High-order fuzzy-neuro expert system for time series forecasting

Knowledge-Based Systems 46 (2013) 12–21

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Knowledge-Based Systems

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High-order fuzzy-neuro expert system for time series forecasting

Pritpal Singh ⇑, Bhogeswar BorahDepartment of Computer Science and Engineering, Tezpur University, Tezpur 784 028, Assam, India

a r t i c l e i n f o

Article history:Received 18 July 2012Received in revised form 6 December 2012Accepted 25 January 2013Available online 21 March 2013

Keywords:Fuzzy time seriesHigh-orderTemperatureStock exchangeIntervalFuzzy logical relationArtificial neural network

0950-7051/$ - see front matter � 2013 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.knosys.2013.01.030

⇑ Corresponding author.E-mail addresses: [email protected] (P. Singh),

a b s t r a c t

In this article, we present a new model based on hybridization of fuzzy time series theory with artificialneural network (ANN). In fuzzy time series models, lengths of intervals always affect the results of fore-casting. So, for creating the effective lengths of intervals of the historical time series data set, a new ‘‘Re-Partitioning Discretization (RPD)’’ approach is introduced in the proposed model. Many researchers sug-gest that high-order fuzzy relationships improve the forecasting accuracy of the models. Therefore, in thisstudy, we use the high-order fuzzy relationships in order to obtain more accurate forecasting results.Most of the fuzzy time series models use the current state’s fuzzified values to obtain the forecastingresults. The utilization of current state’s fuzzified values (right hand side fuzzy relations) for predictiondegrades the predictive skill of the fuzzy time series models, because predicted values lie within the sam-ple. Therefore, for advance forecasting of time series, previous state’s fuzzified values (left hand side offuzzy relations) are employed in the proposed model. To defuzzify these fuzzified time series values,an ANN based architecture is developed, and incorporated in the proposed model. The daily temperaturedata set of Taipei, China is used to evaluate the performance of the model. The proposed model is alsovalidated by forecasting the stock exchange price in advance. The performance of the model is evaluatedwith various statistical parameters, which signify the efficiency of the model.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Advance prediction of events like temperature, rainfall, stockprice, population growth, and economy growth, are major scien-tific issues in the area of forecasting. Forecasting of all these eventsare tedious tasks because of their dynamic nature. Forecasting ofall these events with 100% accuracy may not be possible, but theforecasting accuracy and the speed of forecasting process can beimproved. So, in this article, we present a novel forecasting model,which is developed by hybridizing fuzzy time series theory withartificial neural network (ANN). The main aim of designing sucha hybridized model is explained next.

For fuzzification of time series data set, determination oflengths of intervals of the historical time series data set is veryimportant. In most of the fuzzy time series models [1–5], thelengths of the intervals were kept the same. No specific reason ismentioned for using the fix lengths of intervals. Huarng [6] showsthat the lengths of intervals always affect the results of forecasting.So, for creating the effective lengths of intervals, a new ‘‘Re-Parti-tioning Discretization (RPD)’’ approach is incorporated in the pro-posed model.

After generating the intervals, time series data set is fuzzifiedbased on the fuzzy time series theory. Most of the previous fuzzy

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[email protected] (B. Borah).

time series models [1,7,2–4,8] use first-order fuzzy relationshipsto get the forecasting results. Many researchers show that high-or-der fuzzy relationships improve the forecasting accuracy of themodels [9–14]. Therefore, in this work, we employ the high-orderfuzzy relationships for obtaining the forecasting results.

Song and Chissom [1] adopted the following method to forecastenrollments of the University of Alabama:

YðtÞ ¼ Yðt � 1Þ � R; ð1Þ

where Y(t � 1) is the fuzzified enrollment of year (t � 1), Y(t) is theforecasted enrollment of year ‘‘t’’ represented by fuzzy set, ‘‘�’’ is themax–min composition operator, and ‘‘R’’ is the union of fuzzy rela-tions. This method takes much time to compute the union of fuzzyrelations R, especially when the number of fuzzy relations is morein (1) [15,16]. In 1996, Chen [3] used simplified arithmetic opera-tions for defuzzification operation by avoiding this complicatedmax–min operations and their method produced better results thanSong and Chissom models [1,7,2]. Most of the existing fuzzy timeseries models use Chen’s defuzzification method [3] in order to ob-tain the forecasting results. However, forecasting accuracy of thesemodels are not good enough. Also, previous fuzzy time series mod-els use the current state’s fuzzified values for forecasting. This ap-proach, no doubt, improves the forecasting accuracy, but itdegrades the predictive skill of the fuzzy time series models, be-cause predicted values lie within the sample. So, for obtaining theforecasting results out of sample (i.e., in advance), we use the

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P. Singh, B. Borah / Knowledge-Based Systems 46 (2013) 12–21 13

previous state’s fuzzified values (left hand side of fuzzy relations) inthis model. To defuzzify these fuzzified values, an ANN based archi-tecture is developed, and incorporated in this model. So, we haveentitled this model as ‘‘High-order fuzzy-neuro time series forecast-ing model’’. The proposed model has the advantage that it can pro-duce good forecasting results. We demonstrate the application ofthe proposed model using the following two real-world data set:

1. Daily average temperature data set of Taipei, China.2. Daily stock exchange price data set of Bombay Stock Exchange

(BSE), India.

