(PDF) Forecasting energy consumption using ensemble ARIMA–ANFIS hybrid algorithm

Electrical Power and Energy Systems 82 (2016) 92–104 Contents lists available at ScienceDirect Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes Forecasting energy consumption using ensemble ARIMA–ANFIS hybrid algorithm Sasan Barak a,c,⇑, S. Saeedeh Sadegh b a Faculty of Economics, Technical University of Ostrava, Ostrava, Czech Republic b Department of Industrial Engineering, Tarbiat Modares University, Tehran, Iran c Department of SAEQM, University of Bergamo, Via dei Caniana 2, 24127 Bergamo, Italy a r t i c l e i n f o a b s t r a c t Article history: Energy consumption is on the rise in developing economies. In order to improve present and future Received 11 May 2015 energy supplies, forecasting energy demands is essential. However, lack of accurate and comprehensive Received in revised form 8 March 2016 data set to predict the future demand is one of big problems in these countries. Therefore, using ensemble Accepted 9 March 2016 hybrid forecasting models that can deal with shortage of data set could be a suitable solution. In this paper, the annual energy consumption in Iran is forecasted using 3 patterns of ARIMA–ANFIS model. In the first pattern, ARIMA (Auto Regressive Integrated Moving Average) model is implemented on 4 Keywords: input features, where its nonlinear residuals are forecasted by 6 different ANFIS (Adaptive Neuro Fuzzy Energy forecasting ARIMA Inference System) structures including grid partitioning, sub clustering, and fuzzy c means clustering ANFIS (each with 2 training algorithms). In the second pattern, the forecasting of ARIMA in addition to 4 input AdaBoost features is assumed as input variables for ANFIS prediction. Therefore, four mentioned inputs beside Ensemble algorithm ARIMA’s output are used in energy prediction with 6 different ANFIS structures. In the third pattern, due to dealing with data insufficiency, the second pattern is applied with AdaBoost (Adaptive Boosting) data diversification model and a novel ensemble methodology is presented. The results indicate that proposed hybrid patterns improve the accuracy of single ARIMA and ANFIS models in forecasting energy consumption, though third pattern, used diversification model, acts better than others and model’s MSE criterion was decreased to 0.026% from 0.058% of second hybrid pattern. Finally, a comprehensive comparison between other hybrid prediction models is done. Ó 2016 Elsevier Ltd. All rights reserved. Introduction tion in the last decade, energy demand management is very important for achieving economic success, environment preserva- Energy is vital important for development of every country tion and suitable planning for existing resources that result in from the social, economic and environmental perspective. It has self-sufficiency and economic development. Therefore, various magnificent effect on industrial and agricultural products, health, techniques have been used for energy demand management to fore- sanitary, population, education and human life quality [1]. cast future energy demands accurately [4]. However, energy fore- As energy is a crucial input to industrial part of country, energy casting is difficult, because it is affected by rapid development of demand increases along the industrial function increase. Rapid economy, technology, government decisions and other factors [5]. changes in industry and economy strongly affect energy consump- As far as energy prediction is concerned, especially in developing tion. Therefore, energy consumption is an important economical countries such as Iran, lack of data is a critical problem in forecast- index that represents economic development of a city or a country ing. Moreover, missing values and lack of a standard and precise sys- [2]. According to the international energy agent report, there should tem for data collection raised other issues in such countries [6]. This be many transformations in amount and type of future energy study proposes a diversified hybrid ARIMA (Auto Regressive Inte- consumption (year 2030). As over the past decade global energy grated Moving Average)–ANFIS (Adaptive Neuro Fuzzy Inference consumption has increased rapidly because of population and eco- System) model to deal with such problems in energy consumption. nomic growth [3,4]. According to wide growth of energy consump- The contribution of the paper is summarized as follows: ⇑ Corresponding author at: Faculty of Economics, Sokolská trˇída 33, Ostrava, Developing a hybrid ARIMA–ANFIS algorithm based on three Czech Republic. Tel.: +420 722459247; fax: +420 597 322 008. different patterns. E-mail addresses:

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(S. Barak). http://dx.doi.org/10.1016/j.ijepes.2016.03.012 0142-0615/Ó 2016 Elsevier Ltd. All rights reserved. S. Barak, S.S. Sadegh / Electrical Power and Energy Systems 82 (2016) 92–104 93 Using diversification method to deal with data insufficiency. and forecasting of energy consumption, with use of small set of Finally, comparing all patterns with different prediction models. (inadequate) data, population and GDP as inputs. They used annual data from Iran and some other countries from 1995 to 2005 and This paper is organized as follows. A comprehensive literature the results showed the superiority of proposed hybrid model com- for energy forecasting models such as ARIMA, fuzzy and ANFIS pared to single models. The application of fuzzy models in energy models as well as Ensemble models is reviewed in