j ourna l h o mepage: www.elsev ier .com/ locate /asoc
uzzy lagged variable selection in fuzzy time series withenetic algorithms
agdas Hakan Aladaga,∗, Ufuk Yolcub, Erol Egrioglub, Eren Basb
Department of Statistics, Hacettepe University, Ankara 06800, TurkeyDepartment of Statistics, Ondokuz Mayis University, Samsun 55139, Turkey
r t i c l e i n f o
rticle history:eceived 26 July 2011eceived in revised form 21 January 2012ccepted 22 March 2014vailable online xxx
a b s t r a c t
Fuzzy time series forecasting models can be divided into two subclasses which are first order and highorder. In high order models, all lagged variables exist in the model according to the model order. Thus,some of these can exist in the model although these lagged variables are not significant in explainingfuzzy relationships. If such lagged variables can be removed from the model, fuzzy relationships will bedefined better and it will cause more accurate forecasting results. In this study, a new fuzzy time series
eywords:orecastinguzzy time seriesenetic algorithmsartial high order modelariable selection
forecasting model has been proposed by defining a partial high order fuzzy time series forecasting modelin which the selection of fuzzy lagged variables is done by using genetic algorithms. The proposed methodis applied to some real life time series and obtained results are compared with those obtained from othermethods available in the literature. It is shown that the proposed method has high forecasting accuracy.
The definition of fuzzy time series was firstly introduced by Songnd Chissom [26]. In the study of Song and Chissom [26], fuzzyime series were divided into two classes which are time variantnd time invariant and for the solution of both types of fuzzy timeeries, first order and higher order fuzzy time series forecastingodels were defined. The forecasting models based on first order
uzzy time series were proposed by Song and Chissom [27] andong and Chissom [28] for the time invariant and the time variantuzzy time series, respectively.
Fuzzy time series approaches are generally compose of threeain stages such as fuzzification, determination of fuzzy relation-
hips and defuzzification. In the literature, new methods have beenmproved by making contributions to each of these main stages. Inhe stage of fuzzification, different approaches have been used toet better results. While fixed interval lengths are used in Songnd Chissom [28–30], Chen [7], Hurang [19], Chen [8], Tsaur et al.
Please cite this article in press as: C.H. Aladag, et al., Fuzzy lagged variSoft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.028
32], Singh [27], and Egrioglu et al. [14,15], dynamic length of inter-al lengths are employed in Huarng and Yu [21], Davari et al. [13],olcu et al. [33], Kuo et al. [23,24], Park et al. [26], and Hsu et al. [18]
in order to partition the universe of discourse. Besides, clusteringmethods are used in the stage of fuzzification by Cheng et al. [12], Liet al. [25], Aladag et al. [4], Egrioglu et al. [16], Chen and Tanuwijaya[9], Bang and Lee [5].
In the stage of defuzzification, Song and Chissom [30] utilizedfeed forward artificial neural networks. Chen [7], Huarng [19], andHuarng and Yu [21] used centralization method. And, Aladag et al.[2] and Cheng et al. [11] preferred to use a method based on adap-tive expectation and centralization methods.
The stage of the determination of fuzzy relationships is alsoquite effective on the forecasting performance of fuzzy time seriesapproach. For establishing fuzzy relationships, Song and Chissom[26] used fuzzy relationship matrix and Sullivan and Wodall [31]employed transition matrices based on Markov chain. Chen [7] pro-posed an approach using the fuzzy logic group relationship tables.Cheng et al. [11], Huarng [19], Huarng and Yu [21], Yu [35] andEgrioglu et al. [16] used fuzzy logic group relationship tables todetermine fuzzy relationships. Huarng and Yu [22] and Aladaget al. [1] preferred to use feed forward artificial neural networksto determine fuzzy relationships. In the study of Aladag et al. [3],Elman recurrent neural networks were utilized for the process ofdetermining the fuzzy relationships. To determine the fuzzy rela-tionships, Yu and Huarng [36,37] and Yolcu et al. [34] proposed
able selection in fuzzy time series with genetic algorithms, Appl.
different approaches in which feed forward artificial neural net-works using membership values are used.
