generation no less than that in father generation. Step 3 Under the constraint of a constant size of population n, perform roulette wheel selection n − 1 times with the dynamically adjusted reproduction probability of individuals. Denote r(r = n− 1, n− 2, · · · , 1) as the number of individuals to be selected for next generation, srk the having-been-selected times of the k-th individual, and the reproduction probability of the k-th individual is adjusted through the following equation Pkadj =
n × Pk − srk , r
(9)
where Pkadj is the adjusted reproduction probability of the k-th individual. Pkadj (k = 1, 2, · · · , n) sum up to 1, therefore they are the reproduction probability distribution of individuals, which is proved by Equation (10) in the first round of roulette wheel selection. The proof in the remainder rounds of roulette wheel selection can be operated similarly. Pmax × n − 1 + n−1
n
j=1,j=max
Pj × n n−1
=
n
Pj × n 1 − n−1 n−1 j=1
=
1 n × Pj − n − 1 j=1 n−1
=
1 n − = 1. n−1 n−1
n
(10)
122 2.5
YANG FENGMEI, et al.
Crossover
Crossover is the major approach to generating new trees of forecast equations with higher fitness. Randomly select two individuals, for example, yt = yt−4 − yt−1 + yt−2 /yt−5 and yt = yt−3 + yt−2 ∗ yt−3 + yt−2 /yt−8 depicted in Figure 3, to crossover with the crossover ratio of Pc . The procedures are described as: Randomly select a non-leaf node N od1 in the tree of parent 1 and N od2 in the tree of parent 2, interconvert the sub-tree with the root of N od1 and the sub-tree with the root of N od2, and the two new trees representing for forecast equations are generated as described in Figure 3. The significance of selecting only non-leaf node is to prevent the bloat phenomenon of trees. Parent 1
Parent 2
+
yt-4
+
/ yt-1
yt-2
+
Nod2
Nod1
yt-5
u
yt-2
Offspring 2
+
+
/
yt-3
Offspring 1 crossover
yt-2
yt-4
yt-8
/
+ yt-1
u
yt-3
yt-3
yt-2
yt-2
/ yt-5
yt-2
yt-8
yt-3
Figure 3 An example of crossover
2.6
Mutation
Mutation is an effective approach to help the algorithm avoid sinking into the local minimum. Every individual will mutate with a predefined mutation probability Pm . Once an individual is selected to mutate, we firstly create a new tree and randomly select one non-leaf node N odm in the original tree, then substitute the sub-tree with the root of N odm with the new created tree. Figure 4 presents an example of mutation. Original tree
New created tree
+
yt-3
yt-6
/ yt-4
+
Nodm
u
yt-2
yt-5
Mutated tree mutation
+ u
/ yt-2
yt-8
yt-3
+ yt-4
yt-6
/ yt-2
Figure 4 An example of mutation
yt-8
FORECASTING TIME SERIES WITH GP-LSM
3 3.1
123
SARIMA and BP-ANN Method SARIMA Method
The well-known the autoregressive integrated moving average (ARIMA) model, has been widely used in analyzing non-stationary univariate time series data. In applying this technique, it is required that the time series data are either stationary, or can be differenced to make them stationary. ARIMA model is generally referred to as an ARIMA (p, d, q) model, where p, d, and q are integers greater than or equal to zero that refer to the order of autoregressive, integrated, and moving-average parts of the model, respectively[19] . The partial auto-correlation function (PACF) and auto-correlation function (ACF) are usually used to evaluate p and q. High-order p and q values can cause over-fitting, but raising the orders of p and q generally improves goodness of fit. Some timeseries include seasonal and trend characteristics. Via extending the ARIMA (p, d, q) model to absorb seasonal factors, we get a seasonal autoregressive integrated moving average (SARIMA) model, proposed by Box and Jenkins[20] , which was popular in this context. This model often specified as SARIMA (p, d, q)(P, D, Q)s, its general form is expressed as follows: ⎧ ⎪ φp (B)Φ P (B s )(1 − B)d (1 − B s )D xt = θq (B)ΘQ (B s )εt , ⎪ ⎪ ⎪ ⎨ (11) E(εt ) = 0, VAR(εt ) = σε2 , E(εt εs ) = 0 (t = s), ⎪ ⎪ ⎪ ⎪ ⎩ E(xs εt ) = 0), ∀s < t, ⎧ ⎪ ⎨φp (B) = 1 − φ1 B − φ2 B 2 − · · · − φp B p , (12) ⎪ ⎩θq (B) = 1 − θ1 B − θ2 B 2 − · · · − θq B q , where B is the backshift operator B(xt ) = xt−1 , ΦP (B s ) is the P -degree seasonal autoregressive back-sift operator, D denotes the order of seasonal integrated parts, xt denotes t-time forecast value, εt denotes t-time error, ΘQ (B s ) is the seasonal Q-degree moving average back-sift operator. 3.2
BP-ANN Method
Artificial neural network (ANN) is a nonlinear model containing many simple computational units, and BP-ANN is one of the most widely used. Commonly back propagation (BP) network comprises input, output, and hidden layers (in Figure 5). Neurons between adjacent layers are interconnected by a weighting factor. BP network optimizes the connection weights between neurons by BP algorithm.
