lable at ScienceDirect

Energy 36 (2011) 6981e6992

Contents lists avai

Energy

journal homepage: www.elsevier .com/locate/energy

A hybrid fuzzy mathematical programming-design of experiment frameworkfor improvement of energy consumption estimation with small data setsand uncertainty: The cases of USA, Canada, Singapore, Pakistan and Iran

A. Azadeh a,b,*, M. Saberi c,d, S.M. Asadzadeh a,b, M. Khakestani a,b

aDepartment of Industrial Engineering, Center of Excellence for Intelligent Based Experimental Mechanics, College of Engineering, University of Tehran, P.O. Box 11365-4563, IranbDepartment of Engineering Optimization Research, College of Engineering, University of Tehran, P.O. Box 11365-4563, IrancDepartment of Industrial Engineering, University of Tafresh, Irand Institute for Digital Ecosystems & Business Intelligence, Curtin University of Technology, Perth, Australia

a r t i c l e i n f o

Article history:Received 6 January 2011Received in revised form6 June 2011Accepted 10 July 2011Available online 5 November 2011

Keywords:Hybrid frameworkFuzzy regressionSmall data setsUncertaintyEnergy consumptionDesign of experiment

* Corresponding author. Department of Industriallence for Intelligent Based Experimental MechanUniversity of Tehran, P.O. Box 11365-4563, Tehran 1482084164; fax: þ98 21 8208 4162.

E-mail addresses: ali@azadeh.com, aazadeh@ut.ac

0360-5442/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.energy.2011.07.016

a b s t r a c t

Utilization of small data sets for energy consumption forecasting is a major problem because it couldcreate large noise. This study presents a hybrid framework for improvement of energy consumptionestimationwith small data sets. The framework is based on fuzzy regression, conventional regression anddesign of experiment (DOE). The hybrid framework uses analysis of variance (ANOVA) and minimumabsolute percentage error (MAPE) to select between fuzzy and conventional regressions. The significanceof the proposed framework is three fold. First, it is flexible and identifies the best model based on theresults of ANOVA and MAPE. Second, the framework may identify conventional regression as the bestmodel for future energy consumption forecasting because of its dynamic structure, whereas in the case ofuncertainty and ambiguity, previous studies assume that fuzzy regression provides better solutions andestimation. Third, it is ideal candidate for short data sets. To show the applicability of the hybridframework, the data for energy consumption in Canada, United States, Singapore, Pakistan and Iran from1995 to 2005 are considered and tested. This is the first study which introduces a hybrid fuzzyregression-design of experiment for improvement of energy consumption estimation and forecastingwith relatively small data sets.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Data collection is a major issue in several countries includingdeveloping countries such as Iran and Pakistan due to lack of data,shortage of data, missing values and lack of a robust and standarddata collection system. Furthermore, small data sets are not onlythe subsequent of availability to the data but more the ability ofreliance on the available data. For example, Iran has experienced an8-year long War against Iraq and the economic and societalconditions today are extremely different with the ones during warperiod. Consequently, one cannot rely on the war-period data toconstruct forecasting models which relate energy consumption toeconomic and societal variables e.g. GDP and population. It meansthat the access to reliable data is limited here and it makes the

Engineering, Center of Excel-ics, College of Engineering,178-43111, Iran. Tel.: þ98 21

.ir (A. Azadeh).

All rights reserved.

available data set small. Economic recession, energy crisis andpolitical changes also are other important examples that cast doubton the justification of using all available data for forecastingpurposes. Hence, available data are limited to a small set.

In several forecasting studies, the focus is on the development ofmore sophisticatedmodels for energy forecasting and availability ofdata is taken as granted. The author in [1] has used 180 rows of datain electricity and oil consumption to show the results of hiscombined linear-ANN model for forecasting in variance-instableenvironment. Zhang et al. [2] forecasted china transport energydemand for 2010, 2015 and 2020 based on partial least-squareregressionwith thedata from1990 to2006. The real annual values ofTurkey hydroelectric generation for 1970e2006 are used to predictit in the years 2007e2012 [3]. Singular spectral analysis method isproposed for short-term load forecasting [4]. It uses 844 dailyobservations of electricity consumption to estimate its parameters.

However, in some real world cases, the available and reliabledata set is small in size. More formally, the small data set can bedefined as follows: small data set refers to a set of available andreliable data set which is not large enough to be used for training/

A. Azadeh et al. / Energy 36 (2011) 6981e69926982

estimation of artificial intelligence/econometric models. In otherword, with regard to the number of unknown parameters in themodel, the number of available and reliable data is small inproportion to the data set which would let a robust and uniqueestimation of model parameters. Indeed, here the focus is on data-driven forecasting approaches such as ANN, neuro-fuzzy, and linearor nonlinear conventional regressions.

Regression analysis is one of the most used statistical tools toexplain the variation of a dependent variable Y in terms of thevariation of explanatory variables X as: Y¼ f(X) where f(X) forregression analysis is not required to be linear. It refers to a set ofmethods by which estimates are made for the model parametersfrom the knowledge of the values of a given inputeoutput data set.The goal of the regression analysis is:

(a) To find an appropriate mathematical model, and(b) To determine the best fitting coefficients of the model from the

given data

The use of classical regression is bounded by some strictassumptions about the given data. This model can be applied only ifthe given data are distributed according to a statistical model andthe relation between X and Y is crisp. Overcoming such limitations,fuzzy regression is introduced which is an extension of the classicalregression and is used in estimating the relationships betweenvariables where the available data are very limited and impreciseand variables are interacting in an uncertain, qualitative, and fuzzyway [5].

Fundamental differences between fuzzy regression and classicalregression are as follows: Fuzzy regression can be used to fit fuzzydata and crisp data into a regression model, whereas classicalregression can only fit crisp data. Classical regression analysis isbased on some assumptions. As one of the assumptions, theunobserved error terms should mutually be independent andidentically distributed. Lack of such assumption affects the effec-tiveness of the analysis. In this case, fuzzy regression can bereplaced. In contrast to the classical regression that is based onprobability theory, fuzzy regression is based on possibility theoryand fuzzy set theory. In classical regression, the unfitted errorsbetween a regression model and observed data are assumed asobservation error that is a random variable. In fuzzy regression, thesame unfitted errors are viewed as the fuzziness of the modelstructure [6,7].

The goal of fuzzy regression analysis is to find a regressionmodel that fits all observed fuzzy data within a specified fittingcriterion. Different fuzzy regression models are obtained depend-ing on the fitting criterion used. Fuzzy regression is based onminimizing the total squares errors of the spread value as the fittingcriterion. Moreover, with fuzzy regression, a mathematicalprogramming approach is developed such that the predictabilitycan be improved and the computation complexity can bedecreased.