Many real life fuzzy time series contains complex relation-ships which cannot be explained by only using first order
agged variables. Since first order fuzzy time series forecastingodels do not include high order lagged variables, these models
an be insufficient. In the high order models used in the litera-ure, all lagged variables take place in the model depending onhe model order. In this case, fuzzy lagged variables which areot significant in explaining fuzzy relationships will be also in theodel and the forecasting performance will be affected negatively.
n modeling, it is a well-known fact that variable selection phases an important issue which directly affects the forecasting per-ormance of a model. However, variable selection has never beenerformed in the literature for any fuzzy time series approaches.his motivated us to suggest a new fuzzy time series approach inhich variable selection is done using genetic algorithms in order
o increase forecasting accuracy. In the literature, there has been noariable selection in fuzzy time series method is performed. Bahre-our et al. [6] dealt with determination of model order. However,ariable selection was not performed in Bahrepour et al. [6]. In thistudy, by defining partial high order fuzzy time series forecastingodel, a new fuzzy time series forecasting method is proposed. In
he proposed method, fuzzy lagged variables are selected by usingenetic algorithms. Since only significant lagged variables are usedn the model, the constructed model is a partial high order fuzzyime series forecasting model.
Basic definitions of fuzzy time series are given in the second sec-ion of the paper. Genetic algorithms are briefly summarized in thehird section. In the fourth section, the algorithm of the proposed
ethod is presented. In Section 5, the proposed method is com-ared to other methods in the literature. Finally, in the last section,he obtained results are discussed.
. Fuzzy time series basic definitions
The fuzzy time series was firstly introduced in Song and Chissom28]. The fuzzy time series, time variant and time invariant fuzzyime series definitions are given below [28].
efinition 1. Let Y(t) (t = . . ., 0, 1, 2,. . .), a subset of real numbers,e the universe of discourse on which fuzzy sets fj(t) are defined.
f F(t) is a collection of f1(t), f2(t), . . . then F(t) is called a fuzzy timeeries defined on Y(t).
efinition 2. Suppose F(t) is caused by F(t − 1) only, i.e.,(t − 1) → F(t). Then this relation can be expressed as F(t) = F(t − 1)◦
(t, t − 1) where R(t, t − 1) is the fuzzy relationship between F(t − 1)nd F(t), and F(t) = F(t − 1)◦ R(t, t − 1) is called the first order modelf F(t). “◦” represents max–min composition of fuzzy sets.
efinition 3. Suppose R(t, t − 1) is a first order model of F(t). If forny t, R(t, t − 1) is independent of t, i.e., for any t, R(t, t − 1) = R(t − 1,
− 2), then F(t) is called a time invariant fuzzy time series otherwiset is called a time variant fuzzy time series.
Song and Chissom [29] firstly introduced an algorithm based onhe first order model for forecasting time invariant F(t). In [29], theuzzy relationship matrix R(t, t − 1) = R is obtained by many matrixperations. The fuzzy forecasts are obtained based on max–minomposition as below:
(t) = F(t − 1)◦R (1)
The dimension of R matrix is dependent number of fuzzy setshich are partition number of universe and discourse. If we wantsing more fuzzy sets, we need more matrix operations for obtain
matrix. In this situation, the method suggested by Song andhissom [29] is more complex.
Please cite this article in press as: C.H. Aladag, et al., Fuzzy lagged variSoft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.028
efinition 4. Let F(t) be a fuzzy time series. If F(t) is caused by(t − 2), F(t − 1),. . ., and F(t − n) then this fuzzy logical relationships represented by
Fig. 1. Crossover operator.
F(t − n), . . ., F(t − 2), F(t − 1) → F(t) (2)
and it is called the nth order fuzzy time series forecasting model.