124
YANG FENGMEI, et al.
Input layer
Hidden layer
Output layer
Figure 5 BP-ANN structure
BP-ANN model has been an important approach to the nonlinear problem, thanks to its sharp nonlinear mapping ability. Hecht-Nielsen[21] debated that if only there are enough nodes in the hidden layer, a BP-ANN with 3 layers can approach to any integrable continuous nonlinear function. The objective function of BP-ANN model is described as E=
1 1 Yi − Yi 2 = Yi − f (W · X + θ)2 , 2 2
(13)
where f is the activation transfer function in the hidden layer that is often the sigmoid function, W is the matrix of the weights between the input nodes and the hidden nodes, X is a matrix composed of the input vectors, and θ is the bias vector of the hidden layer. BP-ANN model divides the data set into training set, test set, and validation set, uses BP weight-learning algorithm to generate the optimal weight distribution by minimizing the objective function on the training set, and then checks the performance of BP-ANN model with the optimal weight distribution on validation set, and finally selects the BP-ANN model with the best performance to forecast on the test set. The structure of BP-ANN, the learning rate, the lag order, and so on, are factors affecting the performance of BP-ANN (in [5]). The structure of BP-ANN denotes the number of layers, the number of nodes in each layer, and the distribution of the weights. The learning rate denotes the speed of the weight update. The larger the learning rate is, the faster the weight update is, and unfortunately so is the possibility of missing the optimum solution. The lag order denotes the number of period ahead of the being forecasted value compared with the independent variables. For instance, the lag order can be denoted as In = 1, 3, 5, when we utilize xt−1 , xt−3 , and xt−5 to forecast xt .
4
Empirical Analysis
Containerization plays an important role in the drastic growth of international trade. The furious global competition in increasingly more open and volatile markets has made a major impact on the port operations, construction and upgrading of port facilities. Therefore, accurately forecasting container throughput is significant to the port management. This paper
FORECASTING TIME SERIES WITH GP-LSM
125
forecasts Tianjin and Qingdao ports’ container throughput with 3 different forecast methods and compares their performance. 4.1
Data Description and Evaluation Criteria 1300 1100 900 700 500 300 100 2011-7 2011-1 2010-7 2010-1 2009-7 2009-1 2008-7 2008-1 2007-7 2007-1 2006-7 2006-1 2005-7 2005-1 2004-7 2004-1 2003-7 2003-1 2002-7 2002-1 2001-7 2001-1 Tianjin Port
Figure 6
Qingdao Port
The monthly Tianjin and Qingdao ports’ container throughput in the period 2001–2011 (Unit: Thousand TEU)
The Tianjin and Qingdao ports’ container throughput data used in this study are monthly and are obtained from CEIC macroeconomic data base (http://www.ceicdata.com/), covering the period from January 2001 to December 2011 with n = 132 observations in total, as illustrated in Figure 6. This paper takes the monthly data from January 2001 to December 2010 as the in-sample (training period) data set with 120 observations for training and validation purposes, then take the remainder data from January 2011 to December 2011 as the out-ofsample (test period) data set with 12 observations, which is used to evaluate the performance of prediction based on some evaluation criteria. In order to save space, the original data are not listed here, and the detailed data can be obtained from the website or from the authors. The problem about GP-LSM and BP-ANN are solved by using Matlab version 7.0, and that about SARIMA is worked out with EVIEWS version 6.0. For comparison, three main evaluation criteria, the root of mean squared error (RMSE), the mean absolute percentage error (MAPE), and the correct direction forecast rate (CDFR) are used in this paper. Given T pairs of the observed values yi and predicted values yi , RMSE is the measure of the accuracy of yi simulating yi , and is calculated as T 1
RMSE = (yi − yi )2 . (14) T i=1
126
YANG FENGMEI, et al.