In general, there are two approaches of fuzzy regression due todifferent fitting criterions [8e10]. The first approach is based onminimizing fuzziness as an optimal criterion which first proposedby Tanaka [8]. Different researchers used Tanaka’s approach tominimize the total spread of the output [9,11,12]. As pointed out byWang and Tsuar [13], the advantage of this approach is itssimplicity in programming and computation, but it has been criti-cized to provide toowide ranges in estimationwhich could not givemuch help in application [5] and not to utilize the concept of leastsquares [14]. The second approach uses least squares of errors asa fitting criterion to minimize the total square error of the output.Different aspects of this approach were investigated by Celmins[15,16], Diamond [17], Savic and Pedrycz [18] and Chang and Ayyub

[19]. Celmins [15] defines a compatibility measure between fuzzydata and amodel and uses this measure as amodel-fitting criterion.Diamond [17] developed a fuzzy least-square method. Savic andPedrycz [18] proposed a combined approach for fuzzy least-squareregression analysis (FLSRA) by integrating minimum fuzzinesscriterion into the ordinary least-squares regression. Chang andAyyub [19] discussed reliability issues of Fuzzy least-squareregression analysis (FLSRA), such as standard error and correla-tion coefficient. This approach, though providing narrower range,costs too much of computation time [13]. Therefore, a naturalextension of fuzzy regression would be the integration of the leastsquares’ concept into fuzzy regression.

Approaches used for energy forecastingmay be divided into twomain categories, i.e. econometric approach and artificial intelli-gence approach. Classical and fuzzy regressions are categorizedunder econometric approaches. Artificial neural network (ANN) asan intelligence approach has found considerable attention inenergy forecasting (see for example [20e22]). Although ANN doesnot need a predetermined production function between outputsand inputs however, the main disadvantage of ANN is that it usuallyrequires a large set of data for training and validation and it is hardto apply such model with small data sets. Authors in [23] comparedthe results of fuzzy regression and Artificial Neural Network (ANN)in predicting monthly electricity consumption of Iran and showedthat fuzzy regression forecast the monthly demand with lowerMAPE.

Fuzzy regression models have been successfully applied tovarious engineering problems such as Ergonomics [24] and QualityControl [25]. The application of this method in the context ofenergy demand forecasting is well treated in the literature.

In [12], in an uncertain environment, a study of the computerand peripheral equipment sales in the United States was discussedusing fuzzy linear regression introduced by Tanaka et al. [7]. Thewell-known fuzzy rule-based TakagieSugenoeKang (TSK) modelcombined with a set of fuzzy regressions was proposed to inves-tigate the impact of the climate change on the short-term elec-tricity consumption duration in Iran [26]. The paper introducesa type III TSK fuzzy inference machine combined with a set of linearand nonlinear fuzzy repressors in the consequent part to modeleffects of the climate change on the electricity demand. An inte-grated fuzzy system, data mining and time series framework toestimate and predict electricity demand for seasonal and monthlychanges were presented in [27]. The authors argued that theapplication of their framework is appropriate and capable ofhandling situations with non-stationary data and used theirframework to forecast the electricity consumption in Iran andChina. In [28] a new approach was introduced to find the param-eters of a linear fuzzy regression, with fuzzy outputs and crisp inputdata. The proposed method was used to forecast the annual end-use energy in the ResidentialeCommercial sector of Iran.

Although previous studies have provided satisfactory results inforecasting energy demand, however none of them have dealtwith small data sets. Moreover, intelligent approach such as ANNor neuro-fuzzy requires large data sets to be trained and tested.Also, simple models could not treat the problems associated withsmall data sets. To the best knowledge of the authors, this is thefirst study that introduces a flexible approach for energyconsumption estimation and forecasting with small data sets.Small data sets are often the case in developing countries.Furthermore, small data sets are usually associated with fuzzinessand uncertainty. On the other hand, there is a small chance thatthey may be associated with certainty and crispness. This is whya hybrid and flexible framework composed of both fuzzy andclassical regressions together with design of experiment seems tobe ideal in such situations.

A. Azadeh et al. / Energy 36 (2011) 6981e6992 6983

1.1. Manufacturing technology and energy consumption

Exploring the literature in the field of energy consumptionshows that in several countries there is a positive relationshipbetween gross domestic production (GDP) and energy consump-tion. Improving some of the main features of manufacturing tech-nology is directly related to energy consumption. Also, thelimitations of energy resources and strictly increasing energyconsumption trend show the need to design accurate devices forconsumption of energy in manufacturing sector in particular andindustrial sector in general. Hence, there is a need to focus on trendof energy consumption in the future, particularly in manufacturingsector. For example, if the future energy consumption is accuratelyestimated and forecasted with high and acceptable confidence,then, the needed power or natural gas can be estimated and thefacilities needed for fulfillment of these demands can be designedand constructed in timely manner for higher efficiency andproductivity. The energy consumption in some countries such asUSA and Canada hasmore heterogeneity than Pakistan for example.Moreover, as shown in Fig. 1, energy consumption in thesecountries has uncertain structure and usually a short history ofsteady past trend is in hand to forecast the future. This makeseconometric models (fuzzy or classical regressions) the idealcandidate for forecasting purposes. Please note that Fig. 1 repre-sents annual energy consumption in case studies so the heteroge-neity in energy consumption figures is discussed based on annualvariation in energy consumption not daily/weekly/monthly varia-tions. Moreover, the trend for each country is based on annual datafrom 1995 to 2005.

Fig. 1. Energy consumption in the

Industrial sector and as themain part of it, manufacturing sectorhave a considerable participation in total energy use in almostevery industrialized country around the world. For example, theindustrial sector in the United States has the greatest share amongthe other sectors in total energy use (Fig. 2). Overall, accurateforecasting of energy consumption in manufacturing sector wouldbe quite beneficial with respect to strictly increasing technologicaland industrial growth in 21st century.

The remainder of the paper comprises the followings: Section 2introduces the hybrid fuzzy regression DOE framework for totalenergy consumption estimation with small data sets. Methodologyof the hybrid framework is also presented. Section 3 presents anexperiment showing the applicability of the proposed frameworkfor the cases of the USA, Canada, Singapore, Pakistan and Iran.Results and discussions about the experiment are located in Section4. Finally, Section 5 presents proper conclusions of the study.