Section ‘Literat is reviewed by [19]. The review indicates that fuzzy based models ure review’. In Section ‘The background’ details of ARIMA, ANFIS in energy field provide realistic estimates. and AdaBoost (Adaptive Boosting) models are described and our proposed algorithms with 3 patterns are explained in Section ‘The ANFIS proposed model’. In Section ‘Application and results’, the proposed models have been evaluated using energy consumption data from ANFIS (Adaptive Neuro Fuzzy Inference System) model is one of Iran. Finally, conclusions are given in the last section. the most popular artificial intelligent models that have got advan- tages of both neural network and fuzzy model. The first application of ANFIS in time-series prediction is the Jang’s work [20]. In ANFIS, Literature review the relations between variables are shown by fuzzy If-Then rules. Therefore it can interpret the obtained results, which is not possi- Increasing global energy demand requires intelligent forecast- ble with the structures such as neural network [21]. It is also one of ing algorithms and models. Suganthi and Samuel [4] have surveyed the best models in estimation function among other neuro-fuzzy many different models in the field of energy forecasting and intro- models [22]. duced two types of models including the following: (1) Traditional Ying and Pan [23] applied ANFIS model to forecast annual regio- forecasting models such as time series, regression, econometrics nal electricity load in Taiwan with data of years from 1981 to 2000. models, and ARIMA. (2) Soft computing models such as fuzzy logic, According to MAPE criteria and statistical results, ANFIS model was genetic algorithm, neural network, support vector regression mod- found to perform better than regression, neural network, support els for forecasting national and regional energy demand. vector machines, genetic model and fuzzy hybrid systems. Efendi- Time series models are the simplest models for trend analysis in gil et al. [10] compared neural network and ANFIS model for fore- energy forecasting. Some time series approaches such as tradi- casting demand with incomplete data. Results showed that ANFIS tional statistical models including the following: moving average, could be used in demand forecasting with limited data. Akdemir exponential smoothing and ARIMA are linear forecasting methods and Çetinkaya [24] proposed an ANFIS model to forecast the annual [7]. energy demand in Turkey with use of population, income level, ARIMA model is one of the most popular time series models and peak load and energy demand data for 27 years. In spite of small has been broadly used [4,7]. Pappas et al. [8] proposed ARIMA number of data, good results were obtained. Al-Ghandoor et al. model for forecasting Greek electricity consumption and compared [25] forecasted energy demand in Jordan’s transportation, with the proposed model with three analytical time-series models. the use of two models: ANFIS and quadratic exponential smooth- Results showed that ARIMA model is more efficient than the other ing. Annual data from 1985 to 2009 were used to forecast energy time-series models. demand for years from 2010 to 2030 and results expressed effi- From the other point of view, statistical forecasting methods ciency of ANFIS model in energy demand forecasting. Thus, most usually require normal data, while large data sets are trendy or of the results showed that ANFIS had good results in energy seasonal data pattern are often inadequate or noisy [2,9,10]. ARIMA demand modeling and forecasting. models are linear but real time series rarely has linear structure. Energy demand is forecasted based on economic and non- economic indexes. The nonlinearity of these indexes and energy ARIMA–ANFIS demand have led to a search in the field of artificial intelligence approaches such as neural network and fuzzy models [11]. These Recently, hybrid ANFIS models have been successfully used. methods are used because of high flexibility and power of forecast- Azadeh et al. [26] proposed a hybrid ANFIS model for forecasting ing, estimating and overcoming with noisy data [12]. monthly electricity demand in Iran and yielded good results com- Pao [13] forecasted Taiwan energy consumption by neural net- pared to time series model, genetic algorithm and neural network. works and linear models. Neural network has functioned better Li et al. [27] compared neural network and genetic-ANFIS hybrid than the linear models. model to forecast daily energy demand of a hotel. Results showed But depending on situation, accuracy of ANN methods good performance of hybrid model, though hybrid model was com- decreases because of several reasons. Forecasting accuracy of plicated. Li and Hu [28] proposed an ARIMA–fuzzy system model ANN depends on learning data set and their adequacy. Moreover for time series forecasting. First, a Sugeno fuzzy model was applied ANN methods sometimes get stuck in local minimum, so choosing on input–output data to produce fuzzy rules. Then, ARIMA model proper data set, is too critical in neural network models and these was embedded in answer part of fuzzy rules and obtained good models get good results only when the number of data is high [14]. results. Babu and Reddy [29] proposed a hybrid model of ARIMA Fuzzy models have good results in varying situations with inad- and ANN based on moving-average filter model. Then, using a sim- equate data. Recently, fuzzy logic has been widely used to deal ulated data set and experimental data sets such as sunspot data, with