# First-Order ARMA Type Fuzzy Time Series Method Based on Fuzzy Logic Relation Tables

Authors: Cem Kocak
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Fuzzy time series approaches have an important deficiency according to classical time series approaches. This deficiency comes from the fact that all of the fuzzy time series models developed in the literature use autoregressive (AR) variables, without any studies that also make use of moving averages (MAs) variables with the exception of only one study (Egrioglu et al. (2013)). In order to eliminate this deficiency, it is necessary to have many of daily life time series be expressed with Autoregressive Moving Averages (ARMAs) models that are based not only on the lagged values of the time series (AR variables) but also on the lagged values of the error series (MA variables). To that end, a new first-order fuzzy ARMA(1,1) time series forecasting method solution algorithm based on fuzzy logic group relation tables has been developed. The new method proposed has been compared against some methods in the literature by applying them on Istanbul Stock Exchange national 100 index (IMKB) and Gold Prices time series in regards to forecasting performance.


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## Body

## 1. Introduction

Fuzzy time series approaches not necessitating many of the limitations seen in classical time series approaches such as linearity, stationarity, and number of observations have increased the interest towards these approaches. Fuzzy time series concept first mentioned in the literature by Song and Chissom [1] was based on the fuzzy set theory of Zadeh [2]. Song and Chissom [3, 4] divided fuzzy time series into two groups, namely, time variant and time invariant. A vast majority of the studies in the literature are methods proposed for solving time invariant fuzzy time series. Because models are significantly effective on the forecasting performances during the determination of fuzzy relations stage, different approaches have been proposed in the literature. In the studies of Song and Chissom [1, 3, 4], relations are determined with complex matrix operations. In order to eliminate this complexity, a new first-order fuzzy time series model has been proposed where fuzzy logic group relation tables are used in Chen’s [5] study with simplified operations not necessitating complex matrix processes. This approach of Chen [5] is used in many studies due to its positive effect on forecasting performance. Therefore, Chen [6] developed a new approach by using the fuzzy logic relation tables also in high-order fuzzy time series models. Because the methods proposed in the studies of Chen [5, 6] necessitate obtainment of many fuzzy logic group relation tables, they require numerous operations. Thus, studies where fuzzy relations are determined with artificial neural networks are commonly seen. Some Studies where artificial neural networks are used for determining fuzzy relations may be listed as the studies of Huarng and Yu [7], Aladag et al. [8], Yu and Huarng [9], and Yolcu et al. [10].In majority of the studies in the literature, interval lengths are specified intuitively. In his study, Huarng [11] has proposed two separate approaches based on average and distribution to specify optimal interval length. The optimal interval lengths determined by the approach of Huarng [11] may be obtained to be very large values. Thus, Egrioglu et al. [12] proposed approaches based on the optimisation of interval length. Different from these studies, Huarng and Yu [13] proposed an approach based on ratio with interval length varying exponentially instead of determining a fixed interval length in the solution of first-order fuzzy time series. Determining the ratio in the study of Huarng and Yu [13] requires many complex calculations. Therefore, Yolcu et al. [14] proposed a new approach that improves the approach of Huarng and Yu [13], based on the optimisation of ratio.There are 3 most commonly used models in the analysis of single variable time series in classical time series approach. These are autoregressive (AR), moving averages (MAs), and mixed autoregressive moving averages (ARMAs) models. However, fuzzy time series methods developed in the literature focus on the AR model of the classical time series theory, without any study conducted on the utilisation of MA and ARMA models except Uslu et al. [15], Alpaslan et al. [16], and Aladag et al. [17]. These studies [15–17] on the other hand are methods that have been proposed for the solution of seasonal time series. No study has been made on the inclusion of error variable to the model for nonseasonal fuzzy time series with the exception of Egrioglu et al. [18] study. This study [18] is the first study developed as first order fuzzy ARMA type model based on particle swarm optimization for the solution of nonseasonal time series in the literature. All of the models in the literature have mentioned issues such as the use of universe of discourse partitioning, membership order, model order, and artificial intelligence approaches. However, the fuzzy time series models proposed in the literature only including AR variables which may lead to a model specification error. In many modelling of the real life time series, MA variables are also required. In this sense, use of only AR variables for the solution of fuzzy time series requiring also MA variables for modelling becomes insufficient regarding the forecasting performance.For the purpose of eliminating the adverse effects mentioned, a solution algorithm for a new first-order fuzzy ARMA(1,1) time series forecast model where fuzzy relations are determined based on fuzzy logic group relations has been proposed in this study. The logic for determining fuzzy relations in the proposed method is an approach similar to that of the study of Chen [5], aiming to show that the forecasting performance can be significantly improved when the model specification error in the method of Chen [5], accepted as a fundamental approach in literature, is eliminated. For many real life time series, performance can be increased through the use of high-order fuzzy time series models due to the ability to realise solution with more information. However, because the proposed fuzzy ARMA(1,1) model uses a second variable (error variable), it utilises more information to the extent it eliminates the model error, and although just a first order model, it occurs to have a better forecasting performance than that of high order fuzzy time series methods.In this paper, the time series which were used in application are examined according to long-range dependence. Time series can be classified short range or long-range dependence time series. Many time series have long-range dependence. The long-range dependence time series are forecasted different methods like autoregressive fractionally integrated moving average (ARFIMA). ARFIMA models have fractionally differencing parameter. First studies were concerned with estimation of fractional differencing parameter in fractional white noise processes.R/S statistic was proposed in Hurst [19]. The other important studies about fractional differenced processes are Li and Zhao [20, 21], Li [22], Stanley et al. [23], Werner [24], Beran [25], Ivanov et al. [26], Podobnik et al. [27], Zevallos and Palma [28], and Bhansali and Kokoszka [29].In the second section of the study, basic definitions regarding fuzzy time series have been provided. In the third section, the proposed fuzzy time series forecasting model has been defined and the solution algorithm has been provided. In the fourth section, the proposed method has been applied to Istanbul Stock Exchange (IMKB) national 100 index time series and gold prices in 2009 taken from Turkish Republic Central Bank (TCMB) website, comparing it to some other methods in the literature regarding forecasting performance. An in-depth comparison has been made in this section. In the fifth section, the study has been summed up by discussing the results obtained.