2. Method: the hybrid framework

The economic indicators used in this paper are population andGross Domestic Production in the last periods. The proposedframework uses ANOVA to select either fuzzy regression or classicalregression model for future energy estimation. Fuzzy regressionand classical regression are tested through a randomized completeblock design at a¼ 5% significance level. The null hypothesis of thisdesign is that the forecasts of fuzzy regression, the forecasts ofclassical regression, and actual energy consumption as treatmentare not statistically different. If the null hypothesis is accepted thenthe preferredmodel is the onewhich has lowerMAPE. Otherwise, if

selected countries (1015 Btu).

Fig. 2. Total energy consumption by end-use sectors in the USA. (Source: http://www.eia.doe.gov/emeu/aer/ep/ep_frame.html.)

A. Azadeh et al. / Energy 36 (2011) 6981e69926984

the null hypothesis is rejected, Tukey Simultaneous Tests is used tocompare treatment means and to select the preferred model. Thefuzzy regression models are solved by the Lingo software.

Flexibility in using ANOVA or MAPE for selection betweenregression models is the first unique feature of this study. Inaddition, this study does not presume a preferred model and selectit dynamically with accordance to the data in the case. The signif-icance of the proposed framework is three fold. First, it is flexibleand identifies the best model based on the results of ANOVA andMAPE. Second, the proposed framework may identify classicalregression as the best model for future energy consumption fore-casting because of its dynamic structure, whereas in the case ofuncertainty and ambiguity, previous studies assume that fuzzy

No

Collection of inputvariables: X1 and X2 and output variable: Y for the

targeted short period

Is thhypo

Perform Tukey Simultaneous Tests for identification of either

the preferred model

Fitting the bemodel

Developmeregression mo

the best rm

Forecasticonsumption

fuzzy regrclass ica l r eg

Perform ANfor fuzzy r

classical regactual data

Fig. 3. The hybrid fuzzy regression DOE framework for tota

regression provides better solutions and estimation. Third, it isideal candidate for short data sets. Fig. 3 depicts the hybridframework of this study. The reader should note that all steps of thehybrid framework are based on standard and scientific methodol-ogies which are Fuzzy regression, classical regression, analysis ofvariance (ANOVA), Tukey Simultaneous Tests and MAPE. Further-more, the Fuzzy regression modeling is based on which regressionmodel is selected for the data set. The best model is distinguishedby modeling, running and testing various regression models andselecting the model with lowest error.

2.1. Model variables

The following variables are defined to estimate annual energyconsumption:

- Y: Energy consumption in each country- X1: Annual population of each country- X2: Gross Domestic Production (GDP)

2.2. Error estimation methods

There are four basic error estimation methods which are listedbelow:

- Mean Absolute Error (MAE)- Mean Square Error (MSE)

Yes e null thesis of

Selection of fuzzy regression or classical

regression based on the lowest value of MAPE

st regression to data

nt of fuzzy del based on egression ode l

ng energy by best fitted ession and ress ion m ode ls

OVA F-test egression, ression and (test data)

l energy consumption estimation with small data sets.

A. Azadeh et al. / Energy 36 (2011) 6981e6992 6985

- Root Mean Square Error (RMSE)- Mean Absolute Percentage Error (MAPE)

They can be calculated by the following equations:

MAE ¼

Pnt¼1

jxt � x0t jn

MSE ¼

Pnt¼1

ðxt � x0Þ2

n

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnt¼1

ðxt � x0Þ2

n

vuuut

MAPE ¼

Pnt¼1

����xt � x0

xt

����n

(1)

All methods, except MAPE have scaled output. As input dataused for the model estimation, preprocessed and raw data, havedifferent scales. MAPE method is the most suitable one to estimatethe errors.

2.3. Design of experiment

To examine the differences of the estimated results of fuzzyregression and classical regression when compared with actualdata, the examiner needs to first determine all sources of variabilityon the response (here Energy Consumption) and second design anexperiment to study the significance of the variability sources. Theexperiment should be designed such that variability arising fromextraneous sources can be systematically controlled. Time is thecommon source of variability in the experiment that can besystematically controlled through blocking. The experimentdesigned in this study is a randomized complete block design(RCBD). Moreover, the blocks provide sufficient replications in theexperiment. In this case, the interactions between treatments andblocks are treated as the random error component.

The hypothesis is:

H0: m1 ¼ m2 ¼ m3H1: mismj; i; j ¼ 1;2;3; isj (2)

where m1, m2 and m3 are the average estimation values obtainedfrom actual data.

A detailed description of the procedure of ANOVA for RCBD,mainly adopted from [29], will be given here. The procedure isusually summarized in an ANOVA table (Table 1).

In RCBD of Table 1, there are a treatments, b blocks and abobservations in total because there are no replications in the design.Fa,(a�1),(a�1)(b�1) is a value from F-distributionwith (a� 1) degrees offreedom in nominator and (a� 1)(b� 1) degrees of freedom in

Table 1Analysis of variance for a randomized complete block design.

Source of variation Sum of squares Degrees of freedom

Treatments SSTreatments a� 1

Blocks SSBlocks b� 1

Error SSError (a� 1)(b� 1)

Total SSTotal ab� 1

denominator and the probability that F variable is greater thanFa,(a�1),(a�1)(b�1) is a.

The formulas for computing the sum of squares (SS) in Table 1are presented in Equations (3)e(6).

SSTotal ¼Xai¼1

Xbj¼1

y2ij � y2$$ab

(3)

SSTreatments ¼ 1b

Xai¼1

y2i$ �y2$$ab

(4)

SSBlocks ¼ 1a

Xbj¼1

y2$j �y2$$ab

(5)

SSError ¼ SSTotal � SSTreatments � SSBlocks (6)

where yij is the observation in ith treatment and jth Block. Dot (.)means sum over the index as y$$ is the sum of all yijs, yi$ is the sumof yijs in ith treatment, and y$j is the sum of yijs in jth Block.