3. Genetic algorithms
The genetic algorithms were first proposed by Holland [17].Genetic algorithms have successfully solved many complex opti-mization problems. It does not require the specific mathematicalanalysis of optimization problems. Genetic algorithms is based onthe principle while good generations survive, the bad generationsdisappear. The genetic algorithms have some parameters such aspopulation size, evaluation function, crossover rate, mutation rateand maximum generation number. The genetic algorithms searchoptimal solution with many chromosomes. In one chromosome,there are many gens. The genes are decision variables when geneticalgorithms are used for optimization. These genes are representedwith some codes. In genetic algorithm binary code system (0.1)is used commonly. But there are other code systems in the liter-ature like real-code or decimal code. This code system plays animportant role for the problem. Therefore the researcher must bedetermined it carefully. Genetic algorithms are generally startingfrom a random population. The population size can be determinedby researchers due to the characteristic of the problem. After firstpopulation is randomly generated, various techniques can be uti-lized in the production of the next generation. When the geneticalgorithm cycle finishes the last generation’s fittest chromosomeis taken as the optimal solution for the problem. The genetic algo-rithm does not provide the variety after a point so it needs sometechniques to ensure the genetic variety. These techniques providea new variety for the algorithm and also they lead to create newgenerations.
Some of these techniques are crossover, mutation and naturalselection which can be summarized as follows:
Crossover: The system randomly selects two chromosomesfrom a population and randomly picks a crossover point fromthe two selected chromosomes in order to swap genes after thiscrossover point. This operation is called as the crossover operation.Crossover operator is applied to a population with the probabilityPC . Crossover rate controls the frequency of using crossover opera-tor. Crossover operator is applied to PC.l.n pieces of chromosomesin a population. In here PC is the crossover rate, l is the number ofgenes in a chromosome and n is the number of chromosomes ina population. Crossover rate should be selected carefully. Becausehigh crossover rate performs the population change quickly andlow crossover rate reduce the searching velocity. And also, thecrossover operation is depending on the crossover rate. A ran-dom number is generated from the uniform distribution. Then, thecrossover operation is performed if random number is bigger thanthe crossover rate. The crossover operation is performed as in Fig. 1.
Mutation: First of all, the researcher has to determine the muta-tion rate. Then, one chromosome is randomly selected. If a realvalue generated from the interval (0,1) is smaller than or equal to
able selection in fuzzy time series with genetic algorithms, Appl.
the mutation rate, the mutation operation will be performed with arandomly selected gene from the chromosomes. There are variousmutation operations based on the characteristic of the problem.Mutation operator is applied to Pm.l.n pieces of chromosomes in a
opulation. In here Pm is the mutation rate, l is the number of genesn a chromosome and n is the number of chromosomes in a pop-lation. Mutation rate should be selected carefully like crossoverate. Because high mutation rate enlarge the searching space andhis can obstruct to find the suitable solution. If mutation rate isaken low, chromosomes resemble each other more and more andlgorithm is chocked. A chromosome structure with four genes isiven in Fig. 2. In this figure the second gene of this chromosomes mutated. This means a new gene (the second gene) is gener-ted randomly instead of the old gene. The mutation operation iserformed as in below.
Natural selection: Each chromosome of any generation is evalu-ted by using an evaluation function. All chromosomes are orderedccording to their corresponding evaluation function values. Theest chromosomes are transferred the next generation. Some chro-osomes among the worst ones are discarded from generations.
hen, the new chromosomes instead of discarded chromosomesre placed to the new generation. The important rule in naturalelection is protecting the population size.
. The proposed method
To solve fuzzy time series faced in real life fuzzy time series fore-asting models are frequently used which is given in Eq. (2). In theseodels, it is unavoidable that fuzzy lagged variables which can-
ot make any contribution to the explanatory power of the modelill be found in the model. This situation will also cause to reduce
he forecasting performance of the model. Moreover, it could causeperation congestion. Although Bahrepour et al. [6] dealt with theelection of model order, fuzzy lagged variables selection was notxecuted. Therefore, in Bahrepour et al. [6], all fuzzy lagged vari-bles are also take place in the model so similar problems exists. Itould be wiser to solve the problem of variables selection instead
f trying to determine only model order. That will significantlyncrease the forecasting performance. The model which arises fromhe result of the selection of fuzzy lagged variables is called as par-ial high order model which is given in Definition 5. We would likeo note that in this study, Definition 5, which describes a partialigh order fuzzy time series model, is firstly introduced.