## 2. Definition of Fuzzy Time Series

Fuzzy time series concepts and definitions have been developed in accordance to the lagged variables of times series (AR, autoregressive) in all studies conducted in the literature. Main time series definitions developed using AR variables are listed below.Definition 1.
LetX(t)(t=…,0,1,2,…), a subset of real numbers, be the universe of discourse on which fuzzy sets fj(t) are defined. If F(t) is a collection of f1(t),f2(t),… then F(t) is called a fuzzy time series defined on X(t) [1, 3, 4].Definition 2.
Let us consider the fuzzy relation betweenR(t,t-1), F(t-1), and F(t). For any t value, if R(t,t-1) is independent from t, then R(t,t-1)=R(t-1,t-2). In this case, F(t) is called the time invariant fuzzy time series, while otherwise called as time variant fuzzy time series [1].Definition 3.
If theF(t) fuzzy time series is only affected by one lagged F(t-1) fuzzy time series, then the fuzzy relation between F(t-1) and F(t) is expressed as
(1)F(t-1)⟶F(t).
This is called as a first-order fuzzy time series forecasting model. Then this relation can be expressed as
(2)F(t)=F(t-1)∘R(t,t-1).
The “∘” operator in (2) had been determined as the max-min operator by Song and Chissom [1, 3, 4].Definition 4.
IfF(t) fuzzy time series is affected by the lagged fuzzy time series of F(t-1),F(t-2),…,F(t-p), then the fuzzy relation between F(t) fuzzy time series and F(t-1),F(t-2),…,F(t-p) fuzzy time series may be expressed as
(3)F(t-p),…,F(t-2),F(t-1)⟶F(t)
and is called the pth order fuzzy time series forecasting model [6].


## 3. The Proposed Method

Based on the definition of fuzzy AR(1) given in Definition3, the definition of the main fuzzy time series to represent the fuzzy ARMA(1,1) model is expressed as follows.Definition 5.
LetF(t) be a fuzzy time series and let ε(t) be the fuzzy error series obtained from F(t) fuzzy time series. If F(t) is affected by one lagged F(t-1) and one lagged ε(t-1) fuzzy time series, then the relationship can be expressed as
(4)F(t-1),ε(t-1)⟶F(t).
This is called as first-order fuzzy autoregressive moving averages (ARMA(1,1)) time series forecasting model [18].In this study, an algorithm has been proposed for solving the ARMA(1,1) fuzzy time series forecasting model defined in (4). In the algorithm proposed, initially the AR(1) fuzzy time series model defined in (1) is estimated. Later on, errors are calculated by taking the differences between the observed values of the times series and the forecasts obtained through the solution of fuzzy AR(1). By using these errors, the fuzzy ARMA(1,1) model defined in (4) is estimated. The algorithm of the proposed approach is given below.Algorithm 66.
The proposed method’s algorithm.Step 1 (the universe of discourse(U) and subintervals (ui,i=1,2,…b) are defined).
The beginning and the ending points of the universe of discourse for time series are determined. ThenU is divided into subintervals according to appropriate interval length. Definition of interval length is up to the researcher. It should not be forgotten that the interval length to be determined affects the number of subintervals. If the smallest value of the time series is taken as Xmin, the largest value as Xmax, and two arbitrary values as D1 and D2, the universal set may be defined as the closed interval of
(5)U=[Xmin-D1,Xmax+D2].ui subintervals determined for i=1,2,…b are the subintervals of the universal set U, which is defined as
(6)U={u1,u2,…,ub}.
For example, for Xmin=43 and Xmax=96 when U is selected as [40, 100] and interval length is selected as 10, subintervals are specified as u1=[40,50], u2=[50,60], u3=[60,70], u4=[70,80], u5=[80,90], and u6=[90,100].Step 2.
For the time series, fuzzy sets are defined according to the universal set(U) and the divisions of (ui). These fuzzy sets are expressed as
(7)Ai=fAi(u1)u1+fAi(u2)u2+⋯+fAi(ub)ub,fori=1,2,…,b.
For i=1,2,…b,
(8)fAi(ui)={1,k=i0.5,k=i-1,i+10,otherwise.
For example, according to (7) and (8), the fuzzy set of A3 can be expressed as 0/u1+0.5/u2+1/u3+0.5/u4+0/u5+0/u6.Step 3 (observations are fuzzified).
Subintervals (ui) where each observation occurs are defined. Then the fuzzy set Ai where the defined sub-interval has the highest membership value is determined. The fuzzy value of the observation is this Ai fuzzy set defined.Step 4.
For the purpose of determining fuzzy relations, fuzzy logic relations are identified and a fuzzy logic group relation table is formed.