The rest of this paper is organized as follows. In Section 2, wepresent related works for fuzzy time series models. In Section 3,we review the theory of fuzzy set with an overview of fuzzy timeseries. In Section 4, we give an overview of ANN along with itsapplication in the proposed model. In Section 5, description ofdata set is provided. Section 6 shows the application of a new ap-proach to find the length of intervals in the universe of discourse.The architecture of the proposed model and its training phasesare presented in Sections 7 and 8 respectively. The performanceof the model is assessed with various statistical parameters,which are discussed in Section 9. Empirical analysis for forecast-ing the daily temperature is presented in Section 10. Section 11shows the application of the proposed model for forecastingthe stock exchange price. The conclusions are discussed inSection 12.

2. Related works

Forecasting using fuzzy time series is applied in several areasincluding forecasting university enrollments, sales, road accidentsand financial forecasting. In a conventional time series models,the recorded values of a special dynamic process are representedby crisp numerical values. But, in a fuzzy time series model, the re-corded values of a special dynamic process are represented by lin-guistic values. Based on fuzzy time series theory, first forecastingmodel was introduced by Song and Chissom [1,7,2]. They pre-sented the fuzzy time series model by fuzzy relational equationsinvolving max–min composition operation and applied the modelto forecast the enrollments in the University of Alabama. In 1996,Chen [3] used simplified arithmetic operations avoiding the com-plicated max–min operations and their method produced betterresults. Later, many studies provided some improvements to thefuzzy time series methods in determining the lengths of intervals,fuzzification process and defuzzification techniques. Hwang et al.[4] used the differences of the available historical data as fuzzytime series rather than direct usage of raw numeric values. Sahand Degtiarev also used a similar approach in [17]. Huarng triedto improve the forecasting accuracy based on determination ofthe length of intervals [6] and heuristic approaches [5]. Lee andChou [18] forecasted the university enrollments with the averageerror rate less than Chen’s method [3] by defining the supportsof the fuzzy numbers that represent the linguistic values of the lin-guistic variables more appropriately.

Yu [19] proposed weighted fuzzy time series model to resolveissues of recurrence and weighting in fuzzy time series forecasting.Cheng et al. [8] used entropy minimization to create the intervals.They also used trapezoidal membership functions in the fuzzifica-tion process. Chang et al. [20] presented cardinality-based fuzzytime series forecasting model, which builds weighted fuzzy rulesby calculating the cardinality of fuzzy relations. To enhance theperformance of fuzzy time series models, Chen et al. [21] incorpo-rates the concept of the Fibonacci sequence in the existing modelsas proposed by Song and Chissom [1,2] and Yu [19]. To obtain less

number of intervals, Cheng et al. [22] proposed a model using fuzzyclustering technique to partition the data effectively. The K-meansclustering algorithm has been applied to partition the universe ofdiscourse in [23]. Chou et al. [24] forecasted the tourism demandbased on hybridization of rough set with fuzzy time series. Singhand Borah [25] forecasted the university enrollments with the helpof new proposed algorithm by dividing the universe of discourse ofthe historical time series data into different length of intervals. Re-cent advancement in fuzzy time series forecasting models can befound in [26–29].

Recently, many researchers have proposed various hybridiza-tion based models to solve complex problems in forecasting. Forexample, Hadavandi et al. [30] presented a new approach basedon genetic fuzzy systems and ANNs for building a stock priceforecasting expert system to improve the forecasting accuracy.Cheng et al. [31] proposed a new stock price forecasting modelbased on hybridization of genetic algorithm with rough set the-ory. Kuo et al. [32] hybridized the particle swarm optimizationwith fuzzy time series to adjust the lengths of intervals in theuniverse of discourse. Aladag et al. [33] introduced a new ap-proach to define fuzzy relation in high order fuzzy time seriesusing feed forward neural networks. Teoh et al. [34] proposed afuzzy-rough hybrid forecasting model, where rules (fuzzy logicalrelationships) are generated by rough set algorithm. Pal and Mitra[35] proposed a rough-fuzzy hybridization scheme for case gener-ation. They used the fuzzy set theory for linguistic representationof patterns and then obtained the dependency rules by using therough set theory. For advance prediction of dwelling fire occur-rence in Derbyshire (United Kingdom), Yang et al. [36] employedthree approaches: logistic regression, ANN and Genetic Algorithm.Keles et al. [37] proposed a model for forecasting the domesticdept by Adaptive Neuro-Fuzzy Inference System. Chang et al.[38] developed a hybrid model by integrating K-mean clusterand fuzzy neural network to forecast the future sales of a printedcircuit board factory. Huarng and Yu [39], and Yu and Huarng[40] presented a new hybrid model based on neural networkand fuzzy time series to forecast TAIEX. Kuo et al. [41] and Huanget al. [42] introduced a new enrollments forecasting model basedon hybridization of fuzzy time series and particle swarmoptimization.