efinition 5. Let F(t) be fuzzy time series. If F(t) is caused by(t − m1),. . ., F(t − mk) (1 ≤ m1 < . . . < mk), where mi (i = 1,2,. . ., k)ake integer values, fuzzy logical relationship can be representedy
(t − m1), F(t − m2), . . ., F(t − mk) → F(t) (3)
nd it is called the kth order partial fuzzy time series forecastingodel. First of all, mi values have to be determined. Determination
rocess of these values can be called as the selection of fuzzy laggedariables. Which fuzzy lagged variables should be employed in thisartial model, which is firstly proposed in this study, is an important
ssue. In this study, a novel approach in which a partial high orderorecasting model is used is proposed. In the proposed approach,o deal with variable selection problem, genetic algorithms are uti-
Please cite this article in press as: C.H. Aladag, et al., Fuzzy lagged variSoft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.028
ized to determine the fuzzy lagged variables which will be includedn the partial model. In other words, to select the variables which
ill exist in the partial high order model, genetic algorithms aresed in the proposed forecasting approach. The flowchart of the
1 1 0 1 0 0 0 0
Fig. 3. The structure of a chromosome.
proposed approach can be seen in Fig. 4. Also, algorithm of theproposed method can be given as follows:
Step 1. The first step is the fuzzification step. The universe of dis-course U can be defined for time series as given in (4) then, andthe universe of discourse can be partitioned into “c” subintervalsaccording to a given interval length as seen in (5). Then, fuzzy sets Aj(j = 1,2,. . ., c) can be defined according to the generated subintervalsui (i = 1,2,. . ., c) as given in (6).
U = [Dmin, Dmax] (4)
u1 = [Dmin, Dmin + la],
u2 = [Dmin + la, Dmin + 2 × la],
... uc = [Dmin + (c − 1) × la, Dmax]
(5)
A1 = 1u1
+ 0.5u2
, A2 = 0.5u1
+ 1u2
+ 0.5uS
, . . ., Ac−1 = 0.5uc−2
+ 1uc−1
+ 0.5uc
, Ac = 0.5uc−1
+ 1uc
(6)
Finally, each of real observations of time series is mapped into afuzzy set which has the highest membership value in the intervalincludes this real observation. Thus, F(t) is generated.
Step 2. In this step, parameters of the genetic algorithm are deter-mined. These parameters are the number of chromosome cn, thenumber of gene in a chromosome gn, crossover ratio cr, muta-tion ratio mr, maximum iteration number maxin, and the numberof chromosome, which will be eliminated in each iteration, dcn.The parameter gn also represents the maximum model order. TheLength of all chromosomes equals to maximum model order andall of lengths are equal. Maximal order of the model is decided dueto the time series. If the number of observations of the time series isgreat enough, maximum model order can be increased as much asrequested. Evaluation function is a measure which is used to eval-uate the forecasting performance. Evaluation function is modifiedversion of AIC:
AICC = log(
SSE
n
)+ 2k
n − k − 1(7)
where k is model order and sum square error (SSE) is calculated asfollow:
SSE =∑n
t=1(xt − �xt)
2
where xt is crisp time series, �xt is defuzzified forecasts and n is thenumber of forecasts.
Step 3. The initial population is generated in the third step. Foreach of cn chromosomes, gn genes are randomly generated. To gen-erate a gene value, an value is randomly generated from theinterval [0,1]. If ≤ 0.5 the gene value is taken 0, otherwise it istaken 1. An example structure of a chromosome is given in Fig. 3when the number of gene is 8.
Chromosomes contain the inputs of the high order partial model.
able selection in fuzzy time series with genetic algorithms, Appl.