For example, where the fuzzy logic relations are asA2→A3, A2→A3, A2→A5, according to Chen method [5], the fuzzy logic group relation for A2 fuzzy value is as A2→A3,A5. In the proposed method, this relation occurs to be as A2→A3,A3,A5. Thus, a little improvement has been realised in the fuzzy AR(1) model of Chen [5] with the proposed method.Step 5 (fuzzy forecasts are obtained).
AsF(t-1)=Ai and F(t)=Aj, 3 possible situations regarding forecast obtainment are as follows.
Situation 1. If a relation of Ai→Aj,…,Aj is valid on the fuzzy group relation table where Ai affects only a pieces of Aj, then the fuzzy forecast is Aj. For example, if the group relation for A1 is as A1→A2 that this relation is repeated a few times in the time series, then the fuzzy forecast is specified as A2.
Situation 2. If a relation of Ai→Aj,…,Aj,Ak,…,Ak,Al,…,Al is valid on the fuzzy group relation table where Ai affects a pieces of Aj, bpieces Ak and c pieces Al, then the fuzzy forecast is Aj,…,Aj,Ak,…,Ak,Al,…,Al, comprising of a+b+c pieces of fuzzy values. For example; if the group relation for A2 is as A2→A2,A3,A3,A5,A5,A5, then the fuzzy forecast is specified as A2,A3,A3,A5,A5,A5.
Situation   3. If Ai→empty on the fuzzy group relation table, the fuzzy forecast occurs to be Ai. For example, if the group relation for A3 is A3→empty, then the fuzzy forecast is A3.Step 6 (defuzzification process is executed).
In this step, centralisation method is used. When the fuzzy forecast for Situation 1 and Situation 3 defined in Step5 is Aj, then the defuzzy forecast should be the middle point of the uj sub-interval that has the highest membership value within the fuzzy set Aj. For Situation 2, the defuzzy forecast is calculated with the weighted average formula below, by using the middle points (mj,mk,…,ml) of the uj,uk,…,ul intervals that have the highest membership value of each of the Aj,Ak,…,Al fuzzy sets. Consider
(9)x^(t)=a×mj+b×mk+c×mla+b+c.
On an example time series, when solutions are made according to the proposed method from Step 1 to Step 6 as per the determined subintervals of u1=[40,50], u2=[50,60], and u3=[60,70], the solutions displayed in Table 1 are obtained.Table 1
An example to the fuzzy AR(1) solution of the proposed method.Step 7.
Errors are calculated by taking the differences between the observed time series values and the defuzzified forecast values obtained in Step6. Real values of the time series are x(t) and the defuzzified forecast values obtained in Step 6 are x^(t); the error series e(t) is calculated as follows:
(10)e(t)=x(t)-x^(t).Step 8.
For the errors, the universe of discourse set is defined as(V) and subintervals are defined as (vi,i=1,2,…b).
The same partition of the universe of discourse processes done in Step1 is made for the error series.Step 9.
Fuzzy sets based on the universal set (V) and partitions (vi) are defined for the errors. The fuzzy sets are expressed as
(11)Bj=fBj(v1)v1+fBj(v2)v2+⋯+fBj(vc)vc,forj=1,2,…,c.
For j=1,2,…c,
(12)fBj(vj)={1,k=j0.5,k=j-1,j+10,otherwise.
For example, according to (11) and (12), the fuzzy set of B3 can be specified as 0/v1+0.5/v2+1/v3+0.5/v4+0.5/v5.Step 10 (error seriese(t) is fuzzified).
Subintervals (vj) for each observation are determined. Then the fuzzy set Bj where the determined sub-interval has the highest membership value is defined. The fuzzy value of the observation is this fuzzy set Bj.Step 11 (fuzzy relations are determined and fuzzy logic group relation table is formed).
The fuzzy valueAi of the fuzzy relations time series and the fuzzy value Bj of the error series are determined by taking the fuzzy values into consideration together. For the one lagged fuzzy value of the tth observation being F(t-1)=Ai, one lagged error value being ε(t-1)=Bj, and the fuzzy value being F(t)=Ak, the fuzzy relation given in (4) occurs to be (Ai,Bj)→Ak. Thus, the fuzzy values are formed of (Ai,Bj) ordered pairs, and a relation in the manner of one lagged time series and error affecting the time series is mentionable. For example, when the fuzzy logic relations are as (A2,B3)→A2, (A2,B3)→A2, (A2,B3)→A4, the fuzzy logic group relation for (A2,B3) fuzzy value occurs to be (A2,B3)→A2,A2,A4.Step 12 (fuzzy forecasts are obtained).
AsF(t-1)=Ai ve ε(t-1)=Bj and F(t)=Ak, 3 possible situations regarding forecast obtainment are as follows.
Situation 1. If a relation of (Ai,Bj)→Ak,…,Ak is valid on the fuzzy group relation table where (Ai,Bj) affects only a pieces of Ak, then the fuzzy forecast is Ak. For example, if the group relation for (A1,B2) is as (A1,B2)→A2 that this relation is repeated within a few times in the time series, then the fuzzy forecast is determined as A2.
Situation 2. If a relation of (Ai,Bj)→Ak,…,Ak,Al,…,Al,Am,…,Am is valid on the fuzzy group relation table where (Ai,Bj) affects a pieces of Ak, b pieces Al and c pieces Am, then the fuzzy forecast is  Ak,…,Ak,Al,…,Al,Am,…,Am, comprising of a+b+c pieces of fuzzy values. For example, if the group relation for (A1,B2) is as (A1,B2)→A2,A3,A3,A5,A5,A5, then the fuzzy forecast is determined as A2,A3,A3,A5,A5,A5.