3. Fuzzy sets and fuzzy time series

In 1965, Zadeh [43] introduced fuzzy sets theory involving con-tinuous set membership for processing data in presence of uncer-tainty. He also presented fuzzy arithmetic theory and itsapplication [44–46]. In this section, we will briefly review fuzzysets theory from [43] and fuzzy time series concepts from [1,7,2].

Definition 3.1 (Fuzzy Set [43]). A fuzzy set is a class with varyingdegrees of membership in the set. Let U be the universe ofdiscourse, which is discrete and finite, then fuzzy set A can bedefined as follows:

A ¼ flAðx1Þ=x1 þ lAðx2Þ=x2 þ � � �g ¼ RilAðxiÞ=xi; ð2Þ

where lA is the membership function of A, lA: U ? [0,1], and lA(xi)is the degree of membership of the element xi in the fuzzy set A.Here, the symbol ‘‘+’’ indicates the operation of union and the sym-bol ‘‘/’’ indicates the separator rather than the commonly used sum-mation and division in algebra respectively.

When U is continuous and infinite, then the fuzzy set A of U canbe defined as:

A ¼Z

lAðxiÞ=xi

� �; 8xi 2 U; ð3Þ

14 P. Singh, B. Borah / Knowledge-Based Systems 46 (2013) 12–21

where the integral sign ‘‘R

’’ stands for the union of the fuzzy single-tons, lA(xi)/xi. Fuzzy time series concept was proposed in [1,7,2] andthe main difference between the traditional time series and the fuz-zy time series is that the values of the former are crisp numerical val-ues, while the values of the latter are fuzzy sets. The crisp numericalvalues can be represented by real numbers whereas in fuzzy sets, thevalues of observations are represented by linguistic values. The def-inition of fuzzy time series is briefly reviewed as follows:

Definition 3.2 (Fuzzy Time Series [1,7,2]). Let Y(t) (t = 0,1,2, . . .) bea subset of R and the universe of discourse on which fuzzy sets li(t)(i = 1,2, . . .) are defined and let F(t) be a collection of li(t)(i = 1,2, . . .). Then, F(t) is called a fuzzy time series on Y(t)(t = 0,1,2, . . .).

From Definition 3.2, we can see that F(t) is a function of time t,and li(t) are linguistic values of F(t), where li(t) (i = 1,2, . . .) arerepresented by fuzzy sets, and the values of F(t) can be differentat different times because the universe of discourse can be differ-ent at different times. Fuzzy time series can be divided into twocategories which are the time-invariant fuzzy time series and thetime-variant fuzzy time series.

If F(t) is caused by F(t � 1), i.e., F(t � 1) ? F(t), then this relation-ship can be represented as follows:

FðtÞ ¼ Fðt � 1Þ � Rðt; t � 1Þ; ð4Þ

where R(t, t � 1) is fuzzy relationship between F(t) and F(t � 1).Here, R is the union of fuzzy relations and ‘‘�’’ is max–min compo-sition operator. It is also called the first-order model of F(t).

Definition 3.3 (Fuzzy time-variant and time-invariant series[15]). Let F(t) be a fuzzy time series, and R(t, t � 1) be a first-ordermodel of F(t). If R(t, t � 1) = R(t � 1,t � 2) for any time t, and F(t)only has finite elements, then F(t) is referred as a time-invariantfuzzy time series. Otherwise, it is referred as a time-variant fuzzytime series.

Definition 3.4 (Fuzzy logical relationship [1–3]). Assume thatF(t � 1) = Ai and F(t) = Aj. The relationship between F(t) andF(t � 1) is referred as a fuzzy logical relationship (FLR), which canbe represented as:

Ai ! Aj; ð5Þ

where Ai and Aj refer to the previous state and current state of the FLRrespectively.

Definition 3.5 (Fuzzy logical relationship group [1–3]). Assume thefollowing FLRs:

Ai ! Ak1;

Ai ! Ak2;

� � �Ai ! Akm:

Chen [3] suggested that the FLRs having the same previous state aregrouped into a same fuzzy logical relationship group (FLRG). So,based on Chen’s model [3], these FLRs can be grouped into the sameFLRG as:

Ai ! Ak1;Ak2; . . . ;Akm:

Definition 3.6 (High-order FLR [14]). Assume that F(t) is caused byF(t � 1),F(t � 2), . . . ,F(t � n) (n > 0), then high-order FLR can beexpressed as:

Fðt � nÞ; . . . ; Fðt � 2Þ; Fðt � 1Þ ! FðtÞ: ð6Þ

4. ANN and its application

ANNs are massively parallel adaptive networks of simple non-linear computing elements called neurons which are intended toabstract and model some of the functionality of the human nervoussystem in an attempt to partially capture some of its computa-tional strengths [47]. The neurons in an ANN are organized into dif-ferent layers. Inputs to the network are existed in the input layer;whereas outputs are produced as signals in the output layer. Thesesignals may pass through one or more intermediate or hidden lay-ers which transform the signals depending upon the neuron signalfunctions.