In other words, they consist of fuzzy lagged variables. If a gene in achromosome is 1, fuzzy lagged variable which corresponds to thisgene will be included in the model. If it is 0, then it will not be foundin the model.
tep 4. Evaluation function values are calculated in this step. AICCalues are computed by repeating the Steps 4.1–4.5 for each chro-osome in the current population.
tep 4.1. Model order is determined by examining the gene valuesf a chromosome. For instance, the first, the second and the fourthene values are 1 for the chromosome given in Fig. 1. Thus, theollowing high order partial fuzzy time series model correspondso the model represented by this chromosome.
(t − 1), F(t − 2), F(t − 4) → F(t) (8)
In the model above, m1 = 1, m2 = 2, and m3 = 4 and it is a thirdrder partial fuzzy time series forecasting model.
tep 4.2. Fuzzy logic relations and fuzzy logic group relations areomposed by using generated high order partial model. For exam-le, when the fuzzy time series given in Table 1 is examined, fuzzy
ogic relations and fuzzy logic group relations will be as follows forhe model presented in (8).
tep 4.3. Fuzzy forecasts are calculated with the help of fuzzyogic group relations. Three different situations could arise in thealculation of fuzzy forecasts. For example, if
ase 1. If Ai1 , Ai2 , ..., Aik→ Aj fuzzy logic group relation only
xists, the forecast value equals Aj.
ase 2. If Ai1 , Ai2 , ..., Aik→ Aj2 , Aj2 , ..., Ajs fuzzy logic group rela-
ion exists, the forecast value equals Aj1 , Aj2 , ..., Ajs ..
ase 3. If Ai1 , Ai2 , . . ., Aik, → # fuzzy logic group relation exists,
he forecast value equals Ai1 , Ai2 , . . .Aik.
tep 4.4. Defuzzificated forecasts are calculated. Centralizationethod is applied for the defuzzification. For Case 1 which is given
n Step 4.3, when fuzzy forecast is Aj, the corresponding defuzzifi-ated forecast is the midpoint of the interval uj which has theighest membership value in fuzzy set Aj. For Case 2 and 3, when the
uzzy forecast composes more than one fuzzy set, the defuzzifiedorecast is the mean value of the middle points of intervals whichas the highest membership value of corresponding fuzzy sets.
When predictions are calculated for Case 3, defuzzified forecastsre computed by using Master Voting (MV) scheme as follows:
MV scheme procedure proposed by Kuo et al. [23] obtains fore-
Please cite this article in press as: C.H. Aladag, et al., Fuzzy lagged variSoft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.028
asts of test set by giving weights to the past linguistic values inhe current state. MV scheme gives the highest weight to the latestast linguistic value and the weight of other past linguistic value
s increased by one. In accordance with the MV scheme procedure,
PRESSputing xxx (2014) xxx–xxx
forecast value of untrained test set values can be calculated withfollowed equation.
In (10), Wh and � represent the highest weight determined bythe user, and fuzzy relationship order, respectively. Also, mt1 andmtk
(2 ≤ k ≤ �) are mean values of the intervals which related to thelast and the other linguistic values in the current state, respectively.
Step 4.5. AICC value is calculated by using the formula given in(7).
Step 5. Chromosome (chromosomebest) which is the best solu-tion found so far.
Step 6. Natural selection process is applied by using sequencemethod. In this method, all chromosomes are sequenced accordingto their AICC values and dcn chromosomes are discarded with thehighest AICC value. Then, dcn chromosomebests are added to thegeneration by using elitist strategy.
Step 7. Crossover process is performed in this step. In order to per-form this process, a mating pool is randomly generated. For eachpair of chromosomes in the mating pool, a random value is gener-ated from uniform distribution on the interval (0,1). If the producedrandom value is smaller than the ratio cr, crossover is applied. Sin-gle point crossover method is used in the application of crossoverprocess. In this method, a crossover point is randomly chosen andgenes of these chromosomes are exchanged at this random point.
Step 8. Mutation process is executed in this step. A random valueis generated from uniform distribution on the interval (0,1) foreach chromosome. If the generated value is smaller than ratio mr,mutation operation is applied. In the mutation process, a gene in achromosome is randomly chosen. If the gene value is 0, it is trans-formed to 1. Otherwise, it is transformed to 0.