2.4. Fuzzy regression models

There are two main approaches in fuzzy regression modeldevelopment e fuzzy linear regression (FLR) and fuzzy least-squares regression (FlSR) [30,31]. Fuzzy linear regression was firstintroduced by Tanaka et al. in 1982 [7] and its variations suggestedby Tanaka [32], Sakawa and Yano [10,33], Peters [9], Kim and Bishu[34]. Sakawa and Yano also introduced fuzzy data in the formula-tion. They considered the possibility and necessity conditions forfuzzy equality as defined by Dubois and Prade. The fuzzy least-squares regression (FLSR) was firstly introduced by Diamond [17]and Celmins [15,16]. The developed models of this approach aresimilar to Savic and Pedrycz [18], Bardossy and Duckstein [35],Chang and Lee [11] and Tanaka and Lee [36]. The basic Tanakamodel assumes a fuzzy linear function as shown in Model (7):

~Y ¼ ~A0X0 þ ~A1X1;.; ~ANXN ¼ ~AX (7)

where X¼ [X0, X1. XN]T is a vector of independent variables,~A ¼ ½~A0;

~A1;.; ~AN �T is a vector of fuzzy coefficients presented in theform of symmetric triangular fuzzy numbers denoted by~A ¼ ðaj; cjÞ where aj is its central value and cj is the spread value.Thus, Model (7) can be rewritten as Model (8):

~Yl ¼ ða0; c0Þ þ ða1; c1ÞX1 þ/þ ðaN; cNÞXN (8)

The above fuzzy regression analysis assumes the crisp input andoutput data, while the relation between the input and output datais defined by a fuzzy function. By applying the Extension Principle,it derives the membership function of estimated value. Each valueof the dependant variable can be estimated as a fuzzy number

Mean squares F-value P-value

SSTreatments

a� 1MSTreatments

MSErrorPðF � value > Fa;ða�1Þ;ða�1Þðb�1ÞÞ

SSBlocksb� 1

SSErrorða� 1Þðb� 1Þ

Table 2Raw data for the United States.

Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Consumption (1015 Btu) 91.17 94.18 94.77 95.18 96.82 98.98 96.33 97.86 98.21 100.35 100.69Population (Millions) 266.56 269.67 272.91 276.12 279.29 282.34 285.02 287.68 290.34 293.03 295.73GDP 2.5 3.7 4.5 4.2 4.4 3.7 0.8 1.6 2.5 3.6 3.1

A. Azadeh et al. / Energy 36 (2011) 6981e69926986

~Yi ¼ ðYLi ;Y

h¼1i ;YU

i Þ, i¼ 1, 2. M, where the lower bound, centralvalue and the upper bound are shown in Model (9):

Y li ¼ PN

j¼0

�aj � cj

�xij

Yh¼1i ¼ PN

j¼0ajx

Yui ¼ PN

j¼0

�aj þ cj

�xij

(9)

Thus, the proposed model of Tanaka becomes Model (10) asfollows:

Z ¼ MinPNj¼0

cj

PNj¼0

ajxijþð1�hÞPNj¼0

cj��xij��� yiþð1�hÞei ci ¼ 1;2;.;M

PNj¼0

ajxij�ð1�hÞPNj¼0

cj��xij��� yi�ð1�hÞei ci ¼ 1;2;.;M

cj �0; a˛R; xi0 ¼ 1; 0�h�1

(10)

where the term h is referred to as a measure of goodness of fit ora measure of compatibility between data and a regression modeland yi is the center and ei is the spread of the ith collected data.Fuzzy linear regression (FLR) has been criticized, especially in theoriginal formulation of Tanaka et al. [7]. As Jozsef [37] pointed outthe solution of Tanaka’s model is xj-scale dependant and many cjsmay be zero. To rectify this problem, Tanaka et al. [38] and Reddenand Woodall [39] proposed the following objective function asshown by Model (11):

Z ¼ MinPNj¼0

cjPMi¼1

jxijj!

yj �Pni¼1

Pixij � ð1� hÞ Pni¼1

cixij

(11)

Other development of original Tanaka models has been dis-cussed in Tanaka and Lee [36].

Savic and Pedrycz [18] noted that not all data points are allowedto influence the estimated parameters in fuzzy linear regression(FLR). The fuzzy linear regression may tend to become multi-co-linear as more independent variables are collected [13,40].Furthermore, the original Tanaka model was extremely sensitive to

Table 3Raw data for Canada.

Year 1995 1996 1997 1998 19

Consumption (1015 Btu) 12.20 12.54 12.66 12.36 12Population (Millions) 29.62 29.98 30.31 30.63 30GDP 2.5 3.7 4.5 4.2 4

the outliers [9] and prediction intervals becomewider as more dataare collected [39,41]. On the other hand, fuzzy least-squaresregression (FLSR) has had very few criticisms because of its simi-larity to traditional least-squares regression. However, FLSR issensitive to outliers and it should be used only when enough dataare available which results in losing one of the advantages of thefuzzy regression. Özelkan [31], Özelkan et al. [42] and Özelkan andDuckstein [43] developed a bi-objective fuzzy regression (BOFR)model which is capable of solving the problems of fuzzy linearregression (FLR), mentioned above, especially the problem of dataoutliers as shown by Model (12).

Min�dominated�V ;Ep

�Subject to : PN

j¼0ajxij�ð1�hÞPN

j¼0cj��xij��

!�ðyi�ð1�hÞeiÞ� εL;i

ðyiþð1�hÞeiÞ� PN

j¼0ajxijþð1�hÞPN

j¼0cj��xij��

!� εR;i

εL;i;εR;i �0 ci ¼ 1;2;.;M

(12)

where V denotes the vagueness measure defined as the spread ofthe prediction to be minimized and Ep is the deviation fromoutliers which are brought in Model (13) and εl,i, εR,i are relaxationvariables.03p3N is the compensation level. The expression“min-dominated” indicates the non-inferior solution-findingprocess.

V¼PMi¼1

PNj¼0

ajxijþð1�hÞPNj¼0

cj��xij��

!� PN

j¼0ajxij�ð1�hÞPN

j¼0cj��xij��

!!

Ep¼PNi¼1

�εpL;iþε

pR;i

(13)

Moreover, Özelkan [31] and Özelkan and Duckstein [43] provedthat fuzzy linear regression (FLR) models of Tanaka et al. [7],Tanaka [32], Peters [9] and classic crisp (non-fuzzy) regressionmodel are specific cases of their model. Although having severaladvantages compared with other fuzzy linear regression (FLR)models, the Özelkan’s model still has some drawbacks, such as thecentral tendency property does not exist fully and explicitly in themodel [36].

99 2000 2001 2002 2003 2004 2005

.94 12.93 12.97 13.33 13.74 14.03 14.31

.96 31.28 31.59 31.90 32.21 32.51 32.81

.4 3.7 0.8 1.6 2.5 3.6 3.1

Table 4Raw data for Singapore.

Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Consumption (1015 Btu) 1.185 1.352 1.451 1.499 1.485 1.522 1.611 1.592 1.682 1.893 2.023Population (Millions) 3.54 3.67 3.80 3.90 3.97 4.04 4.12 4.20 4.28 4.35 4.43GDP 8.2 7.8 8.3 �1.4 7.2 10.1 �2.4 4.2 3.1 8.8 6.6

Table 5Raw data for Pakistan.

Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Consumption (1015 Btu) 1.581 1.669 1.692 1.738 1.814 1.856 1.810 1.875 1.978 2.051 2.252Population (Millions) 127.62 130.82 133.99 137.18 140.36 143.96 147.65 150.42 152.94 155.85 158.78GDP 5 4.8 1 2.6 3.7 4.3 2 3.2 4.8 7.4 7.7

Table 6Raw data for Iran.

Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Consumption (1015 Btu) 1.131 1.125 1.041 1.056 1.074 1.084 1.134 1.144 1.012 1.125 1.244Population (Millions) 60.78 61.34 61.91 62.41 62.83 63.27 63.75 63.94 63.99 64.33 64.74GDP 2.7 7.1 3.4 2.7 1.9 5.1 3.7 7.5 7.2 5.1 4.4

A. Azadeh et al. / Energy 36 (2011) 6981e6992 6987

Using the idea from Özelkan and Duckstein [44] in dealing withoutlier Nasrabadi et al. developed a multi-objective fuzzy linearregression model to overcome the shortcomings of exciting fuzzyregression approaches. Asmentioned by Peters [9] and Özelkan andDuckstein [43], outliers can be models by introducing soft bound-aries to the fuzzy linear regression (FLR) model. Moreover, Changand Ayyub model [14] model (14) is used for the purpose of thisstudy. This model is an extension of Tanka’s model and resolves theoutlier problem associated with previous models.

Z ¼ minPni¼1

Pmj¼1

cixij

yj �Pni¼1

Pixij þ ð1� hÞ Pni¼1

cixij

(14)

3. Experiment

The proposed framework is applied to energy consumptionestimation and forecasting in United States, Canada, Pakistan, Iranand Singapore from 1995 to 2005. It is furthermore used to identifythe preferred model to forecast and estimate energy consumptionin these countries by the hybrid mechanism of the proposedframework which is based on fuzzy regression, classical regression,F-test, Tukey Simultaneous Tests and MAPE. The raw data withrespect to the 2 independent variables for these countries areshown in Tables 2e6. The data from 1995 to 2001 are used as train

Table 7Error estimation in Canada.

Year Actual data Fuzzy regression Classical regression

2002 13.325 13.330 13.0902003 13.736 13.600 13.2302004 14.029 14.030 13.3852005 14.308 14.110 13.480

MAPE 0.006 0.0395

data and data 2002e2005 is used as test data. The reader shouldnote that the data for both train and test are relatively small. Model(14) which is the extension of Tanaka model is chosen as the bestfitted fuzzy regression model because it covers the outlier prob-lems. The outlier problem is specifically important in treatment ofsmall data sets. The fuzzy regression models were developed andtested by the Lingo software (Appendix 1). The link to this model isalso shown in Appendix 2.

4. Results and analysis

The estimated results of fuzzy regression, classical regressionand actual data are compared by ANOVA F-test. The experimentwas designed such that variability arising from time can besystematically controlled through blocking. Therefore a blockeddesign of ANOVA is applied according to model (2) and the resultsare shown in Tables 7e16.

4.1. Canada

As can be seen from Table 7, the results of both fuzzy and clas-sical regression methods with respect to energy consumptionestimation are relatively close to actual data. Hence, the results ofboth methods are verified by actual data for Canada.

According to ANOVA results in Table 8, with a¼ 0.05 the nullhypothesis is rejected for Canada because p-value oftreatment¼ 0.003< 0.05 and therefore, further analysis needs tobe performed to foresee which treatment pairs caused the rejection

Table 8Analysis of variance for Canada.

Source of variation Sumof squares

Degreesof freedom

Meansquares

F-value P-value

Treatments (regressions) 0.7132 2 0.3566 19.04 0.003Blocks (years) 0.9123 3 0.3041Interaction (error) 0.1124 6 0.0187

Total 1.7379 11

Table 10Analysis of variance for the USA.

Source of variation Sumof squares

Degreesof freedom

Meansquares

F-value P-value

Treatments (regressions) 1.960 2 0.980 7.95 0.021Blocks (years) 10.679 3 3.560Interaction (error) 0.739 6 0.123

Total 13.378 11

Table 11Error estimation in Singapore.

Year Actual data Fuzzy regression Classical regression

2002 1.592 1.640 1.6522003 1.682 1.681 1.7042004 1.893 1.850 1.7292005 2.023 1.973 1.785

MAPE 0.0195441 0.0637625

Table 12Analysis of variance for Singapore.

Source of variation Sum ofsquares

Degrees offreedom

Meansquares

F-value P-value

Treatments (regressions) 0.01497 2 0.00748 1.35 0.329Blocks (years) 0.16276 3 0.05425Interaction (error) 0.03335 6 0.00556

Total 0.21107 11

Table 9Error estimation in the USA.

Year Actual data Fuzzy regression Classical regression

2002 97.858 98.98 97.792003 98.21 99.46 98.632004 100.351 100.42 99.522005 100.691 100.91 99.96

MAPE 0.0067640 0.0051281

Table 14Analysis of variance for Pakistan.

Source of variation Sum ofsquares

Degrees offreedom

Meansquares

F-value P-value

Treatments (regressions) 0.05774 2 0.02887 5.62 0.042Blocks (years) 0.09660 3 0.03220Interaction (error) 0.03083 6 0.00514

Total 0.18517 11

Table 15Error estimation in Iran.

Year Actual data Fuzzy regression Classical regression

2002 1.144 1.17 1.1142003 1.012 1.173 1.1122004 1.125 1.152 1.0952005 1.244 1.259 1.089

MAPE 0.054 0.069

Table 16Analysis of variance for Iran.

Source of variation Sum ofsquares

Degrees offreedom

Meansquares

F-value P-value

Treatments (regressions) 0.01533 2 0.00767 2.45 0.167Blocks (years) 0.01569 3 0.00523Interaction (error) 0.01879 6 0.00313

Total 0.04981 11

Table 17Summary of ANOVA and MAPE results.

Country P-value MAPE

Fuzzy regression Classical regression

Canada 0.003 0.00600 0.00395The USA 0.021 0.00670 0.00510Singapore 0.329 0.01954 0.06376Pakistan 0.042 0.01130 0.06700Iran 0.167 0.05400 0.06900

A. Azadeh et al. / Energy 36 (2011) 6981e69926988

of null hypothesis. Furthermore, we use Tukey Simultaneous Test asfollows and conclude that fuzzy regression method provides thebest estimation for Canada:

Comparing treatments 1 and 2: p-value¼ 0.6899> 0.05 thenm1¼ m2Comparing treatments 1 and 3: p-value¼ 0.0030< 0.05 thenm1s m3Comparing treatments 2 and 3: p-value¼ 0.0067< 0.05 thenm2s m3

Table 13Error estimation in Pakistan.