A simple neural network architecture as proposed by Lippmann[48] is shown in Fig. 1.

In Fig. 1, Z1,Z2, . . . ,Zn are the set of input neurons, which trans-mitting information or signals to another output neuron, say Y.Each input neuron Z1,Z2, . . . ,Zn has an interconnection links withanother neuron. Each interconnection link of input neurons areassociated with some weights as W1,W2, . . . ,Wn. For this neuralnetwork architecture, the net input can be calculated as:

Yinput ¼ z1w1 þ z2w2 þ � � � þ znwn ¼Xn

i¼1

ziwi; ð7Þ

where z1,z2, . . . ,zn are the activations or output of input neurons Z1,Z2, . . . ,Zn, and w1,w2, . . . ,wn are the weights associated withz1,z2, . . . ,zn.

For output neuron Y, output y can be determined by applyingactivation function on the net input Yinput as:

y ¼ f ðYinputÞ: ð8Þ

The back-propagation neural network (BPNN) is one of the sig-nificant developments in the area of ANN [49,50]. The BPNN canconsist of multi-layer feed-forward neural network with one inputlayer, limited number of hidden layers and one output layer. InFig. 2, an architecture of the BPNN is shown, which consists of onlyone hidden layer. The main objective of using the BPNN with mul-ti-layer feed-forward neural network is to minimize the output er-ror obtained from the difference between the calculated output(o1 ,o2, . . . ,on) and target output (n1,n2, . . . ,nn) of the neural networkby adjusting the weights. So, in the BPNN, each information is sentback again in the reverse direction until the output error is verysmall or zero. The BPNN is trained under the process of threephases as:

1. Using the feed-forward neural network for training process ofinput information. Adjustment of weights and nodes are madein this phase.

2. To calculate the error.3. Update the weights.

Due to a large number of additional parameters [51], e.g., initialweight, learning rate, momentum, epoch, activation function, etc.,the ANN model has great capability to learn by making properadjustment of these parameters, to produce the desired output.In this study, we use the BPNN algorithm to defuzzify the fuzzified

Fig. 1. A simple neural network architecture.

Fig. 2. A multi-layer BPNN architecture.

Table 1Historical data of the daily average temperature from June 1996 to September 1996 inTaipei (Unit: �C).

Day Month

June July August September

1 26.1 29.9 27.1 27.52 27.6 28.4 28.9 26.83 29.0 29.2 28.9 26.44 30.5 29.4 29.3 27.55 30.0 29.9 28.8 26.66 29.5 29.6 28.7 28.27 29.7 30.1 29.0 29.28 29.4 29.3 28.2 29.09 28.8 28.1 27.0 30.3

10 29.4 28.9 28.3 29.911 29.3 28.4 28.9 29.912 28.5 29.6 28.1 30.513 28.7 27.8 29.9 30.214 27.5 29.1 27.6 30.315 29.5 27.7 26.8 29.516 28.8 28.1 27.6 28.317 29.0 28.7 27.9 28.618 30.3 29.9 29.0 28.119 30.2 30.8 29.2 28.420 30.9 31.6 29.8 28.321 30.8 31.4 29.6 26.422 28.7 31.3 29.3 25.723 27.8 31.3 28.0 25.024 27.4 31.3 28.3 27.025 27.7 28.9 28.6 25.826 27.1 28.0 28.7 26.427 28.4 28.6 29.0 25.628 27.8 28.0 27.7 24.229 29.0 29.3 26.2 23.330 30.2 27.9 26.0 23.531 26.9 27.7


time series data set. Paradigms adopted for building the basicarchitecture for the proposed neural network is explained next.

Designing the right neural network architecture is a heuristicbased approach and also a very time consuming process. The per-formance of the neural network architecture depends on numberof layers, number of nodes in each layer and number of intercon-nection links with the nodes [52]. Since, a neural network withmore than three layers generate arbitrarily complex decision re-gions. Therefore, a single hidden layer with one input layer andone output layer is considered here in designing the architecture.The number of nodes in input layer will depend on order of FLRs.For example, for third-order FLR, there would be three nodes in in-put layer; for fourth-order FLR, there would be four nodes in inputlayer, and so on. The minimum number of nodes in hidden layer isdetermined by the following equation:

Hiddennodes ¼ Inputnodes � 1; ð9Þ

where Hiddennodes and Inputnodes represent the number of nodes inhidden and input layers respectively. A neural network architecturefor the fifth-order FLRs is shown in Fig. 3.