Step 9. Steps from 4 to 8 are repeated until a predeterminednumber of iterations (maxin) is reached.
Step 10. Optimal solution is taken as chromosomebest.
We would like to note again that the flowchart of the algorithmof the proposed approach is also presented in Fig. 4.
5. The application
In the implementation, four real life time series were employedto evaluate the forecasting performance of the proposed approach.These time series are Taiwan Stock Exchange CapitalizationWeighted Stock Index (TAIEX) and, three different time seriesfrom Index 100 in stocks and bonds exchange market of Istanbul(ISBEMI).
In order to evaluate the generalization ability of the proposedmethod, holdout cross validation method was utilized in the imple-mentation. In this validation method, data set is divided into twosubsets which are training and test sets. When the best model andfuzzy relations are determined, the training set is only used. Then,the generalization ability of methods can be evaluated according toa performance measure calculated over the test set.
There are some parameters that have to be determined whenthe proposed method is used. Like in most of fuzzy time seriesapproaches available in the literature, selection of these param-eters depends on some factors which can change due to data. In
able selection in fuzzy time series with genetic algorithms, Appl.
fuzzification phase of the proposed method length of interval isdetermined. Other parameters are parameters of genetic algorithmthat is utilized to determine fuzzy lagged variables for fuzzy timeseries forecasting model.
Length of interval and para meters of genetic algorith m initial values are
determined.
According to defined leng th of interval, uni verse of discourse is
partitioned.
Initial population is generated.
Fitness function values for chromosome s in population are
computed by using steps 4 .1-4.5 .
Natural se lec tion, crossov er and mutation ope rations are applied to
chromosomes in p opulation.
Optimal solution i s ch romosomebest.
YES
NO Is maximum iteration nu mber
reached?
itdvhillds
g
Fig. 4. The flowchart of the algorithm of the proposed approach.
In the fuzzification step of the proposed approach, length ofnterval is decided. Determination of length of interval is an impor-ant issue in fuzzy time series. The effects of length of interval wereiscussed in details by Huarng [19]. In the literature, length of inter-al has been generally determined by trial and error although thereave been some systematic approaches to find the best length of
nterval. In the implementation part of this study, different intervalengths were examined and the best one among them was selectedike in most studies available in the literature. Examined values areetermined due to the range of related time series. Details can be
Please cite this article in press as: C.H. Aladag, et al., Fuzzy lagged variSoft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.028
een in the following paragraphs.It is a well-known fact that the selection of parameters of
enetic algorithms still remains a problem in the literature. May be,
Fig. 5. TAIEX Data.
utilizing experiment design could be a good way to determine theseparameters. However, it could not be possible because of very highcomputational cost for most applications. Therefore, in the litera-ture, possible values of a parameter have been restricted and thebest value of it has been selected among these values specified dueto related problem. Another way is that some values for a parameterwhich were examined in previous studies have been used. In a sim-ilar way, in the proposed method, selection of the parameters cn,cr, mr, maxin, dcn, and gn was done due to the related time seriesor previous studies. In the literature, it is known that good solu-tions can be obtained for numerical optimization problems whencn is picked as 30. Thus, we also took cn as 30. As mentioned inStep 2 of the proposed algorithm presented in the previous sec-tion, gn equals to maximal order of the model and this order canbe determined due to the number of observations of related timeseries. The parameter dcn can be decided proportional to numberof chromosomes. The value of maxin can be selected dependingon related time series. Other parameters, namely cr, and mr, weretaken as 0.1 and 0.01, respectively like in other studies available inthe literature.
5.1. TAIEX application
In order to show the performance of the proposed method, theTaiwan Stock Exchange Capitalization Weighted Stock Index databetween 01.01.2004 and 31.12.2004 was analyzed. The graph of thetime series is shown in Fig. 5. The first 205 observations between01.01.2004 and 31.10.2004 were used as training set and the last45 observations were used as test set. The results of the proposedmethod were compared to those obtained from some fuzzy timeseries methods proposed by Song and Chissom [29], Chen [7], Chen[8], Huarng and Yu [21], Huarng et al. [20], Yu and Huarng [37],Aladag et al. [1] and Chen and Chen [10]. When the methods pro-posed by Song and Chissom [27], Chen [7], Chen [8] and Aladag et al.[1] were used, the number of fuzzy sets were taken as 10, 15, 20 and25 and lengths of interval are determined for these numbers. Theresults for the methods proposed by Chen and Chen [10], Huarnget al. [2], Yu and Huarng [37] were taken from the study conductedby Chen and Chen [10].