Situation   3. If (Ai,Bj)→empty on the fuzzy group relation table, the fuzzy forecast occurs to be Ai. For example, if the group relation for (A1,B2) is (A1,B2)→empty, then the fuzzy forecast is A1.Step 13 (defuzzification process is made).
In this step, centralisation method is used. When the fuzzy forecast for Situation 1 and Situation 3 defined in Step5 is Aj, then the defuzzy forecast should be the middle point of the uj sub-interval that has the highest membership value within the fuzzy set Aj. For Situation 2, the defuzzy forecast is calculated with the weighted average formula given in (9).
On an example time series, when solutions are made according to the proposed method as per the determined time series subintervals ofu1=[40,50], u2=[50,60], and u3=[60,70] and error series subintervals of v1=[-5,0], v2=[0,5], and v3=[5,10], solutions displayed in Table 2 are obtained.Table 2
An application of the proposed method on an example time series.


## 4. Application

The performance indicators of the root mean square error (RMSE), mean average percentage error (MAPE), and direction accuracy (DA) values used for comparison of the results obtained are as follows:(13)RMSE=∑t=1n(Yt-Y^t)2n,MAPE=1n∑t=1n|Yt-Y^tYt|,DA=1n-1∑t=1n-1{1,(Yt+1-Yt)(Y^t+1-Yt)>0,0,otherwise.
Data is divided into two within the applications, assigning the first part as training set and the second part as test set obtained through taking the last observations into consideration of which number was predetermined. By looking up the fuzzy relation table obtained for the training set as per the Steps 5 and 12 of the proposed method, the fuzzy forecast of the test set and from there the defuzzy forecasts of the test set are calculated by utilizing Steps 6 and 13. After, the RMSE, MAPE, and DA values for the test set are calculated. Therefore, the future performances of the methods are determined. As the forecasts with the lowest RMSE value calculated for the test set provide the best result of the used method, the future performances of the forecasts are obtained along with the aid of MAPE and DA values.Solution of vast majority of some fuzzy forecasting methods in the literature is realised according to the specified number of fuzzy sets, and some are realised according to interval lengths. For the purpose of maintaining consistency during the comparison of forecast performances, the interval length to be used at fuzzification stage is determined as to have the number of fuzzy sets 5 as the lowest and 35 as the highest for each application data and for all methods. Therefore, in case the universal set division is realised in accordance to interval length, the interval lengths to be tried have been specified by calculation with the formula below:(14)IntervalLength=max(data)-min(data)NumberofFuzzySets.
The operations below have been realised when making solution via the proposed method.(i)
The lastk number of data has been specified as test set, aiming to increase the future performances.(ii)
During the fuzzification of time series stage of the proposed method, different time series interval lengths for the division of universal setU have been tried. Data have been solved from Step 1 to Step 6 according to these intervals lengths. And so, a lot of forecasts have been obtained. The test set forecast with the smallest RMSE value among these forecasts has been determined as the best result of the fuzzy AR(1) model.(iii)
The error value of the first observation of data has been assumed as 0, while the error values of other observations have been calculated with the formula (9) by using the training set and test set forecasts obtained through the best result of the fuzzy AR(1) model. Thus the error series has been obtained.(iv)
Different time series interval lengths and different error series interval lengths have been tried by solving data from Step1 to Step 13. Among these trials, the test set forecast with the smallest RMSE value has been determined as the best result of the fuzzy ARMA(1,1) model.For the purpose of comparing the proposed method with the other fuzzy time series methods in the literature, 2 different data sets comprising of less observations (smaller sample size) and more observations (larger sample size) have been used. One of these data sets is the IMKB time series seen in Figure1 comprising of 53 observations between the dates of 01.10.2010 and 23.12.2010. The second data set is the gold prices time series seen in Figure 2 comprising of 248 observations received from Turkish Republic Central Bank (TCMB) website between the dates of 02.01.2009 and 31.12.2009.Figure 1
Graph of IMKB time series.Figure 2
Graph of gold prices.In the solution of the IMKB data given in Figure1 via the methods in the literature by taking the last 7 observations and the last 15 observations as test set, one has the following.(i)
During the fuzzification stage of the Song and Chissom [1] first-order fuzzy time series method, among the 31 different results obtained from increasing the fuzzy set number between 5 and 35, the test set forecast that has the minimum RMSE value has been determined as the best result of the method from this results.(ii)
For each of the 2nd-, 3rd-, 4th-, and 5th-order models of Chen [5] first-order fuzzy time series forecast method, Chen [6] high order fuzzy time series forecast method, and Aladag et al. [8] high order fuzzy time series forecast method, the RMSE values have been found for different intervals lengths being increased 100 units between 300 and 2300. The test set forecasts that have the minimum RMSE values among these 21 trials have been determined as the best results of the methods.(iii)