high level of uncertainty issues [15,16]. Accuracy of energy electricity price data, and stock market data, the proposed hybrid forecasting is usually impressed by data uncertainty and interde- ARIMA–ANN model was applied along with individual ARIMA pendency between model’s variables. These relations are elimi- and ANN models and some existing hybrid ARIMA–ANN models. nated by classification of fuzzy model [17]. Mamlook et al. [18] Table 1 shows some features and results of the explained studies: forecasted short term electricity consumption of Jordan by fuzzy As can be seen from Table 1, the performance of hybrid ANFIS model and found that fuzzy model performed much better than model in energy forecasting, is so brilliant especially in lack of data the usual statistical forecasting models. and varying situations and more precise results have been Yet, probabilistic consumption pattern cannot be correctly fore- achieved after hybridization with other models. Also, using ensem- casted just by fuzzy based or time-series models. Azadeh et al. [6] ble model as a new concept has improved the result of energy fore- proposed fuzzy-regression hybrid model to improve estimation casting [40]. 94 S. Barak, S.S. Sadegh / Electrical Power and Energy Systems 82 (2016) 92–104 Table 1 Review of hybrid energy forecasting implementations. Number Study Study field Energy market Forecasting method Forecasting scope Error percent 1 [8] Electricity Greek ARIMA Daily Best model: 1.4% 2 [13] Electricity Taiwan Neural network Monthly 4.02% Time series 8.88% 3 [30] Electricity Turkey Neural network Annual 1.15% 4 [31] Electricity Turkey Fuzzy logic Annual – 5 [32] Electricity Taiwan Weighted Neuro fuzzy Monthly 6.43% 6 [14] Electricity A small region Neuro fuzzy Annual 17.62% 7 [33] Electricity Australia ARIMA, Neural network, 30 min ARIMA: 4.23% Neuro fuzzy Neural network: 3.23% Neuro fuzzy: 0.92% 8 [10] All kind of demand 3 Company in Istanbul Neuro fuzzy, Neural Monthly Neuro fuzzy: 4.88% network 7.05% 2.41% 10 [25] Transportation energy Jordon ANFIS Annual 0.155% 11 [6] Gas consumption 4 Middle East countries ANFIS–SFA hybrid Annual Bahrain: 1.8% Saudi Arabia: 1.4% Syria: 7.5% UAE: 1.6% 12 [12] Electricity Iran ARIMA-fuzzy regression Monthly 0.82% 13 [34] Electricity market PJM interconnected Hybrid: Mid term 0.144 clearing price (MCP) electric market SVM–ARMAX 14 [35] Wind energy Iran Hybrid ant colony (ACO) Short term (daily) MAPE: 3.53% and particle swarm (PSO) 15 [36] Wind power Portugal ANFIS–EPSO Short term MAPE: 3.75% 16 [37] Electricity Turkey Gray model Annual MAPE: 3.28% 17 [38] Energy production and China Gray model–Markov chain Annual 0.69% consumption 0.93% 18 [39] Gas consumption Iran ELFIS–ANN–ANFIS Hourly ELFIS NMSE: 0.26% 19 [40] Heating energy Norwegian University of Ensemble of neural networks Daily MAPE: 5.25% consumption Science and Technology building Ensemble models AdaBoost in bankruptcy forecasting and achieved remarkable results. Taking a step forward, we develop a novel combination Recently, studies in machine learning have shown the predic- of AdaBoost methodology with hybrid ARIMA–ANFIS model to tion with a series or ensemble of models is better than a single improve the forecasting result of energy consumption prediction. model and practice of one model improves by hits of other models. An ensemble methodology uses advantages of some predictive models to achieve better results. Ensemble method has two kinds The background of learning approach: learning without interaction between the learning agents (ensemble learning) and learning with interaction ARIMA model during the learning step (co-learning) [41]. In another point of view, ensemble models are classified based An ARIMA model [ARMA (p, q)] for x time series that includes n on data diversification. Some algorithms of this classification are as instances is predefined as [50]: follows: K-fold cross validation, Bagging, Boosting, and Random p q forests. X X v k v Tk xk ¼ A i xk i þ Bj vk j þ vk E ¼R ð1Þ In K-fold cross validation method, training and validation data i¼1 j¼1 sets are divided to K equal parts. One part is considered as valida- tion data and K 1 and other parts are considered as training data. where the m-dimensional vector v k is uncorrelated random noise, This is done in K times and each time one new part is considered as not necessarily Gaussian, with zero-mean and covariance matrix validation data and others as training set [42]. Boosting and Bag- R, h = (p, q) is the order of the predictor and A1, . . ., Ap and B1, . . ., ging methods combine weak models and provide better prediction Bq are the m m coefficient matrices of the multivariate (MV) models. Bagging is appropriate for improving tree algorithms while ARMA model [50]. ARIMA model has three components: (1) auto Boosting can be used for many algorithms such as additive models regressive (AR), (2) integrated average (IA) and (3) moving average with high-dimensional predictors [43]. (MA). The structure of ARIMA model consists of four steps: (1) Adaptive Boosting (AdaBoost) proposed by Freund and Schapire model identification, (2) parameter estimation, (3) model recogni- [44] is an effective ensemble method that enjoys weighted average tion and (4) model verification and forecasting [28]. method for combination of learning algorithms. In ARIMA (p, d, q), p expresses the number of autoregressive Recently, AdaBoost has been successfully used in many fields of terms, q is the number of lagged forecast errors and d is the num- study, and few of them