The neural network as shown in Fig. 3 have five nodes (Ii, i =1,2, . . . ,5) in input layer. The arrangement of nodes in input layeris done in the following sequence:

Yðt � 5Þ;Yðt � 4Þ;Yðt � 3Þ;Yðt � 2Þ;YðtÞ ! Yðt þ 1Þ: ð10Þ

Here, each input node take the previous days (t � 5, t � 4, . . . , t)fuzzified time series values (e.g., A5,A4, . . . ,A1) to predict 1 day(t + 1) advance daily temperature value ‘‘At+1’’. In Eq. (10), each‘‘t’’ represent the day for considered fuzzified time series values.

5. Description of data set

For verifying the model, the daily average temperature data set[15] from June 1996 to September 1996 in Taipei is employed. Thisdata set is shown in Table 1. Taipei, which is the capital of theRepublic of China, is situated at the northern tip of the island ofChina. It is the political, economic, and cultural center of China.So, advance prediction of daily temperature of Taipei is very advan-tageous for the inhabitant of Taipei.

Fig. 3. A BPNN architecture for the fifth-order FLRs.

6. Re-Partitioning Discretization (RPD) approach

In this section, we propose a new discretization approach re-ferred to as ’’RPD’’ for determining the universe of discourse ofthe historical time series data set and partitioning it into differentlengths of intervals. To explain this approach, the daily averagetemperature data set from June 1, 1996 to June 30, 1996, shownin Table 1, is employed. Each step of the approach is explainednext.

Step 6.1. Compute range (R) of a sample, S = {x1,x2, . . . ,xn} as:

R ¼ Maxvalue �Minvalue; ð11Þ

where Maxvalue and Minvalue are the maximum and minimum valuesof S respectively.From Table 1, Maxvalue and Minvalue for the June temperature data set(S) are 30.9 and 26.1 respectively. Therefore, the range R for thisdata set is computed as:

R ¼ 30:9� 26:1 ¼ 4:8:

Step 6.2. Split the data range R into M equally spaced classes,where M can be defined as [53]:

M ¼ 1þ logn2

� �; ð12Þ

where n is the size of the sample S.Based on Eq. (12), we can compute M as:

M ¼ 1þ 1:4770:3010

¼ 5:907; where sample size n ¼ 30:

Step 6.3. Obtain width of an interval (H) as:

H ¼ RM: ð13Þ

Based on Eq. (13), we can calculate the width as:


H ¼ 4:85:907

¼ 0:8126:

Step 6.4. Define the universe of discourse U of the sample S as:

U ¼ ½Lb;Ub�; ð14Þ

where Lb = Minvalue � H, and Ub = Maxvalue + H.Based on Table 1, we have the universe of discourse of the sample Sas:

U ¼ ½26:1� 0:8126;30:9þ 0:8126� ¼ ½25:287;31:713�:

Step 6.5. Compute mid-point (Umid) of the universe of discourseU as:

Umid ¼Lb þ Ub

2: ð15Þ

The Umid of the sample S is obtained as:

Umid ¼25:287þ 31:713

2¼ 28:5:

Step 6.6. Find sub-sets of the sample S such that:

A ¼ fx 2 Sjx 6 Umidg; ð16ÞB ¼ fx 2 Sjx P Umidg: ð17Þ

From Table 1, we have obtained the elements of A and B as:

A ¼ f26:1;27:1;27:4;27:5;27:6;27:7;27:8;27:8;28:4;28:5g;

B ¼ f28:7;28:7;28:8;28:8;29;29;29;29:3;29:4;29:4;29:5;29:5;29:7;30;30:2;30:2;30:3;30:5;30:8;30:9g:

Step 6.7. Define sub-boundaries for A and B as:

UA ¼ ½Amin;Amax�; ð18ÞUB ¼ ½Bmin;Bmax�; ð19Þ

where UA and UB are the sub-boundaries for A and B respectively.Here, Amin and Amax represent the minimum and maximum valuesof the sub-set A respectively. Similarly, Bmin and Bmax representthe minimum and maximum values of the sub-set B respectively.From Eqs. (18) and (19), we can define the sub-boundaries for A andB as:

Table 2Intervals and their corresponding elements for the June daily temperature data set.