Year Actual data Fuzzy regression Classical regression

2002 1.875 1.925 1.8702003 1.978 1.972 1.8842004 2.051 2.052 1.8942005 2.252 2.218 1.925

MAPE 0.0113 0.067

4.2. USA

As can be seen from Table 9, the results of both fuzzy andclassical regression methods with respect to energy consumptionestimation are relatively close to actual data. Hence, the results ofboth methods are verified by actual data for the USA.

According to ANOVA results in Table 10, with a¼ 0.05the null hypothesis is rejected for US because p-value oftreatment¼ 0.021<0.05, and therefore, further analysis needs tobe performed to foresee which treatment pairs caused the rejectionof null hypothesis. Furthermore, we use Tukey Simultaneous Testand conclude that both fuzzy and classical regression methodsprovide the good estimation for US. However, classical regression isselected because it provides slightly smaller relative error (Table 9).

Comparing treatments 1 and 2: p-value¼ 0.0813> 0.05 thenm1¼ m2Comparing treatments 1 and 3: p-value¼ 0.4859> 0.05 thenm1¼ m3Comparing treatments 2 and 3: p-value¼ 0.0188< 0.05 thenm2s m3

Table 18Comparison of the hybrid framework versus other methods.

Methods/approaches Features

CrispData

Non crispdata

Handlingdata linearity

Handling datanonlinearity

Conductingexperiments

Relativeerror estimation

Identifying the bestmodel and solution

Flexibility Smalldata set

Classical regression models O O O OFuzzy regression models O O O O OGenetic algorithm O O OArtificial neural network O O OParticle swarm optimization O O OThe hybrid fuzzy regression-design

of experiment frameworkO O O O O O O O O

A. Azadeh et al. / Energy 36 (2011) 6981e6992 6989

4.3. Singapore

As can be seen from Table 11, the results of both fuzzyand classical regression methods with respect to energyconsumption estimation are relatively close to actual data.Hence, the results of both methods are verified by actual data forSingapore.

According to ANOVA results in Table 12, with a¼ 0.05 the nullhypothesis is accepted for Singapore because: p-value oftreatment¼ 0.329> 0.05 and selection of the fuzzy regression orclassical regression model is based on the lower value of MAPE.Therefore, fuzzy regression method has resulted in lower relativeerror and provides better fitness when compared with actual data(Table 11).

4.4. Pakistan

As can be seen from Table 13, the results of both fuzzy andclassical regression methods with respect to energy consumptionestimation are relatively close to actual data. Hence, the results ofboth methods are verified by actual data for Pakistan.

According to ANOVA results in Table 14, with a¼ 0.05 the nullhypothesis is rejected for Pakistan because p-value oftreatment¼ 0.042< 0.05, and therefore, further analysis needs tobe performed to foresee which treatment pairs caused the rejectionof null hypothesis. Furthermore, we use Tukey Simultaneous Testand conclude that both fuzzy and classical regression methodsprovide the good estimation for US. However, fuzzy regression isselected because it provides slightly smaller relative error(Table 13).

Comparing treatments 1 and 2: p-value¼ 0.9984> 0.05 thenm1¼ m2Comparing treatments 1 and 3: p-value¼ 0.0635> 0.05 thenm1¼ m3Comparing treatments 2 and 3: p-value¼ 0.0594> 0.05 thenm2¼ m3

4.5. Iran

As can be seen from Table 15, the results of both fuzzyand classical regression methods with respect to energyconsumption estimation are relatively close to actual data.Hence, the results of both methods are verified by actual data forIran.

According to ANOVA results in Table 16, with a¼ 0.05 the nullhypothesis is accepted for Iran because p-value oftreatment¼ 0.167> 0.05 and selection of the fuzzy regression orclassical regression model is based on the lower value of MAPE.

Therefore, fuzzy regression method has resulted in lower relativeerror and provides better fitness when compared with actual data(Table 15).

4.6. Technical note

Definitely, there is no need to forecast what we alreadyknow. However, this study selects the data in the years2002e2005 to test the forecasting accuracy of the models andto compare them with the preferred model. Moreover, ANOVAis used to examine if there is differences between whatforecasted by regressions and actual data. In the case ofrejected H0, it is concluded that the differences between theresults of classical regression and fuzzy regression are statisti-cally significant and therefore we cannot rely on MAPE to selectthe preferred model. In this case, Tukey Simultaneous Testsmethod is used to identify which model is closer to actual dataat a level of significance. On the other hand, when H0 cannotbe rejected, it is inferred that there is not significantdifferences between the forecasting results of regression models.As a result, we rely on MAPE to decide the preferred forecastingmodel.

5. Conclusion

This research presented a hybrid framework to estimate andpredict energy consumption with small data sets. To show theapplicability of the proposed framework, annual energyconsumption in Canada, United States, Pakistan, Singapore andIran from 1995 to 2005 was used. Then, ANOVA was applied tocompare the proposed fuzzy regression, classical regression andactual data. It was found that the null hypothesis is true for Iranand Singapore and MAPE was used to identify which model iscloser to the actual data and consequently fuzzy regression esti-mations are closer to actual data. The null hypothesis was false forCanada, Pakistan and US and therefore, Tukey Simultaneous Testswere used to identify which model is closer to the actual data. Itwas shown that the fuzzy regression has better estimated valuesfor energy consumption in Canada and Pakistan. However, theclassical regression has better estimated values for energyconsumption in US.

Table 17 presents the summary of ANOVA and MAPE resultsfor Canada, the USA, Singapore, Pakistan and Iran. As seen, theMAPE results of the selected fuzzy regressions are very close toclassical regressions and are therefore verified by classicalregression.

The proposed framework is highly flexible because it canhandle data nonlinearity, fuzziness, data complexity, crisp dataand both classical and fuzzy regression models. It can furtherhandle noise and outliers. Table 18 presents a comparison

A. Azadeh et al. / Energy 36 (2011) 6981e69926990

between the hybrid framework and existing methods. As shown, itis superior and has several advantages over existing methods.Furthermore, it can conduct both design of experiments andminimum absolute percentage error to show which model (fuzzyor classical) is superior. It also identifies the best classical andfuzzy regression models prior to design of experiment. Finally, theframework is superior to previous approaches because it canhandle relatively data sets such as the five actual examples of thispaper.