As a result of experimentations, parameters of the genetic algo-rithm were determined as cn = 30, cr = 0.1, mr = 0.01, maxin = 500,
dcn = 5, gn = 30. Therefore, there are∑30
k=1
(30k
)= 1.073.741.824
alternative models. For determining length of interval, 60, 65, 70,75, 80, 85, 90, 95, and 100 were examined and the best one was
able selection in fuzzy time series with genetic algorithms, Appl.
found as 75. And, a model which has a satisfactory forecastingperformance could be determined by only 500 iterations. It is pos-sible to reach a suitable solution in a reasonable time without
Fig. 6. The best model obtained from the proposed method.
Table 2The results obtained from all methods.
Method RMSE (training set) RMSE (test set)
Song and Chissom [29] 102.11 77.86Chen [7] 92.79 77.18Chen [8] 19.71 71.98Huarng and Yu [21] 90.43 63.57Huarng et al. [20] – 72.35Yu and Huarng [37] – 67.00
eo
rF
Ta
f
d
pmRiTF
5
ftmo
Fig. 8. The time series graph of Data Set 1, the period between 03.10.2008 and31.12.2008.
Aladag et al. [1] 45.83 69.80Chen and Chen [10] – 57.73The proposed method 44.48 47.95
xamining lots of alternative models. “chromosomebest” whichbtained from the proposed method is shown in Fig. 6.
1th order partial fuzzy time series forecasting model which cor-esponds to chromosomebest can be given as follows:F(t − 9) →(t)where k = 1, m1 = 9.
The results obtained from all methods are summarized inable 2. RMSE values calculated over both training and test setsre shown in this table.
Root mean square error (RMSE) given in (7) is used as evaluationunction in this study.
RMSE =√∑n
t=1(xt−xt )2
n where xt is crisp time series, xt isefuzzified forecasts and n is the number of forecasts.
When Table 2 is examined, it is clearly seen that the pro-osed method has the best forecasting accuracy. Also, the proposedethod produces very consistent results according to obtained
MSE values for training and test sets. In addition, the forecast-ng performance of the proposed method is also examined visually.he graph of the observations and forecasts for test set is given inig. 7.
.2. The data of ISBEMI
We divide the data of ISBEMI into three parts to evaluate the
Please cite this article in press as: C.H. Aladag, et al., Fuzzy lagged variSoft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.028
orecasting performance of the proposed method better. Thesehree time series were separately forecasted by the proposed
ethod and other methods available in the literature. Then thebtained forecasting results were compared. The periods, from
Fig. 7. The graph of the observations and forecasts for test set.
Fig. 9. The time series graph of Data Set 2, the period between 01.10.2009 and31.12.2009.
03.10.2008 to 31.12.2008, from 01.10.2009 to 31.12.2009, and from01.10.2010 to 23.12.2010 of ISBEMI were referred as Data Set 1, 2,
able selection in fuzzy time series with genetic algorithms, Appl.
and 3, respectively. The time series graphs of the data sets are givenin Figs. 8–10.
Fig. 10. The time series graph of Data Set 3, the period between 01.10.2010 and23.12.2009.
.2.1. The application for Data Set 1The last 15 observations of Data Set 1 were employed as test
et. These observations are between 05.12.2008 and 31.12.2008.he parameters of genetic algorithm were taken as follows: cn = 30,r = 0.1, mr = 0.01, maxin = 500, dcn = 5, and gn = 20. To determine theength of interval, 1300, 1400, 1500, 1600, and 1700 were exam-ned and the best one was found as 1500. For these parameters, thebtained chromosomebest was presented in Fig. 11. For compari-on, Data Set 1 was also analyzed with other forecasting approachesvailable in the literature presented in Table 3. For the test setsncluding 15 observations, the forecasting results produced by all
ethods were summarized in Table 3. Also, corresponding RMSEalues calculated over the test are shown in this table. The resultsbtained from other methods were taken from [34]. When Table 3s examined, it is seen that the proposed forecasting approach pro-uces the most accurate results in terms of RMSE measure for Dataet 1.