Corresponding element Mid-point

Interval for UA

[26.1,26.34] (26.1) 26.22[27.06,27.30] (27.1) 27.18[27.30,27.54] (27.4,27.5) 27.42[27.54,27.78] (27.6,27.7) 27.66

UA ¼ ½26:1;28:5�; ð20ÞUB ¼ ½28:7;30:9�: ð21Þ

Step 6.8. Determine deciding factors for A and B as:

DFA ¼Amax � Amin

NA; ð22Þ

DFB ¼Bmax � Bmin

NB; ð23Þ

where DFA and DFB are the deciding factors for A and B respectively.Here, NA and NB represent the total number of elements of A and Brespectively.From Eqs. (22) and (23), the deciding factors for A and B are:

[27.78,28.02] (27.8,27.8) 27.9[28.26,28.50] (28.4,28.5) 28.38

Interval for UB

[28.70,28.81] (28.7,28.7,28.8,28.8) 28.755[28.92,29.03] (29,29,29) 28.975[29.25,29.36] (29.3) 29.305[29.36,29.47] (29.4,29.4) 29.415[29.47,29.58] (29.5,29.5) 29.525[29.69,29.80] (29.7) 29.745[29.91,30.02] (30) 29.965[30.13,30.24] (30.2,30.2) 30.185[30.24,30.35] (30.3) 30.295[30.46,30.57] (30.5) 30.515[30.79,30.90] (30.8,30.9) 30.845

DFA ¼28:5� 26:1

10¼ 0:24; ð24Þ

DFB ¼30:9� 28:7

20¼ 0:11: ð25Þ

Step 6.9. Partition the sub-boundaries UA and UB into differentlength of intervals as:

ui ¼ ½LðiÞ;UðiÞ�; i ¼ 1;2;3; . . . ; 1 6 UðiÞ < Amax; ui

2 UA; ð26Þ

where L(i) = Amin + (i � 1) � DFA, and U(i) = Amin + i � DFA.

v i ¼ ½MðiÞ;VðiÞ�; i ¼ 1;2;3; . . . ; 1 6 VðiÞ< Bmax; v i 2 UB; ð27Þ

where M(i) = Bmin + (i � 1) � DFB, and V(i) = Bmin + i � DFB.Based on Eq. (26), intervals for the sub-boundary UA are:

u1 ¼ ½26:1;26:34�;u2 ¼ ½27:06;27:30�; . . . ;

u6 ¼ ½28:26;28:50�:

Similarly, based on Eq. (27), intervals for the sub-boundary UB are:

v1 ¼ ½28:70;28:81�; v2 ¼ ½28:92;29:03�;v11 ¼ ½30:79;30:90�:

Step 6.10. Allocate the elements to their corresponding inter-vals.Assign the elements of A and B to their corresponding intervalsobtained after partitioning the boundaries UA and UB respec-tively. All these intervals along with their corresponding ele-ments are shown in Table 2. Last column of Table 2represents mid-points of the intervals. Intervals which do notcover historical data are discarded from the list. Intervals forthe remaining three months July, August and September asshown in Table 1 are obtained in a similar way.

7. Architecture of the model

Most of the existing fuzzy time series models as discussed ear-lier use the following six common steps to deal with the forecast-ing problems:

Step 7.1. Partition the universe of discourse into intervals.Step 7.2. Define linguistic terms for each of the interval.Step 7.3. Fuzzify the time series data set.Step 7.4. Establish the FLRs based on Definition 3.4.Step 7.5. Construct the FLRGs based on Definition 3.5.Step 7.6. Defuzzify and compute the forecasted values.

In this article, an improved fuzzy time series forecasting modelis proposed, which is based on the hybridization of fuzzy time ser-ies theory with ANN. This model also employs the high-order FLRsto obtain the forecasting results. Therefore, above steps are modi-fied, which is represented by the data-flow diagram (see Fig. 4).


8. High-order fuzzy-neuro time series forecasting model

We apply the proposed model to forecast the daily temperatureof Taipei from June, 1996 to September, 1996. This model is trainedwith the June daily temperature data set. Each phase of the trainingprocess is explained next.

Phase 8.1. Divide the universe of discourse into differentlengths of intervals.Define the universe of discourse U for the June temperaturedata set. Based on Eq. (14), we have defined the universe of dis-course U as U = [25.287,31.713]. Then, based on the RPDapproach, the universe of discourse U is partitioned into n dif-ferent lengths of intervals: a1, a2,a3, . . . ,an. The experimentalresults are presented in Table 2.Phase 8.2. Define linguistic terms for each of the interval.Assume that the historical time series data set is distributedamong n intervals (i.e., a1,a2, . . . ,an). Therefore, define n linguis-tic variables A1,A2, . . . ,An, which can be represented by fuzzysets, as shown below:

A1 ¼ 1=a1 þ 0:5=a2 þ 0=a3 þ 0=a4 þ 0=a5 þ � � � þ 0=an�2

þ 0=an�1 þ 0=an;

A2 ¼ 0:5=a1 þ 1=a2 þ 0:5=a3 þ 0=a4 þ 0=a5 þ � � � þ 0=an�2

þ 0=an�1 þ 0=an;

A3 ¼ 0=a1 þ 0:5=a2 þ 1=a3 þ 0:5=a4 þ 0=a5 þ � � � þ 0=an�2

þ 0=an�1 þ 0=an;

..

.