are as follows: cost-sensitive classification, ber of non-seasonal differences. Random errors (v k ) are assumed to semi supervised learning, tracking and network intrusion detection be independent, and to have identical distribution with a constant [45,46]. Assaad et al. [47] predicted future values of time series variance. Based on Box and Jenkins method, when the numbers of using neural networks as base learners and AdaBoost ensemble series are less than 240, maximum number of lag is equal to the method. Alfaro et al. [48] compared the results of AdaBoost and number of observations divided by four [51]. To apply ARIMA neural network techniques in the field of forecasting by about model, autocorrelation (ACF) and partial autocorrelation (PACF) thirty percent decrease in generalization error, and deduced the functions should be determined. Order of the AR and MA parame- priority of AdaBoost method results. Heo and Yang [49] used ters can be determined using the partial autocorrelation graph and S. Barak, S.S. Sadegh / Electrical Power and Energy Systems 82 (2016) 92–104 95 the autocorrelation graph of data. Descriptions of steps for creating structure with two inputs and two labels for each input shown in ARIMA model are as follows: Fig. 2 are as follows according to 3 type rules: Model identification: As stationary is essential in ARIMA fore- casting model, data should be often stationary. Differencing is usu- W i ¼ lAi ðxÞ lBi ðxÞ i ¼ 1; 2 ð2Þ ally applied to data to remove trend of data and stabilize the variance [28]. By this way, d parameter is determined. According wi to PACF and ACF figures, time series and probabilistic models’ sta- w i ¼ ; i ¼ 1; 2 ð3Þ w1 þ w2 tionary can be determined. Parameter estimation: One of probabilistic models is made and f 1 ¼ p1 x þ q1 y þ r 1 z w1 f 1 þ w2 f 2 model’s parameters are estimated, in order to minimize Akaike’s )f ¼ ¼w 1f 1 þ w 2f 2 ð4Þ Information Criterion (AICC) [52] and Schwarz’s Bayesian Informa- f 2 ¼ p2 x þ q2 y þ r 2 z w1 þ w2 tion Criterion (BIC) [12,28,50]. Diagnostic checking: In this step, the model’s accuracy and the where x and y are inputs to node i, Ai and Bi are linguistic labels for model’s error stationary are checked [28]. The best model is chosen inputs, and wi is the output of layer 3 and fpi ; qi ; r i g is the parameter according to some forecasting error criteria such as root mean set. square error (RMSE) and mean absolute error (MAE). In Fig. 1, There are 3 kinds of function for fuzzy system creation: Genfis1, the process of choosing the best ARIMA model is shown: Genfis2, and Genfis3. Genfis1 makes fuzzy inference system structure by grid parti- tioning. It makes FIS structure based on constant numbers of mem- ANFIS bership functions and using clustered information models the data behavior in the best way and with the least number of functions. It Takagi–Sugeni–Kang is a fuzzy system with crisp functions that clusters its rules based on fuzzy quality of data sets [53]. are suitable for complex problems. TSK systems are usually used in Genfis2 makes the structure of FIS by subtractive clustering. the shape of a neuro fuzzy system that is called ANFIS. ANFIS is a This method makes a model of data by clustering and a cluster fuzzy inference system that can be trained by a set of input and radius should be determined for it. Radius specifies the confine- output data. ment (limitation) of cluster impact [1,53]. The ANFIS structure shown in Fig. 2 is a five layer network. The Genfis3 makes the FIS structure by fuzzy clustering with use of first layer executes a fuzzification process, the second layer exe- C-mean (FCM). FCM begins to work with an initial guess for cluster cutes the fuzzy AND of the antecedent part of the fuzzy rules, the center. In addition, FCM function assigns a membership degree to third layer normalizes the membership functions, the fourth layer each data point and guides data centers to their correct place in executes the conclusion part of the fuzzy rules, and the last layer data set via updating centers and membership degrees of each data computes the output of the fuzzy system by summing up the out- point repeatedly. This method is done with minimizing a goal func- puts of the four layers. The feed forward equations of the ANFIS tion that represents the distance of each data point to data center Choosing Creating pattern probablistic according to Determining AIC &BIC Drawing ACF likely patterns & PACF MA&AR Checking data figures quantities stationary and determining d parameter Fig. 1. Pattern of choosing the best ARIMA model. Fig. 2. ANFIS structure of type 3, with 2 inputs and one output. 96 S. Barak, S.S. Sadegh / Electrical Power and Energy Systems 82 (2016) 92–104 that has been weighted by membership degree of the data point ANFIS model and finally forecasting is attained from sum of ARIMA (Eq. (5)). and ANFIS models. In the second pattern, the forecasting of ARIMA is used as an N X C Um C j k2 ; input feature to ANFIS model. In other words, the forecasting of X Jm ¼ ij kX i 16m61 ð5Þ i¼1 j¼1 ARIMA in addition to other input features is used in ANFIS predic- tion. Therefore, ARIMA’s output as one of ANFIS inputs, can improve where m is a real number greater than 1 and each of Uij, Xi and Cj ANFIS model’s performance. In the third pattern, because of the lack shows the degree of membership of Xi in the j-th cluster, the i-th of data, the second pattern is applied with AdaBoost model. p-dimensional data and the p-dimensional center of the cluster Two training algorithms including back propagation (BP) and respectively, and ||⁄|| is any norm that shows the similarity between least square gradient descent back propagation (Hybrid BP) are each measured data and the center. With iterative optimization of implemented in ANFIS model to train parameters of membership the above objective function, fuzzy partitioning is done, by updating functions. Hybrid algorithms for TSK-type of fuzzy logic systems membership Uij and cluster centers Cj as follows: are provided in various studies [54] which are a combination of PN m least square and gradient descent back propagation. 