And the MAPE is defined as MAPE = 100 ×
T 1 |yi − yi | . T i=1 yi
(15)
However, RMSE and MAPE cannot provide direct suggestions to decision makers. Many decision makers, such as investors, are much more interested in the direction of the change, for which CDFR is proposed (in [5]). CDFR is expressed as T 1
CDFR = 100 × CDFi , T i=1
where CDFi =
⎧ ⎪ ⎨1, (yi − yi−1 )(yi − yi−1 ) > 0, ⎪ ⎩0, (yi − yi−1 )(y − y ) ≤ 0. i i−1
(16)
(17)
Before forecast, original data should be pre-processed. This paper firstly standardizes the original data using Equation (18) yis =
yi − min y , max y − min y
(18)
where max y and min y respectively stand for the maximum and minimum of y, then pre-treats the new generated data with seasonal adjustment and first order difference, finally acquires the final forecast result with inverse transformation as Equation (19) yi = yis ∗ (max y − min y) + min y. 4.2
(19)
Results and Comparison with 3 Different Methods
Figure 7 visually illustrates the forecast results with 3 different methods of Tianjin port’s container throughput, and Figure 8 apparently demonstrates the forecast results with 3 different methods of Qingdao port’s container throughput. Table 1 compares GP-LSM, SARIMA, and BP-ANN performance levels through 3 evaluation criteria of RMSE, MAPE, and CDFR. From Figure 7, Figure 8, and Table 1, we can draw the following conclusions: (a) GPLSM outperforms the other two methods with regard to both RMSE and MAPE, especially for Tianjin Port; (b) the performance of GP-LSM in CDFR equals to that of the others in Tianjin Port forecast and beats that of SARIMA and BP-ANN in Qingdao Port forecast; (c) GP-LSM performs best in capture the abrupt changes in container throughputs of Tianjin Port and Qingdao Port in 2011 February. For the further verification purpose, this paper forecasts with GP-LSM that the total container throughput of Tianjin port in 2012 will reach to 12419–12615 thousand TEU and that the total container throughput of Qingdao port in 2012 will reach to 14170–14404 thousand TEU.
127
FORECASTING TIME SERIES WITH GP-LSM 1050 1000 950 900 850 800 750 700 2011-12
SARIMA
2011-11
2011-10
2011-9
GP-LSM
2011-8
2011-7
2011-6
2011-5
2011-4
2011-3
2011-2
2011-1
True values
BP-ANN
Figure 7 Forecast results with 3 different methods of Tianjin port (Unit: Thousand TEU)
1200 1150 1100 1050 1000 950 2011-12
SARIMA
2011-11
2011-10
2011-9
GP-LSM
2011-8
2011-7
2011-6
2011-5
2011-4
2011-3
2011-2
2011-1
True values
BP-ANN
Figure 8 Forecast results with 3 different methods of Qingdao port (Unit: Thousand TEU)
Table 1 Comparison of the performance with 3 different methods Tianjin Port
Qingdao Port
RMSE
MAPE(%)
CDFR(%)
RMSE
MAPE(%)
CDFR(%)
GP-LSM
16.3
1.40
67
26.2
2.06
83
SARIMA
21.8
1.96
67
34.8
2.53
75
BP-ANN
16.6
3.66
67
33.7
2.75
75
128
5
YANG FENGMEI, et al.
Conclusion
This paper proposes a forecast method: Genetic programming based on least square method (GP-LSM). Several innovations of this method are described as follows: (a) GP-LSM is able to generate the optimal variable coefficients of the corresponding forecast equation, owing to the utilization of LSM; (b) a new reproduction technique dynamically changing the reproduction probability is proposed to enhance the reproduction efficiency of GP-LSM; (c) the restriction that only the non-leaf nodes are selected for crossover and mutation can prevent the bloat phenomenon of trees. The above all make sure that GP-LSM quickly converges to the global optimal solution. This paper verifies the effectiveness of GP-LSM through empirical analysis of forecasting Tianjin and Qingdao ports’ container throughput. We also use GP-LSM to forecast the global top 20 ports container throughput, and the detailed information can be found in Outlook of Global Top 20 Ports Container Throughput in 2010 and Outlook of Global Top 20 Ports Container Throughput in 2011. In this paper, GP-LSM is applied to forecast Tianjin and Qingdao ports’ container throughput in 2012. It is notable that GP-LSM can be applied to not only the container throughput forecast but also other forecast areas, such as the crude oil price forecast and the foreign exchange rate forecast and so forth.