In summary, this study presented a hybrid framework forforecasting energy consumption based on fuzzy regression,classical regression and DOE for small data sets. The economicindicators used in this paper are population and Gross DomesticProduction (GDP) in the last periods. The proposed frameworkuses ANOVA to select either fuzzy regression or classical regres-sion for future demand estimation. Furthermore, if the nullhypothesis in ANOVA F-test is rejected, Tukey Simultaneous Testsmethod is used to identify which model is closer to actual data ata level of significance. It also uses MAPE when the null hypoth-esis in ANOVA is accepted to select from fuzzy regression orclassical regression model. The significance of the proposedframework is three fold. First, it is flexible and identifies the bestmodel based on the results of ANOVA and MAPE. Second, theproposed model may identify classical regression as the bestmodel for future energy consumption forecasting because of itsdynamic structure, whereas in the case of uncertainty andambiguity, previous studies assume that fuzzy regression

Canada;

!objective function;

min¼c0þ343.7871*c1þ34.60*c2;

!constraints;

!CT 1; p0þ29.61*p1þ2.50*p2-(1-h)* (c0þ29.619*c1þ2.5*

!CT 2; p0þ29.61*p1þ2.50*p2þ(1-h)* (c0þ29.619*c1þ2.5*

!CT 3; p0þ29.983*p1þ3.70*p2-(1-h)* (c0þ29.983*c1þ3.7

!CT 4; p0þ29.983*p1þ3.70*p2þ(1-h)* (c0þ29.983*c1þ3.7

!CT 5; p0þ30.305*p1þ4.5*p2-(1-h)* (c0þ30.305*c1þ4.5*

!CT 6; p0þ30.305*p1þ4.5*p2þ(1-h)* (c0þ30.305*c1þ4.5*

!CT 7; p0þ30.628*p1þ4.2*p2-(1-h)* (c0þ30.628*c1þ4.2*

!CT 8; p0þ30.628*p1þ4.2*p2þ(1-h)* (c0þ30.628*c1þ4.2*

!CT 9; p0þ30.956*p1þ4.4*p2-(1-h)* (coþ30.957*c1þ4.4*

!CT 10; p0þ30.956*p1þ4.4*p2þ(1-h)* (coþ30.957*c1þ4.4

!CT 11; p0þ31.2781*p1þ3.7*p2-(1-h)* (c0þ31.2781*c1þ3

!CT 12; p0þ31.2781*p1þ3.7*p2þ(1-h)* (c0þ31.2781*c1þ3

!CT 13; p0þ31.5928*p1þ0.8*p2-(1-h)* (c0þ31.5928*c1þ0

!CT 14; p0þ31.5928*p1þ0.8*p2þ(1-h)* (c0þ31.5928*c1þ0

!CT 15; p0þ31.9022*p1þ1.6*p2-(1-h)* (c0þ31.9022*c1þ1

!CT 16; p0þ31.9022*p1þ1.6*p2þ(1-h)* (c0þ31.9022*c1þ1

!CT 17; p0þ32.2071*p1þ2.5*p2-(1-h)* (c0þ32.2071*c1þ2

!CT 18; p0þ32.2071*p1þ2.5*p2þ(1-h)* (c0þ32.2071*c1þ2

provides better solutions and estimation. Third, it is ideal forrelatively small data sets.

Future research could extend the present framework by bothconventional and intelligent multivariate nonlinear time series.The significance of the nonlinear test statistics has been deter-mined by a recent study [45] on surrogate data. Thus, artificialneural network (ANN) and adaptive network based fuzzy infer-ence system (ANFIS) may be used to develop frameworks forenergy consumption estimation with nonlinear pattern anduncertainty.

Acknowledgment

The authors are grateful for the valuable comments andsuggestions from the respected reviewers. Their valuablecomments and suggestions have enhanced the strength andsignificance of our paper. This study was supported by a Grant fromUniversity of Tehran (Grant No. 8106013/1/07). The authorsacknowledge the support provided by the University College ofEngineering, University of Tehran, Iran.

Appendix 1. Sample Lingo codes for Canada

c2)<¼12.202;

c2)>¼12.202;

0*c2)<¼12.539;

0*c2)>¼12.539;

c2)<¼12.655;

c2)>¼12.655;

c2)<¼12.361;

c2)>¼12.361;

c2)<¼12.939;

*c2)>¼12.939;

.7*c2)<¼12.932;

.7*c2)>¼12.932;

.8*c2)<¼12.967;

.8*c2)>¼12.967;

.6*c2)<¼13.325;

.6*c2)>¼13.325;

.5*c2)<¼13.736;

.5*c2)>¼13.736;

A. Azadeh et al. / Energy 36 (2011) 6981e6992 6991

Appendix 2

Model :!(Extended Tanaka Model of Fuzzy Regression);

Sets: !Import Sets from Excel;

Input: Response;

Component: Centers,Spreads;

Links(Input,Component): Train;

Endsets

Min ¼ @SUM(Input(I):@SUM(Component(J):Spreads(J)*Train(I,J)));

@FOR(Input(I):@SUM(Component(J):Centers(J)*Train(I,J)þ(1-H)*Spreads (J)*Train(I,J))>¼Response(I));

@FOR(Input(I):@SUM(Component(J):Centers(J)*Train(I,J)-(1-H)*Spreads (J)*Train(I,J))<¼Response(I));

Data: !Import the data from Excel;

Input, Component, Response, Train, H ¼ (@OLE’D:\Lingo8\chang.xls’);

Enddata

End

References

[1] Pao HT. Forecasting energy consumption in Taiwan using hybrid nonlinearmodels. Energy 2009;34(10):1438e46.

[2] Zhang M, Mu H, Li G, Ning Y. Forecasting the transport energy demand basedon PLSR method in China. Energy 2009;34(9):1396e400.

[3] Cinar D, Kayakutlu G, Daim T. Development of future energy scenarioswith intelligent algorithms: case of hydro in Turkey. Energy 2010;35(4):1724e9.

[4] Afshar K, Bigdeli N. Data analysis and short term load forecasting in Iranelectricity market using singular spectral analysis (SSA). Energy 2011;36(5):2620e7.

[5] Wang HF, Tsaur RC. Insight of a fuzzy regression model. Fuzzy Sets and System2000;112:355e69.

[6] Ross TJ. Fuzzy logic with engineering applications. New York: McGraw-Hill;1995.

[7] Tanaka H, Uejima S, Asia K. Linear regression analysis with fuzzy model. IEEETransactions on Systems, Man, and Cybernetics 1982;12(6):903e7.