1 i¼1 U ij X i Since ANFIS prediction in this study is examined with 3 kinds of U ij ¼ m2 1 ; C j ¼ PN m ð6Þ i¼1 U ij Genfis functions (Grid partitioning, Sub clustering, and FCM) with PC kX i X j k k¼1 kX i C j k 2 training algorithms, we calculate 6 kinds of ANFIS structures. n ðKþ1Þ o ðKÞ Modeling steps are described as follows: when maxi;j U ij U ij e is satisfied, iteration will stop, where e is a number between 0 and 1, and k is the step’s number A. By choosing proper quantity for alpha and beta in Eq. (9), all of iteration [1,53]. data are normalized in ½a; a þ bŠ interval. x xmin AdaBoost xnorm ¼ a þ b ð9Þ xmax xmin Ensemble methods are broadly used for classification and B. According to pattern and situation of model’s residuals, and regression and their ability has been shown in a wide range of stationarity or non-stationarity, ARIMA parameters includ- tasks, theoretically and empirically [47]. ing: AR, MA and d are identified and ARIMA model is applied. Boosting uses a series of classifiers to learn the model. In each Linear parts of data are forecasted and model’s error is iteration one classifier (mi ) is learned and new weights are attained by subtracting actual quantity of forecasted assigned to data in order to next classifier ‘‘pay more attention” quantity: to tuples that has been classified wrong. Weight of each classifier’s vote is counted according to its accuracy. Finally, votes of classi- et ¼ yt lt ð10Þ fiers are combined to make the best classifier (m ). In AdaBoost as the most popular Boosting algorithm, a series of where yt is actual quantity and lt is linear part. models are combined and data set is resampled in each model. In this method data are weighted according to their difficulty to be In the first proposed pattern, steps are as follows (see the right learned [45]. side of Fig. 3): 1.1. Since residuals of ARIMA have nonlinear structures, ANFIS The proposed model model is used to forecast ARIMA residuals. So residuals are divided into train and test sets, and then 6 different ANFIS Both ARIMA and ANFIS models have good performance in linear models are trained and tested. Error reduction is used as cri- and nonlinear structures but none of them is comprehensive to be teria for choosing proper model. The formula for this model able to forecast various time series structures. Studies show that is shown in Eq. (11): using dissimilar models improves time series forecasting where data pattern is varying and unstable [7,19]. The use of ARIMA p X q X and soft computing techniques improves precision of energy y k ¼ l k þ nk ¼ ai lk i þ bj v k j þ ðw1 f 1 þ w2 f 2 Þ=ðw1 þ w2 Þ ð11Þ i¼1 j¼1 demand forecasting [4]. Zhang [7] proposed a capable hybrid model that consists of two Eq. (11) considers both linear and nonlinear forecasting parts steps: Step 1 is applying linear model and step 2 is applying non- in forecasting and illustrates final forecasting values by sum linear model using linear model’s residuals. Finally, both models’ of ARIMA and ANFIS results. Model’s error is computed by forecasting results are summed. mean square error (MSE) criteria: yt ¼ lt þ nt ð7Þ n 1X c 2 et ¼ yt lt ð8Þ MSE ¼ q qm ð12Þ n i¼1 i i According to (7), it is supposed that data structure (yt ) contains two parts: Linear part (lt ) and nonlinear part (nt ). First, data are where qci is forecasted value, qm i is real value, and n is the forecasted by linear model and have been checked to see whether number of data. MSE is used as criteria for choosing proper residuals ðet Þ have nonlinear pattern. Then, the residuals have been model. forecasted by a nonlinear model. The model’s final forecasting is 1.2. After obtaining the results of 6 ANFIS models, outputs are attained from sum of linear and nonlinear model results. In Eq. post processed, returned to initial scale and presented as (8), et is nonlinear residual that is yielded by subtracting actual hybrid model’s results. quantity to linear forecasted quantity. Extending the previous researches, in our paper, 3 patterns for In the second pattern, steps ‘‘a” and ‘‘b” are similar to the first time series prediction are presented. In the first pattern, data are pattern model (see the left side of Fig. 3). Steps 2.1–2.3 are as forecasted by ARIMA model, then its residuals are forecasted with follows: S. Barak, S.S. Sadegh / Electrical Power and Energy Systems 82 (2016) 92–104 97 A.Data collection and preprocessing B. Data stationary examining and ARIMA implementation after recognizing AR, MA and d parameters AdaBoost method Using ARIMA forecasts as Dividing residuals to train and implementation one of ANFIS model inputs test and implementation of various ANFIS model Implementation various Paern 1 Choosing the best ANFIS Paern 3 Paern 2 Choosing the best ANFIS ANFIS model and choosing model model the best model Achieving final results of Achieving final results of Achieving final results of hybrid pattern 3 hybrid pattern 2 hybrid pattern 1 C. Choosing the best model Fig. 3. Proposed hybrid ARIMA–ANFIS model. 