Aj ¼ 0=a1 þ 0=a2 þ 0=a3 þ 0=a4 þ 0=a5 þ � � � þ 0=an�2

þ 0:5=an�1 þ 1=an:

Fig. 4. High-order fuzzy-neuro time series forecasting model.

Obtain the degree of membership of each day’s temperature valuebelonging to each Ai. Here, maximum degree of membership offuzzy set Ai occurs at interval ai, and 1 6 i 6 n.

Phase 8.3. Fuzzify the historical time series data. If one day’sdatum belongs to the interval ai, then it is fuzzified into Ai,where 1 6 i 6 n.If one day’s temperature value belongs to the interval ai, thenthe fuzzified temperature value for that day is considered asAi. For example, the temperature value of June 1, 1996 belongsto the interval a1, so it is fuzzified to A1. In this way, we havefuzzified historical time series data set. The fuzzified tempera-ture values are shown in Table 3.Phase 8.4. Establish the high-order FLRs between the fuzzifieddaily temperature values.Based on Definition 3.6, we have established the fifth-orderFLRs between the fuzzified daily temperature values. For exam-ple, in Table 3, the fuzzified daily temperature values for days 1,2, 3, 4, 5 and 6 are A1, A4, A8, A16, A13 and A11, respectively. Here,to establish the fifth-order FLR among these fuzzified values, itis considered that A11 is caused by the previous five fuzzifiedvalues A1, A4, A8, A16 and A13. Hence, the fifth-order FLR is rep-resented in the following form:

Table 3Fuzzifie

Day

123456789

101112131415161718192021222324252627282930

A1;A4;A8;A16;A13 ! A11: ð28Þ

Here, left hand side of the FLR is called the previous state, whereasright hand side of the FLR is called the current state.Previously, most of the fuzzy time series models [9–13] use the cur-rent state’s fuzzified value for defuzzification. The main downside ofusing such fuzzified value for defuzzification is that predictionscope of these models [9–13] lie within the sample. For most ofthe real and complex problems, out of sample prediction (i.e., ad-vance prediction) is very much essential. Therefore, in this model,the previous state’s fuzzified values are used to obtain the forecast-ing results.

d historical data set for the June daily temperature.

June Fuzzified temperature Mid-point

26.1 A1 26.2227.6 A4 27.6629 A8 28.97530.5 A16 30.51530 A13 29.96529.5 A11 29.52529.7 A12 29.74529.4 A10 29.41528.8 A7 28.75529.4 A10 29.41529.3 A9 29.30528.5 A6 28.3828.7 A7 28.75527.5 A3 27.4229.5 A11 29.52528.8 A7 28.75529 A8 28.97530.3 A15 30.29530.2 A14 30.18530.9 A17 30.84530.8 A17 30.84528.7 A7 28.75527.8 A5 27.927.4 A3 27.4227.7 A4 27.6627.1 A2 27.1828.4 A6 28.3827.8 A5 27.929 A8 28.97530.2 A14 30.185

Table 4Fifth-order FLRs for the June fuzzified daily temperature data set.

Fifth-order FLR

A1, A4, A8, A16, A13 ? ?h6iA4, A8, A16, A13, A11 ? ?h7iA8, A16, A13, A11, A12 ? ?h8iA16, A13, A11, A12, A10 ? ?h9iA13, A11, A12, A10, A7 ? ?h10iA11, A12, A10, A7, A10 ? ?h11iA12, A10, A7, A10, A9 ? ?h12iA10, A7, A10, A9, A6 ? ?h13iA7, A10, A9, A6, A7 ? ?h14iA10, A9, A6, A7, A3 ? ?h15iA9, A6, A7, A3, A11 ? ?h16iA6, A7, A3, A11, A7 ? ?h17iA7, A3, A11, A7, A8 ? ?h18iA3, A11, A7, A8, A15 ? ?h19iA11, A7, A8, A15, A14 ? ?h20iA7, A8, A15, A14, A17 ? ?h21iA8, A15, A14, A17, A17 ? ?h22iA15, A14, A17, A17, A7 ? ?h23iA14, A17, A17, A7, A5 ? ?h24iA17, A17, A7, A5, A3 ? ?h25iA17, A7, A5, A3, A4 ? ?h26iA7, A5, A3, A4, A2 ? ?h27iA5, A3, A4, A2, A6 ? ?h28iA3, A4, A2, A6, A5 ? ?h29iA4, A2, A6, A5, A8 ? ?h30i


The fifth-order FLRs obtained for the fuzzified daily temperaturedata are presented in Table 4. In this table, each symbol ‘‘?’’ repre-sents the desired output for corresponding day ‘‘t’’ in the symbol ‘‘hi’’,which would be determined by the proposed model.

Phase 8.5. Defuzzify the fuzzified time series data set.In this model, we use the BPNN algorithm to defuzzify the fuzz-ified time series data set. The neural network architecturewhich is used here for defuzzification operation is presentedin Section 4. The proposed model is based on the high-order

Table 5Advance prediction of the daily temperature for June.