[8] Tanaka H, Watada J. Possibilistic linear systems and their application to thelinear regression model. Fuzzy Sets and System 1988;27:275e89.

[9] Peters G. Fuzzy linear regression with fuzzy intervals. Fuzzy Sets and System1994;63:45e55.

[10] Sakawa M, Yano H. Multiobjective fuzzy linear regression analysis and itsapplication. Electronics and Communications in Japan 1990;73:1e9.

[11] Chang PT, Lee ES. A generalized fuzzy weighted least-squares regression.Fuzzy Sets and Systems 1996;82:289e98.

[12] Heshmaty B, Kandel A. Fuzzy linear regression and its applications toforecasting in uncertain environment. Fuzzy Sets and Systems 1985;15:159e91.

[13] Wang HF, Tsuar RC. Resolution of fuzzy regression model. European Journal ofOperational Research 2000;126:637e50.

[14] Chang Y-HO, Ayyub BM. Fuzzy regression methods e a comparative assess-ment. Fuzzy Sets and Systems 2001;119:187e203.

[15] Celmins A. Least squares model fitting to fuzzy vector data. Fuzzy Sets andSystems 1987;22:245e69.

[16] Celmins A. Multidimensional least-squares model fitting of fuzzy models.Mathematical Modeling 1987;9:669e90.

[17] Diamond P. Fuzzy least squares. Information Sciences 1988;46:141e57.[18] Savic D, Pedrycz W. Evaluation of fuzzy regression models. Fuzzy Sets and

Systems 1991;39:51e63.[19] Chang Y-HO, Ayyub BM. Reliability analysis in fuzzy regression. In: Proc.

annual conf. of the North American fuzzy information processing society(NAFIPS’93), Allentown, PA, USA; 1993. p. 93e7.

[20] Azadeh A, Ghaderi SF, Sohrabkhani S. Annual electricity consumption fore-casting by neural network in high energy consuming industrial sectors.Energy Conversion and Management 2008;49(8):2272e8.

[21] Nasr GE, Badr EA, Younes MR. Neural networks in forecasting electrical energyconsumption: univariate and multivariate approaches. International Journalof Energy Research 2002;26(1):67e78.

[22] Al-Shehri A. Artificial neural network for forecasting residentialelectrical energy. International Journal of Energy Research 1999;23(8):649e61.

[23] Azadeh A, Saberi M, Seraj O. An integrated fuzzy regression algorithm forenergy consumption estimation with non-stationary data: a case study ofIran. Energy 2010;35:2351e66.

[24] Chang PT, Konz SA, Lee ES. Applying fuzzy linear regression to VDT legibility.Fuzzy Sets and Systems 1996;80:197e204.

[25] Lai YJ, Chang SI. A fuzzy approach for multi response optimization: an off-linequality engineering problem. Fuzzy Sets and Systems 1994;63:117e29.

[26] Shakouri H, Nadimi R, Ghaderi F. A hybrid TSK-FR model to study short-termvariations of the electricity demand versus the temperature changes. ExpertSystems with Applications 2009;36(2):1765e72.

[27] Azadeh A, Saberi M, Ghaderi SF, Gitiforouz A, Ebrahimipour V. Improvedestimation of electricity demand function by integration of fuzzy system anddata mining approach. Energy Conversion and Management 2008;49(8):2165e77.

[28] Shakouri H, Nadimi R. A novel fuzzy linear regression model based on a non-equality possibility index and optimum uncertainty. Applied Soft Computing2009;9:590e8.

[29] Montgomery DC. Design and analysis of experiments. 5th ed. New York: JohnWiley & Sons; 2001. p. 126e30.

[30] Diamond P, Tanaka H. Fuzzy regression analysis. In: Stowinski R, editor. Fuzzysets in decision analysis, operations research and statistics. Boston: KluwerAcademic Publishers; 1998.

[31] Özelkan EC. Multi-objective fuzzy regression applied to the calibration rain-fallerunoff models. Unpublished Ph.D. dissertation, Department of Systemsand Industrial Engineering, The University of Arizona; 1997.

[32] Tanaka H. Fuzzy data analysis by possibilistic linear models. Fuzzy Sets andSystems 1987;24:363e75.

[33] Sakawa M, Yano H. Multiobjective fuzzy linear regression analysis for fuzzyinputeoutput data. Fuzzy Sets and Systems 1992;47:173e81.

[34] Kim B, Bishu RR. Evaluation of fuzzy linear regression models by comparingmembership functions. Fuzzy Sets and Systems 1998;100:342e52.

[35] Bardossy A, Duckstein RL. Fuzzy least squares regression: theory and appli-cation. In: Kacprzyk J, Fedrizi M, editors. Fuzzy regression analysis. Heidel-berg: Omnitech Press, Warsaw and Physica-Verlag; 1992. p. 181e93.

[36] Tanaka H, Lee H. Fuzzy linear regression combining central tendencyand possibilistic properties. Proceedings of Fuzzy Conference-IEEE;1997:63e8.

[37] Jozsef S. On the effect of linear data transformations in possibilistic fuzzylinear regression. Fuzzy Sets and Systems 1992;45:185e8.

[38] Tanaka H, Hayashi I, Watada J. Possibilistic linear regression analysis for fuzzydata. European Journal of Operational Research 1989;40:389e96.

[39] Redden DT, Woodall WH. Properties of certain fuzzy linear regression models.Fuzzy Sets and Systems 1994;64:361e75.

[40] Kim KJ, Mosskowitz H, Koksalan M. Fuzzy versus statistical linear regression.European Journal of Operational Research 1996;92:417e34.

[41] Redden DT, Woodall WH. Further examination of fuzzy linear regression.Fuzzy Sets and Systems 1996;79:203e11.

A. Azadeh et al. / Energy 36 (2011) 6981e69926992

[42] Özelkan EC, Duckstein L. Fuzzy regression analysis of tradeoff between dataoutliers and prediction vagueness in fuzzy regression using a bi-objectiveframework. In: Proc. EUFIT’98-6th European congress on intelligent tech-niques and soft computing; 1998. p. 1048e51.

[43] Özelkan EC, Duckstein L. Multi-objective fuzzy regression: a general frame-work. Computer and Operations Research 2000;27:635e40.

[44] Tran L, Duckstein L. Multi-objective fuzzy regression with centraltendency and possibilistic properties. Fuzzy Sets and Systems 2002;130:21e31.

[45] Gao W, Tian Z. Learning Granger causality graphs for multivariate nonlineartime series. Journal of Systems Science and Systems Engineering 2009;18(1):38e52.