References [1] [2] [3]
[4] [5] [6]
[7] [8]
Lam J P and Veall M R, Bootstrap prediction intervals for single period regression forecasts, International Journal of Forecasting, 2002, 18(1): 125–130. Gray A G and Thomson P J, On a family of finite moving-average trend filters for the ends of series, Journal of Forecasting, 2002, 21(2): 125–149. Findley D F, Wills K C, and Monsell B C, Seasonal adjustment perspectives on “Damping seasonal factors: Shrinkage estimators for the X-12-ARIMA program”, International Journal of Forecasting, 2004, 20(4): 551–556. Hyndman R J, Koehler A B, Ord J K, and Snyder R D, Prediction intervals for exponential smoothing state space models, Journal of Forecasting, 2005, 24(5): 17–37. Huang A Q, Xiao J, and Wang S Y, A combined forecast method integrating contextual knowledge, International Journal of Knowledge and Systems Science, 2011, 2(4): 39–53. Zhang J and Thomas L C, Comparison of linear regression model and survival analysis using single and mixture distribution approaches in modeling LGD, International Journal of Forecasting, 2012, 28(1): 204–215. Huang S J and Huang C L, Control of an inverted pendulum using grey prediction model, IEEE Transactions on Industry Applications, 2000, 36(2): 452–458. Hyndman R J, Koehler A B, Snyder R D, and Grose S, A state space framework for automatic forecasting using exponential smoothing methods, International Journal of Forecasting, 2002, 18(3): 439–454.
FORECASTING TIME SERIES WITH GP-LSM [9] [10] [11]
[12] [13]
[14] [15] [16] [17] [18] [19]
[20] [21]
129
Li G Q, Xu S W, and Li Z M, Short-term price forecasting for agro-products using artificial neural networks, Agriculture and Agricultural Science Procedia, 2010, 1: 278–287. Rua A, A wavelet approach for factor-augmented forecasting, Journal of Forecasting, 2011, 30(7): 666–678. Rubio G, Pomares H, Rojas I, and Herrera L J, A heuristic method for parameter selection in LS-SVM: Application to time series prediction, International Journal of Forecasting, 2011, 27(3): 725–739. Koza J R, Genetic Programming: On the Program of Computer by Natural Selection, MIT Press, Cambridge, 1992. Koza J R, Mydlowec W, Lanza G, Yu J, and Keane M A, Reverse engineering of metabolic pathways from observed data using genetic programming, Pacific Symposium on Biocomputing, 2001, 6: 434–445. Fonlupt C, Solving the ocean color problem using a genetic programming approach, Applied Soft Computing, 2001, 1(1): 63–72. Sugimoto M, Kikuchi S, and Tomita M, Reverse engineering of biochemical equations from timecourse data by means of genetic programming, BioSystems, 2005, 80(2): 155–164. Worzel W P, Yu J J, Almal A A, and Chinnaiyan A M, Applications of genetic programming in cancer research, The International Journal of Biochemistry & Cell Biology, 2009, 41(2): 405–413. Tsai H C, Using weighted genetic programming to program squat wall strengths and tune associated formulas, Engineering Applications of Artificial Intelligence, 2011, 24(3): 526–533. Forouzanfar M, Doustmohammadi A, Hasanzade S, and Shakouri G H, Transport energy demand forecast using multi-level genetic programming, Applied Energy, 2012, 91(1): 496–503. Liu P, Chen S H, Yang H H, Hung C T, and Tsai M R, Application of artificial neural network and SARIMA in Portland cement supply chain to forecast demand in Portland cement supply chain to forecast demand, Proceedings of IEEE Forth International Conference on Natural Computation, 2008. Box G E P and Jenkins G M, Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco, 1976. Hecht-Nielsen R, Theory of the back propagation neural network, Proceedings of International Joint Conference on Neural Networks, Washington, 1989.