By using the optimal interval lengths calculated with the distribution-based approach of Huarng [11] and the average-based approaches, solution has been executed as per the first-order fuzzy time series forecast method of Chen [5]. Thus, the best results of the test set from the distribution based approach and the average-based approaches have been obtained via a single trial.(iv)
In the application of the ratio-based approach of Huarng and Yu [13], alpha parameter has been taken as 0.50, obtaining the best result for the test set of the method in one trial.In the solution of the IMKB time series seen in Figure1 via the proposed method by taking the number of test sets 7 and 15, one has the following.(i)
The division of universal setU have been taken different values as increasing the interval length between 300 and 2300 by 100 units and different forecasts have been obtained by solving from Step 1 to Step 6. The test set forecast with the smallest RMSE value among these forecasts has been determined as the best result of the fuzzy AR(1) model. The best fuzzy AR(1) results have been obtained when the interval length is 300 for 7 as the number of test sets and when the interval length is 900 for 15 as the number of test sets.(ii)
The error value of the first observation of data has been assumed as 0, while the error values of other observations have been calculated via the formula (9) by utilising data and forecasts obtained through the best result of the fuzzy AR(1) model. Thus the error series has been obtained for 53 observations.(iii)
Different trials have been made by increasing the interval length between 300 and 2300 by 100 units for the time series and by increasing the interval length between 300 and 2100 by 100 units for the error series by solving from Step1 to Step 13. Among these trials, the test set forecast with the smallest RMSE value has been determined as the best result of the fuzzy ARMA(1,1) model.During the application of the proposed method and the methods in the literature on IMKB time series, the parameters with which the forecasts with the best test set performance for 7 and 15 numbers of test sets occurred to be(i)
for the application of Song and Chissom [1] method, when the number of fuzzy sets is 9 for 7 and 20 for 15,(ii)
for the application of Chen [5] method, when the interval length is 300 for 7 and 900 for 15,(iii)
for the application of distribution-based Huarng [11] approach, when the interval length is 1000, and for the application of average based approach, when the interval length is 200,(iv)
for the application of ratio-based Huarng and Yu [13] approach, when the sample percentile alpha = 0.5,(v)
for the application of Chen [6] method, when the interval length is 2200 in 3rd-order model for 7 and 1400 in 2nd-order model for 15,(vi)
for the application of Aladag et al. [8] method, when the interval length is 600 on 2nd-degree model and unit number of artificial neural network hidden layers is 5 for 7, and when the interval length is 1500 on 2nd-degree model and unit number of artificial neural network hidden layers is 6 for 15,(vii)
for the application of the proposed fuzzy ARMA(1,1) method, when the interval length of time series is 2200 and the interval length of error series is 1400 for 7, and when the interval length of time series is 1400 and the interval length of error series is 400 for 15.Best forecasts and forecast performances of all methods in result of IMKB time series solution for 7 observation test set are summarised in Table3.Table 3
The best results obtained for 7-observation test set of IMKB data.
Date
Test set
Song and Chissom [1]
Chen [5]
Chen [6]
Huarng [11] distribution-based method
Huarng [11] average-based method
Huarng and Yu [13] rational-based method
Aladag et al. [8]
Proposed method*
15.12.2010
65499
65356
66550
65900.0
67400
66500
66784.7
67300
66266.7
16.12.2010
64429
65356
66250
65900.0
65400
66300
66178.0
64900
63700.0
17.12.2010
63524
65975
64450
64800.0
65900
64500
65878.7
64900
63700.0
20.12.2010
63502
64737
63550
64066.7
64900
63500
63514.3
64300
63700.0
21.12.2010
64820
64737
63550
63700.0
64900
63500
63514.3
63700
65900.0
22.12.2010
65440
65975
65800
64800.0
65900
65500
65878.7
65500
65900.0
23.12.2010
66219
65356
66250
65900.0
65400
66300
66178.0
66700
66266.7
RMSE
1161.91
1001.70
928.70
1365.14
1014.73
1317.77
1034.06
606.07
MAPE
0.01387
0.01217
0.01283
0.01773
0.01175
0.01593
0.01350
0.00762
DA
0.50000
0.50000
0.33333
0.50000
0.50000
0.66667
0.66667
0.83333
*The best situation.When Table3 is analyzed, it is seen in result of the solution of IMKB time series for 7 observation test set that the proposed method produced the best forecasting performance with a minimum RMSE value of 606.07, minimum MAPE value of 0.762%, and maximum direction accuracy of 83.33%. The graphs of the last 7 observations of IMKB time series along with the 7-observation test set forecasts obtained with the proposed method are shown together in Figure 3.Figure 3
The graphs of the 7-observation test set of IMKB data and the forecasts of the test set obtained with the proposed method.Best forecasts and forecast performances of all methods in result of IMKB time series solution for 15-observation test set are summarised in Table4.Table 4
The best results obtained for 15-observation test set of IMKB data.