2.1. The ARIMA model’s prediction is used as one of ANFIS The mathematical model of presented AdaBoost is as follows: inputs. Therefore, linear forecasting results are added to other ANFIS inputs and energy consumption is used as mod- Input: Initial training set composed of n examples, denoted as el’s output. For example, where ARIMA output ðlt Þ and the sn ¼ fðx1 ; y1 Þ; ðx2 ; y2 Þ; . . . ; ðxn ; yn Þg other inputs (m) are considered as ANFIS inputs the model’s Initialize: wi1 = 1/n, i.e. formula can be expressed as follows: w1 ¼ w11 ; w21 ; . . . ; wn1 ¼ f1=n; 1=n; . . . ; 1=ng f ¼ ðw1 f 1 þw2 f 2 Þ=ðw1 þw2 Þ for t = 1, 2, . . ., T 1. Take Rt samples randomly from Sn using distribution wt la1 ðlt Þ lb1 ðlt Þðp1 lt þq1 mþr1 Þþ la2 ðlt Þ lb2 ðlt Þðp2 lt þq2 mþr2 Þ ¼ 2. Build a classifier f t using Rt as the training set la1 ðlt Þ lb1 ðlt Þþ la2 ðlt Þ lb2 ðlt Þ ð13Þ 3. Compute: Et = MSE of f t and at = 0.5 ln 1 EtEt 4. Update the weight: witþ1 = normalize (wit ⁄ exp ( at )) where lt is the output of ARIMA model. PT ft 2.2. In this step, data are divided into train and test sets, and then Output: The ensemble prediction: F = t¼1 T and whole PT data are analyzed with 6 structures of ANFIS models. The Et Error = t¼1 T best hybrid model is specified with respect to test criteria and used for forecasting energy consumption. 2.3. After obtaining ANFIS results, outputs are post processed, where sn ¼ fðx1 ; y1 Þ; ðx2 ; y2 Þ; . . . ; ðxn ; yn Þg represents the set of train- returned to initial scale and presented as hybrid model’s results. ing samples, T is the number of iteration, wt ¼ fw1t ; w2t ; . . . ; wnt g shows weight distribution over sample set that is 1/n in the first As far as energy prediction is concerned, especially in develop- iteration and it will be updated in each iteration. Weights for hard ing countries such as Iran, lack of data is a big problem in forecast- samples which classified wrong classifier ðf t Þ will increase in the ing. Therefore, in the third pattern, for improving testing accuracy next iteration. Et represents MSE of f t . of model and encountering with lack of data AdaBoost method is applied to increase data variation. In the last pattern of this study, C. Finally, results of 3 patterns have been compared and the AdaBoost method was implemented based on Fig. 4. best pattern has been chosen. 98 S. Barak, S.S. Sadegh / Electrical Power and Energy Systems 82 (2016) 92–104 Initial training data set (Sn) Weight distribution W1 Weight distribution W2 . . . Weight distribution Wn Training dataset 1 Training dataset 2 Training dataset t Selected from Sn Selected from Sn Selected from Sn According to W1 According to W2 According to Wt First iteration of Second iteration of T’th iteration of weak learner weak learner weak learner AdaBoost First iteration of weak learner Second iteration of Testing dataset combination Prediction result weak learner ... T’th iteration of weak learner Fig. 4. Framework of AdaBoost method for predicting. Fig. 5. Auto correlation and partial auto correlation in Iran’s data before lagging. Application and results the world [55]. On the other hand, in developing countries such as Iran, lack of data is a problem in forecasting [6]. Because of this, it Data set and experiments seems essential to achieve a proper and accurate model for fore- casting future energy consumption in Iran. Therefore, the proposed According to estimations, industrial energy consumption in ensemble based ARIMA–ANFIS hybrid models were used in developing countries is about 45–50% of total commercial energy forecasting. consumption [4]. Iran requires broadly investment in energy field, It is clear from the related literature [56–58] that usually 4 having respectively second and fifth rate in gas and oil reservoirs in independent variables, including the following: population, gross S. Barak, S.S. Sadegh / Electrical Power and Energy Systems 82 (2016) 92–104 99 Table 2 Table 3 Akaike and Schwarz (BIC) criterions comparison for [1, 1, 2] and [2, 1, 1] ARIMA Error criteria results of [1, 1, 2] ARIMA model. models. Model [1, 1, 2] Model [1, 1, 2] [2, 1, 1] RMSE 15.7428 AIC 8.497 8.5263 MAE 12.5907 BIC 8.622 8.6529 domestic production, import and export, are used as inputs to the models in energy consumption forecasting. It seems that these four factors have the most impact on energy consumption in every country. Thus, in this study, annual energy consumption, popula- tion, GDP, export and import data from 1967 to 2012, were used for modeling energy consumption. Data were collected from statis- tics center of Iran. Initial data were normalized according to Eq. (9) to become stationary. In this study, alpha and beta values are zero and one, respectively. In the next section, various models intro- duced in previous sections are implemented and results are analyzed. ARIMA Fig. 7. ARIMA model forecasting residuals. In this section, ARIMA model is identified for forecasting Iran energy consumption and details of ARIMA model implementation are expressed. Results of ARIMA model are achieved using Eviews Table 4 software. MSE criterion results of test and train data in hybrid pattern 1. Identification step: Ascending pattern is seen in correlation fig- Grid partitioning Sub clustering FCM ure of data that shows non-stationary of data (Fig. 5). MSE check data Therefore with making one difference on data, 1 was assigned BP 0.461144459 0.129120718 0.2403921 to d parameter. Hybrid BP 0.37272772 0.111771157 0.1356161 