.2.2. The application for Data Set 2For Data Set 2, the last 15 observations between 11.12.2009 to
1.12.2009 were used as test set. The parameters of genetic algo-ithm were taken as follows: cn = 30, cr = 0.1, mr = 0.01, maxin = 500,cn = 5, and gn = 20. To determine the length of interval, 700, 800,00, 1000, and 1100 were examined and the best one was found as000. When these parameters were used for the proposed method,
Please cite this article in press as: C.H. Aladag, et al., Fuzzy lagged variSoft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.028
he obtained chromosomebest was presented in Fig. 12. For com-arison, Data Set 2 was also forecasted using other forecastingpproaches given in Table 4. For the test sets including 15 obser-ations, the forecasting results produced by all methods were
summarized in Table 4. Also, corresponding RMSE values calcu-lated over the test are shown in this table. The results obtainedfrom other methods were taken from [34]. According to Table 4,it is seen that the best forecasts are obtained when Data Set 2 isforecasted by the proposed approach.
5.2.3. The application for Data Set 3When Data set 3 was analyzed with the proposed method, the
last 15 observations from 01.12.2010 to 23.12.2010 were employedfor test. cn = 30, cr = 0.1, mr = 0.01, maxin = 500, dcn = 5, and gn = 20were employed as parameters of the genetic algorithm. To deter-mine the length of interval, 1300, 1400, 1500, 1600, and 1700 wereexamined and the best one was found as 1500. For these values ofthe parameters, the chromosomebest obtained from the proposedapproach can be seen in Fig. 13. For comparison, Data Set 3 wasalso forecasted using other forecasting approaches given in Table 5.For the test sets including 15 observations, the forecasting resultsobtained from all methods were summarized in Table 5. Also, cor-responding RMSE values calculated over the test were given in thistable. The results obtained from other methods were taken from[34]. When Table 5 is examined, it is observed that the proposedapproach has the best forecasting accuracy for Data Set 3 in termsof RMSE.
6. Discussion and results
able selection in fuzzy time series with genetic algorithms, Appl.
Using first order fuzzy time series forecasting model can beinsufficient since complex relationships included in many real lifefuzzy time series cannot be explained by only using first order
agged variables. In high order forecasting models available in theiterature, all lagged variables have been used in the model depend-ng on the model order. Therefore, fuzzy lagged variables whichre not significant in explaining fuzzy relationships will be also inhe model and the forecasting performance will be affected nega-ively. Variable selection has never been performed for any fuzzyime series method in the literature. On the other hand, it is aell-known fact that variable selection process directly affects the
orecasting performance of a model. In this study, a high orderartial fuzzy time series forecasting model is defined and a newethod in which fuzzy lagged variables in this forecasting model
re determined by using genetic algorithms is proposed. Since fuzzyagged variables of the model are determined, the model order islso decided at the same time. Since the defined high order partialodel does not include some variables which have no significant
Please cite this article in press as: C.H. Aladag, et al., Fuzzy lagged variSoft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.028
ffect on the explanatory power of the model, this model has lessodeling error than that included by other high order models. To
valuate the forecasting performance of the proposed method waspplied to four real time series. For comparison, these time series
were also forecasted with other fuzzy time series forecasting meth-ods available in the literature. As a result of the comparison, itwas observed that the proposed method has the best forecastingaccuracy in terms of RMSE value. Therefore, it can be said thatthe proposed approach can be used as a good alternative fuzzytime series forecasting approach. In the future studies, in order toimprove of forecasting performance of the method proposed in thisstudy, different methods such as artificial neural networks can beused in the stage of determination of fuzzy logic relations.
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