Date
Test set
Song and Chissom [1]
Chen [5]
Chen [6]
Huarng [11] distribution-based method
Huarng [11] average-based method
Huarng and Yu [13] rational-based method
Aladag et al. [8]
Proposed method*
01.12.2010
66156
65695.5
66250
64833.3
65900.0
65300
66328.8
65650
66000
02.12.2010
66939
66438.2
65800
66700.0
65566.7
64100
66164.2
67150
67633
03.12.2010
66860
67088.0
67450
66700.0
67233.3
66700
67143.8
67150
66700
08.12.2010
67705
67088.0
67450
66700.0
67233.3
66700
67143.8
67150
66700
09.12.2010
65914
66252.5
66250
67633.3
65900.0
67700
66328.8
67150
68100
10.12.2010
64759
65695.5
65800
66233.3
65566.7
65900
65791.4
64150
65300
13.12.2010
66380
65695.5
65350
66700.0
65900.0
64700
65258.3
65650
65300
14.12.2010
66510
66438.2
65800
66700.0
65566.7
67100
66164.2
65650
66700
15.12.2010
65499
66438.2
65800
66700.0
67233.3
66500
66164.2
67150
66700
16.12.2010
64429
65695.5
66250
65766.7
65566.7
66300
66328.8
64150
65300
17.12.2010
63524
65695.5
65350
64366.7
65900.0
64500
65258.3
65650
63900
20.12.2010
63502
65138.5
63550
63900.0
64900.0
63500
63684.8
65650
63900
21.12.2010
64820
65138.5
63550
63900.0
64900.0
63500
63684.8
65650
63900
22.12.2010
65440
65695.5
65350
64833.3
65900.0
65500
65258.3
65650
65300
23.12.2010
66219
65695.5
66250
66700.0
65566.7
66300
66328.8
65650
66000
RMSE
919.47
925.42
954.17
1052.90
1283.18
896.96
1060.12
865.28
MAPE
0.01124
0.01081
0.01245
0.01285
0.01558
0.01085
0.01312
0.01029
DA
0.71429
0.57143
0.64286
0.50000
0.57143
0.64286
0.64286
0.71429
*The best result.When Table4 is analyzed, it is seen that the proposed method produced the best forecasting performance with a minimum RMSE value of 865.28, minimum MAPE value of 1.029%, and maximum direction accuracy of 71.43% in result of the solution of IMKB time series for 15-observation test set. The graphs of the last 15 observations of IMKB time series along with the 15-observation test set forecasts obtained with the proposed method are shown together in Figure 4.Figure 4
The graphs of the 15-observation test set of IMKB data and forecasts of the test set obtained with the proposed method.In result of the solutions of IMKB time series, it has been observed that the proposed method significantly increased the future forecasting performance compared to other methods. Also in the graphs within Figures3 and 4, the results of the proposed method are seen to be considerably similar to the test set values.In the solution of the gold prices data given in Figure2 via the methods in the literature by taking the last 30 observations and the last 45 observations as test set, one has the following.(i)
Gold prices solution of Song and Chissom [1], Huarng [11], and Huarng and Yu [13] methods has been conducted just as previously done on the abovementioned IMKB time series.(ii)
For each of the 2nd-, 3rd-, 4th-, and 5th-order models of Chen [5] first-order fuzzy time series forecast method, Chen [6] high-order fuzzy time series forecast method, and Aladag et al. [8] high order fuzzy time series forecast method, the RMSE values have been found for different lengths being increased 100 units between 500 and 3500. The test set forecasts that have the minimum RMSE values among these 31 trials have been determined as the best results of the methods.In the solution of the gold prices data given in Figure2 via the proposed method by taking the last 30 observations and the last 45 observations as test set, one has the following.(i)
During the fuzzification of time series of the proposed method, interval lengths ofU have been tried as increasing the interval length between 500 and 3500 by 100 units by solving from Steps 1 and 6. The test set forecast with the smallest RMSE value among these forecasts has been determined as the best result of the fuzzy AR(1) model.(ii)
The error series have been obtained via the formula (9) for 248 observations with the same calculation made previously in the application of IMKB data.(iii)
Different trials have been made by increasing the interval length between 500 and 3500 by 100 units for the time series and by increasing the interval length between 100 and 1100 by 50 units for the error series by solving from Step1 to Step 13. Among these trials, the test set forecast with the smallest RMSE value has been determined as the best result of the fuzzy ARMA(1,1) model.During the application of the proposed method and the methods in the literature on gold prices time series, the parameters with which the forecasts with the best test set performance for 30 and 45 test sets occurred to be the following:(i)
for the application of Song and Chissom [1] method, when the number of fuzzy sets is 10 for both 30 and 45 test sets,(ii)
for the application of Chen [5] method, when the interval length is 600 for 30 and 1700 for 45,(iii)
for the application of distribution-based Huarng approach [11], when the interval length is 400, and for the application of average based approach, when the interval length is 200,(iv)
for the application of ratio-based Huarng and Yu [13] approach, when the sample percentile alpha = 0.5,(v)
for the application of Chen method [6], when the interval length is 1900 in 5th order model for 30 and 800 in 3rd-order model for 45,(vi)
for the application of Aladag et al. [8], when the interval length is 600 on 5th-degree model and unit number of artificial neural network hidden layers is 5 for 30, and when the interval length is 800 on 3rd degree model and unit number of artificial neural network hidden layers is 4 for 45,(vii)
for the application of the proposed fuzzy ARMA(1,1) method, when the interval length of time series is 800 and the interval length of error series is 2500 for 30, and when the interval length of time series is 900 and the interval length of error series is 1000 for 45.Best forecasts and forecast performances of all methods in result of gold prices time series solution for 30-observation test set are summarised in Table5.Table 5
The best results obtained for 30-observation test set of gold prices.