Pattern estimation step: According to data correlation with 1 lag, MSE train data 1 and 2 are considered as estimated values for AR parameter and 1, BP 0.27423836 0.00157647 0.0261099 2, 3 are calculated as estimated values for MA parameter. So vari- Hybrid BP 5.66e 8 7.87e 9 9.97e 8 ous models are compared and two best models are [1, 1, 2] and [2, 1, 1]. Model recognition: Model‘s equations are written considering Model verification: Residuals’ figure is examined and if it is sta- determined characteristics and models are compared according tionary, model recognition is correct. Results show the auto and to the Akaike [52] and Schwarz (BIC) criterions. AIC and BIC crite- partial autocorrelations for residuals of [1, 1, 2] and [2, 1, 1] models rions for models are compared in Table 2, and [1, 1, 2] model with are stationary; however, [1, 1, 2] model’s residuals have less and AR = 1 and MA = 2 is chosen as the best ARIMA model because of smoother correlation and auto correlation and it has better perfor- low value for AIC and BIC criterions. mance (see Fig. 6). Fig. 6. Auto correlation and partial auto correlation for [1, 1, 2] model residuals. 100 S. Barak, S.S. Sadegh / Electrical Power and Energy Systems 82 (2016) 92–104 Fig. 8. Outputs and errors of hybrid pattern 1, sub clustering type of ANFIS and hybrid BP training algorithm. Table 5 Train and test MSE criterion results of hybrid pattern 2. Grid partitioning Sub clustering FCM MSE check data BP 0.670779481 0.111324451 5.76e 4 Hybrid BP 0.575888121 0.125924396 0.02944284 MSE train data BP 0.088080372 1.95e 7 1.43e 4 Hybrid BP 3.84e 11 4.4e 12 6.90e 10 Fig. 9. Outputs and errors of hybrid pattern 2, FCM type of ANFIS and BP training algorithm. Table 6 Fuzzy rules of ANFIS model of hybrid model 2. 1 If (in1 is in1cluster1) and (in2 is in2cluster1) and (in3 is in3cluster1) and (in4 is in4cluster1) and (in5 is in5cluster1) then (out1 is out1cluster1) (1) 2 If (in1 is in1cluster2) and (in2 is in2cluster2) and (in3 is in3cluster2) and (in4 is in4cluster2) and (in5 is in5cluster2) then (out1 is out1cluster2) (1) 3 If (in1 is in1cluster3) and (in2 is in2cluster3) and (in3 is in3cluster3) and (in4 is in4cluster3) and (in5 is in5cluster3) then (out1 is out1cluster3) (1) 4 If (in1 is in1cluster4) and (in2 is in2cluster4) and (in3 is in3cluster4) and (in4 is in4cluster4) and (in5 is in5cluster4) then (out1 is out1cluster4) (1) 5 If (in1 is in1cluster5) and (in2 is in2cluster5) and (in3 is in3cluster5) and (in4 is in4cluster5) and (in5 is in5cluster5) then (out1 is out1cluster5) (1) 6 If (in1 is in1cluster6) and (in2 is in2cluster6) and (in3 is in3cluster6) and (in4 is in4cluster6) and (in5 is in5cluster6) then (out1 is out1cluster6) (1) 7 If (in1 is in1cluster7) and (in2 is in2cluster7) and (in3 is in3cluster7) and (in4 is in4cluster7) and (in5 is in5cluster7) then (out1 is out1cluster7) (1) 8 If (in1 is in1cluster8) and (in2 is in2cluster8) and (in3 is in3cluster8) and (in4 is in4cluster8) and (in5 is in5cluster8) then (out1 is out1cluster8) (1) S. Barak, S.S. Sadegh / Electrical Power and Energy Systems 82 (2016) 92–104 101 Fig. 10. MSE criterion figure of test data in hybrid pattern 3, grid partitioning ANFIS and BP optimum algorithm. Fig. 13. MSE criterion figure of test data in hybrid pattern 3, FCM ANFIS and BP optimum algorithm. Fig. 11. MSE criterion figure of test data in hybrid pattern 3, sub clustering ANFIS and BP optimum algorithm. Fig. 14. MSE criterion figure of test data in hybrid pattern 3, FCM ANFIS and hybrid BP optimum algorithm. Table 7 Test and train MSE criterion of hybrid model 3. Grid partitioning Sub clustering FCM MSE check data BP 0.193931541 1.19e 3 2.63e 4 Hybrid BP 0.080798813 4.66e 4 3.29e 4 MSE train data BP 0.189186074 4.45e 7 1.37e 4 Hybrid BP 1.74e 11 3.46e 12 4.76e 7 Table 8 Results of the best practice of hybrid patterns 1, 2 and 3. MSE criterion Hybrid pattern 1 Hybrid pattern 2 Hybrid pattern 3 Test data 0.111771157 5.76e 4 2.63e 4 Fig. 12. MSE criterion figure of test data in hybrid pattern 3, grid partitioning ANFIS Train data 7.87e 9 1.43e 4 1.37e 4 and hybrid BP optimum algorithm. 102 S. Barak, S.S. Sadegh / Electrical Power and Energy Systems 82 (2016) 92–104 Table 9 Comparison results. MSE Single Single ANN Zhang Khashei and Bijari Babu and Reddy Hybrid pattern Hybrid pattern Hybrid pattern criterion ARIMA ANFIS [7] [61] [29] 1 2 3 Test data 3.97 0.121771 0.155 0.1431 0.0833 0.0266 0.1117712 5.76e 4 2.63e 4 Train data – 7.97e 09 0.176 0.041 0.009 0.0073 7.87e 9 1.43e 4 1.37e 4 Forecasting step: [1, 1, 2] model is chosen for forecasting because be seen from Fig. 8, learning errors are concurrent around zero of its low RMSE and MAE criterions (see Table 3). and close to normal graph. Testing errors are also close to zero. At second pattern after ARIMA model implementation, ARIMA ARIMA–ANFIS patterns output that expresses linear forecasting part of time series is used as one of the ANFIS inputs (step 2.1). Because ARIMA is very effec- In this section, three different ARIMA–ANFIS hybrid patterns are tive for forecasting linear part of data, using ARIMA results in employed according to the mentioned methods. ANFIS improves time series’ total forecasting pattern. Thus ANFIS In the first pattern, after forecasting energy consumption with model inputs are as follows: population, GDP, import and export ARIMA (step B), its errors which are differences between actual and the other input which is ARIMA forecasting result. With exam- and forecasted consumptions are calculated and forecasted using ining different data divisions, using 80% of data for training and population, GDP, import, and export as inputs. Errors have nonlin- 20% for testing obtained the best results in this sample (step 2.2). ear pattern as can be seen in Fig. 7. Then, combination of ARIMA In Table 5, MSE criterion results of various ANFIS structure for sec- model and proper nonlinear ANFIS model is implemented. ond hybrid pattern (hybrid pattern 2) are calculated. Data were divided into train and test sets. 70% of data are used Based on Table 5, FCM ANFIS structure with BP training algo- as train and 30% as test set and MSE criterion is used to examine rithm has the least test error. As shown in Fig. 9, FCM type of ANFIS model’s efficiency. Six different ANFIS structure models have been with BP has training errors around zero and close to normal graph. made by combining 2 optimization algorithms (BP and hybrid BP) Testing errors are also close to zero that is another reason for effi- and 3 ANFIS types, as mentioned in Section ‘The proposed model’. ciency of the model in forecasting energy consumption. Table 4 contains MSE results for various ANFIS models in test Fuzzy rules that are obtained from ANFIS model of hybrid pat- period. tern 2 are shown in Table 6. Here, in1 to in5 are the model inputs It is clear from Table 4 that sub clustering ANFIS structure with which are (1) the ARIMA output, (2) population, (3) GDP, (4) import, hybrid BP optimization method has the least test error. Train and and (5) export. Out1 is energy consumption that is output of model. test result of the best hybrid pattern 1 is shown in Fig. 8. As can As can be seen, the number of rules is decreased regarding hybrid Fig. 15. ANFIS optimization using GA. S. Barak, S.S. Sadegh / Electrical Power and Energy Systems 82 (2016) 92–104 103 Table 10 Results of optimized ANFIS using GA and PSO. Test result 1 2 3 4 5 Average MSE ANFIS PSO 0.00021 6.02e 05 0.0006893 0.00035 0.00044 0.00035 RMSE ANFIS PSO 0.01447 0.00776 0.0262536 0.01876 0.02105 0.01766 MSE ANFIS GA 0.00053 0.00019 0.0002733 0.00025 0.00025 0.0003 RMSE ANFIS GA 0.02308 0.01379 0.0165311 0.01577 0.0157 0.01697 pattern which shows stationarity of the model and less over train- separately, nevertheless using ensemble methods in forecasting ing and over fitting. According to [59], the dense structure of mod- improves performance of model. According to results, third hybrid els brings about a low level of over fitting and causes a robust pattern that uses AdaBoost method with Genfis3 ANFIS structure prediction. and back propagation training algorithm has better results and At the third hybrid pattern, AdaBoost method is mixed with model’s MSE criterion was decreased to 0.026% from 0.058% of sec- hybrid pattern 2, and proper results were obtained. Results of 10 ond hybrid pattern. Therefore, this model can successfully be used iterations are shown in Figs. 10–14. for energy consumption in Iran. As it is shown in Tables 7 and 8, hybrid pattern 3 improved MSE What makes our study differs from other is using 2 novel pat- criterion and had the best performance between 3 kinds of pro- terns (pattern 2 and 3) in time series prediction. Because of the fact posed hybrid pattern. So, it can be used as a proper model for that lack of data in forecasting field is one of the most critical energy consumption in Iran. The proposed hybrid patterns are issues, the third pattern can be widely used in time series predic- compared with ANN, single ARIMA, ANFIS, and 3 best known liter- tion models while the data are inadequate. ature studies in Table 9. It is inferred from Table 8 that ANFIS has In future studies, new diversification methods as well as using dominant results regarding ANN algorithms, so much as, single other prediction model such as Support Vector Machines can be ANFIS model has better performance than Zhang [7] hybrid model. used to improve results. Also, finding the best values of AdaBoost In the other cases, the hybrid models perform better than the sin- method parameters, with a powerful method such as genetic algo- gle models. Among the hybrid models, the results illustrate that rithm can make remarkable results. Moreover, each of inputs can the hybrid pattern 1 is over-trained. Therefore, the training results be forecasted by ARIMA model and results used as the input of have the best result among all methods while its testing results get hybrid models. at least results among all hybrid models and just get better results than single models. We also compared the results with [60] and Acknowledgments [61] works; however, the third hybrid pattern has the best accu- racy in forecasting energy consumption between all methods. The research was supported through the Czech Science Founda- Until this part of paper, fuzzy inference systems (FIS) and adap- tion (GACR) under project 15-23699S and through SP2016/11, a tive neuro-fuzzy inference systems (ANFIS) have been designed SGS research project of VSB-TU Ostrava, and furthermore by the using classic views such as gradient descend and back propagation. European Social Fund in the framework of CZ.1.07/2.3.00/20.0296. One of improvement and optimization methods in proposed model is optimization of model parameters in learning step. So using Meta-heuristic algorithms in Takagi–Sugeno fuzzy systems may References improve parameter selection in learning steps and finally improve [1] Abbasimehr H, Setak M, Tarokh M. A neuro-fuzzy classifier for customer churn the results. prediction. 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