Date
Test set
Song and Chissom [1]
Chen [5]
Chen [6]
Huarng  [11] distribution-based method
Huarng [11] average-based method
Huarng and Yu [13] rational-based method
Aladag et al. [8]
Proposed method*
17.11.2009
53935
53157
53300
53000.0
53200
53200
53356.4
53300
53400
18.11.2009
54550
53157
53900
53666.7
54000
53900
53894.0
53900
54200
19.11.2009
54495
53157
52400
54066.7
52400
51500
51715.7
54500
54200
20.11.2009
54830
53157
52400
54200.0
52400
51500
51715.7
53900
54200
23.11.2009
55950
53157
55100
54600.0
54800
54900
55017.3
54500
55000
24.11.2009
56285
52294
55700
55266.7
56000
55900
55779.2
55100
55800
25.11.2009
56430
52294
56300
56066.7
56400
56300
56164.1
55700
56600
01.12.2009
57635
52294
56300
56466.7
56400
56500
56551.6
55100
56600
02.12.2009
58330
53157
57500
57000.0
57600
57700
57730.3
55700
57400
03.12.2009
58150
53157
58100
57666.7
58400
58300
58529.8
56300
58200
04.12.2009
56630
53157
58100
58066.7
58000
58100
58128.7
56300
58200
07.12.2009
54820
53157
56900
57400.0
56800
56700
56551.6
55100
56600
08.12.2009
55660
53157
55100
56066.7
54800
54900
51715.7
55100
55000
09.12.2009
55110
52294
55700
55666.7
55600
55700
55779.2
55100
55800
10.12.2009
54180
52294
55100
55266.7
55200
55100
55017.3
54500
55000
11.12.2009
54580
53157
53900
54733.3
54000
54100
53157.9
53900
54200
14.12.2009
54190
53157
52400
54333.3
52400
51500
51715.7
54500
54200
15.12.2009
54120
53157
53900
54200.0
54000
54100
53157.9
53300
54200
16.12.2009
54855
53157
53900
54200.0
54000
54100
53157.9
53900
54200
17.12.2009
54430
53157
55100
54600.0
54800
54900
55017.3
54500
55000
18.12.2009
53750
53157
52400
54466.7
52400
51500
53157.9
54500
54200
21.12.2009
54570
53157
53900
53933.3
53200
53700
53894.0
53300
53400
22.12.2009
53400
53157
52400
53933.3
52400
51500
51715.7
54500
54200
23.12.2009
52990
53157
53300
53666.7
53200
53200
53356.4
53300
53400
24.12.2009
53575
52294
53150
53133.3
53200
53200
53524.7
52700
53400
25.12.2009
53450
53157
53300
53133.3
53200
53200
53356.4
53300
53400
28.12.2009
53795
53157
53300
53266.7
53200
53200
53356.4
53300
53400
29.12.2009
53515
53157
53900
53400.0
53200
53700
53894.0
53900
53400
30.12.2009
53095
53157
53300
53400.0
53200
53200
53356.4
52700
53400
31.12.2009
52920
52294
53300
53400.0
53600
53100
53524.7
53300
53400
RMSE
2410.96
1031.12
857.34
1045.23
1288.80
1412.66
1003.50
707.71
MAPE
0.03339
0.01512
0.01245
0.01530
0.01713
0.01935
0.01382
0.01028
DA
0.55172
0.55172
0.51724
0.55172
0.62069
0.55172
0.48276
0.62069
*The best result.When Table5 is observed, it is seen in result of the solution of gold prices time series for 30 observation test set that the proposed method produced the best forecasting performance with a minimum RMSE value of 707.71, minimum MAPE value of 1.028%, and maximum direction accuracy of 62.07%. The graphs of the last 30 observations of gold prices time series along with the 30-observation test set forecasts obtained with the proposed method are shown together in Figure 5.Figure 5
The graphs of the 30-observation test set of gold prices and forecasts of the test set obtained with the proposed method.Best forecasts and forecast performances of all methods in result of gold prices time series solution for 45-observation test set are summarised in Table6. Furthermore, the graphs of the last 45 observations of gold prices time series along with the 45-observation test set forecasts obtained with the proposed method are shown together in Figure 6.Table 6
The best results obtained for 45-observation test set of gold prices.
Date
Test set
Song and Chissom [1]
Chen [5]
Chen [6]
Huarng [11] distribution-based method
Huarng [11] average-based method
Huarng and Yu [13] rational-based method
Aladag et al. [8]
Proposed method*
17.10.2009
50162
50568
50350
49933.3
50800.0
50420.0
50829.1
50200
50450
18.10.2009
50355
50568
50350
50066.7
50800.0
51100.0
50829.1
50200
50450
19.10.2009
50020
50568
50350
50120.0
50000.0
49900.0
50051.5
50200
50450
20.10.2009
50030
50568
50350
51000.0
50800.0
51100.0
50829.1
50200
50450
23.11.2009
50700
50568
50350
51000.0
50800.0
51100.0
50829.1
50200
50450
24.11.2009
51230
50568
50350
51800.0
51066.7
51500.0
50824.1
51000
50450
25.11.2009
52490
50568
52050
50866.7
51200.0
51500.0
51128.5
51000
52250
01.11.2009
52150
52294
52050
51800.0
52666.7
54300.0
54499.6
52600
52550
02.11.2009
52125
52294
52050
51933.3
50600.0
52100.0
51916.0
51800
52550
03.11.2009
52500
52294
52050
51933.3
50600.0
52100.0
51916.0
51800
52550
04.11.2009
52165
52294
52050
52200.0
52666.7
54300.0
54499.6
52600
52550
07.11.2009
52560
52294
52050
52066.7
50600.0
52100.0
51916.0
51800
52550
08.11.2009
52972
52294
52050
52333.3
52666.7
54300.0
54499.6
52600
52550
09.11.2009
52800
52294
52900
52466.7
52800.0
52900.0
52867.3
52600
52250
10.11.2009
53500
52294
52050
54200.0
52800.0
52900.0
52867.3
52600
53150
17.11.2009
53935
53157
52900
53000.0
53200.0
53200.0
53205.5
53400
54950
18.11.2009
54550
53157
52900
53666.7
54000.0
53900.0
53840.7
53400
54050
19.11.2009
54495
53157
52900
54066.7
52400.0
51500.0
52393.3
54200
54950
20.11.2009
54830
53157
52900
54200.0
52400.0
51500.0
52393.3
54200
54050
23.11.2009
55950
53157
55450
54600.0
54800.0
54900.0
54832.0
54200
54950
24.11.2009
56285
52294
55450
55266.7
56000.0
55900.0
55841.6
54200
55850
25.11.2009
56430
52294
55450
56066.7
56400.0
56300.0
56182.2
55000
55850
01.12.2009
57635
52294
57150
56466.7
56400.0
56500.0
56524.9
55000
56750
02.12.2009
58330
53157
57150
57000.0
57600.0
57700.0
57565.6
55000
57650
03.12.2009
58150
53157
58850
57666.7
58400.0
58300.0
58270.1
55000
58550
04.12.2009
56630
53157
58850
58066.7
58000.0
58100.0
58270.1
55800
58550
07.12.2009
54820
53157
57150
57400.0
56800.0
56700.0
56524.9
55800
56750
08.12.2009
55660
53157
55450
56066.7
54800.0
54900.0
54832.0
55000
54950
09.12.2009
55110
52294
55450
55666.7
55600.0
55700.0
55503.0
54200
55850
10.12.2009
54180
52294
55450
55266.7
55200.0
55100.0
55166.5
54200
54950
11.12.2009
54580
53157
52900
54733.3
54000.0
54100.0
54169.1
54200
54050
14.12.2009
54190
53157
52900
54333.3
52400.0
51500.0
52393.3
54200
54950
15.12.2009
54120
53157
52900
54200.0
54000.0
54100.0
54169.1
54200
54050
16.12.2009
54855
53157
52900
54200.0
54000.0
54100.0
54169.1
54200
54050
17.12.2009
54430
53157
55450
54600.0
54800.0
54900.0
54832.0
54200
54950
18.12.2009
53750
53157
52900
54466.7
52400.0
51500.0
52393.3
54200
54050
21.12.2009
54570
53157
52900
53933.3
53200.0
53700.0
53840.7
53400
54050
22.12.2009
53400
53157
52900
53933.3
52400.0
51500.0
52393.3
53400
54950
23.12.2009
52990
53157
52900
53666.7
53200.0
53200.0
53205.5
53400
53150
24.12.2009
53575
52294
52900
53133.3
52800.0
52900.0
52867.3
52600
53150
25.12.2009
53450
53157
52900
53133.3
53200.0
53200.0
53205.5
53400
52250
28.12.2009
53795
53157
52900
53266.7
53200.0
53200.0
53205.5
53400
53150
29.12.2009
53515
53157
52900
53400.0
53200.0
53700.0
53840.7
53400
54050
30.12.2009
53095
53157
52900
53400.0
53200.0
53200.0
53205.5
53400
53150
31.12.2009
52920
52294
52900
53400.0
53600.0
53100.0
53514.3
53400
53150
RMSE
2009.32
1028.39
787.71
1022.13
1200.24
1041.74
1068.71
719.69
MAPE
0.02556
0.01499
0.01146
0.01499
0.01628
0.01517
0.01328
0.01072
DA
0.54545
0.52273
0.54545
0.52273
0.61364
0.59091
0.54545
0.50000
*The best result.Figure 6
The graphs of the 45-observation test set of gold prices and forecasts of the test set obtained with the proposed method.When Table6 is evaluated, it is seen in result of the solution of gold prices time series for 45 observation test set that the proposed method produced the best forecasting performance with a minimum RMSE value of 719.69, minimum MAPE value of 1.072%.In result of the solutions of gold prices time series, it has been observed that the proposed method significantly increased the future forecasting performance compared to other methods. Also in the graphs within Figures5 and 6, the results of the proposed method are seen to be considerably similar to the test set values.


## 5. Discussion and Conclusion

MA variables are not included in the fuzzy time series forecast models proposed in the literature whereas real life time series are also influenced from MA variables in addition to AR variables. Therefore, redefining fuzzy times series methods as models including also MA variables are a more realistic act. In this study, a solution algorithm for a new first-order fuzzy ARMA(1,1) time series forecast model containing not only AR but also MA variables is proposed based on group relation tables. The method proposed is a basic algorithm similar to Chen [5] approach, aiming at eliminating the model specification error formed due to the exclusion of MA variables. In conclusion of the applications, it has been observed that the proposed method has higher forecasting performance than many of the fuzzy time series forecasting methods commonly used in the literature. It is an important finding that although the proposed method is a basic method based on group relation tables, it may have a higher forecasting performance even than the high order fuzzy time series methods based on artificial neural networks. Therefore, it is obvious that forecasting performance is going to increase significantly when fuzzy ARMA models are developed where fuzzy relations are specified with artificial neural networks and artificial intelligence methods or where membership values are used for specification of fuzzy relations. Thus, the proposed method may be provided with a more systematic structure and higher forecasting performance with improvements that may be done on various stages of the method during future studies. Moreover, there is no linear model assumption in the proposed fuzzy time series method. Thus, the proposed method and other fuzzy time series methods can be applied to nonlinear time series. In the application, proposed method is applied to short range dependent time series. In the future studies, it will be researched about the performance of the proposed method for long range dependent time series.



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*Source: 769125-2013-10-02.xml*