Day Actual temperature Predicted temperature

1 26.1 –2 27.6 –3 29 –4 30.5 –5 30 –6 29.5 28.67 29.7 29.148 29.4 29.549 28.8 29.7

10 29.4 29.4211 29.3 29.3112 28.5 29.2713 28.7 29.0214 27.5 28.9215 29.5 28.5516 28.8 28.5317 29 28.4618 30.3 28.5319 30.2 28.7520 30.9 29.3321 30.8 29.7522 28.7 30.1423 27.8 30.0524 27.4 29.3925 27.7 29.126 27.1 28.4727 28.4 27.7728 27.8 27.6129 29 27.5530 30.2 27.81

FLRs, so to explain the defuzzification operation, we use thenth-order FLRs, where n P 5. The steps involve in the defuzzifi-cation operation are explained next.

Step 8.5.1. For forecasting day Y(t), obtain the nth-order FLR,which can be represented in the following form:

Atn;Atðn�1Þ; . . . ;At1 ! ?hti; ð29Þ

where ‘‘t’’ represent a day which we want to forecast, and ‘‘n’’ is theorder of FLR (n P 5). Here, Atn,At(n�1), . . . ,At1 are the previous state’sfuzzified values from days, Y(t � n), . . . ,Y(t � 2) to Y(t � 1).Step 8.5.2. Find the intervals where the maximum membership

values of the fuzzy sets Atn,At(n�1), . . . ,At1 occur, andlet these intervals be an,an�1, . . . ,a1, respectively. Allthese intervals have the corresponding mid-pointsCn,Cn�1, . . . ,C1.

Step 8.5.3. Replace each previous state’s fuzzified value of (29)with their corresponding mid-point as:

Cn;Cn�1; . . . ;C1 ! ?hti;n P 5: ð30Þ

Step 8.5.4. Use the mid-points of (30) as inputs in the proposedBPNN architecture to compute the desired output‘‘?’’ for the corresponding day ‘‘t’’.

The scaling of mid-points are done before beginning the defuzz-ification operation using min–max normalization [54]. For exam-ple, array of mid-points ‘‘Xi’’ are normalized based on theminimum and maximum values of ‘‘Xi’’. A mid-point ‘‘v’’ of ‘‘Xi’’ isnormalized to ’’�v ’’ by computing:

�v ¼ v �minA

maxA �minAðnewmaxA � newminA

Þ þ newminA; ð31Þ

where minA and maxA are the minimum and maximum values of ar-ray ‘‘Xi’’ respectively. Min–max normalization maps a value ‘‘v’’ to‘‘�v ’’ in the range ½newmaxA ; newminA

�, where newmaxA represents ‘‘1’’and newminA

represents ‘‘0’’.A sample of the results obtained for the June temperature data

set are presented in Table 5. For rest of the months, similar ap-proach is adopted for obtaining the results.

9. Performance analysis parameters

The performance of the proposed model is evaluated with thehelp of means and standard deviations (SDs) of the observed andpredicted values, root mean square error (RMSE) and Theil’s U Sta-tistic. All these parameters are defined as follows:

1. The mean can be defined as:

A ¼Pn

i¼1Ai

n: ð32Þ

2. The SD can be defined as:

SD ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n

Xn

i¼1ðAi � AÞ2

r: ð33Þ

3. The RMSE can be defined as:

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1ðFi � AiÞ2

n

s: ð34Þ

4. The formula used to calculate Theil’s U statistic [55] is:

U ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1ðAi � FiÞ2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn

i¼1A2i

qþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1F2

i

q : ð35Þ

Table 6Advance prediction of the daily temperature from June (1996) to September (1996) in Taipei (Unit: �C).

Day Actual Predicted Actual Predicted Actual Predicted Actual PredictedJune June July July August August September September

1 26.1 – 29.9 – 27.1 – 27.5 –2 27.6 – 28.4 – 28.9 – 26.8 –3 29 – 29.2 – 28.9 – 26.4 –4 30.5 – 29.4 – 29.3 – 27.5 –5 30 – 29.9 – 28.8 – 26.6 –6 29.5 28.6 29.6 29.25 28.7 28.37 28.2 26.817 29.7 29.14 30.1 29.19 29 28.82 29.2 26.978 29.4 29.54 29.3 29.59 28.2 28.69 29 27.59 28.8 29.7 28.1 29.45 27 28.43 30.3 27.69

10 29.4 29.42 28.9 29.26 28.3 28.19 29.9 27.8611 29.3 29.31 28.4 28.95 28.9 28.14 29.9 29.0412 28.5 29.27 29.6 28.77 28.1 28.18 30.5 29.5913 28.7 29.02 27.8 28.76 29.9 27.97 30.2 29.9314 27.5 28.92 29.1 28.45 27.6 28.17 30.3 29.9415 29.5 28.55 27.7 28.67 26.8 28.31 29.5 29.8616 28.8 28.53 28.1 28.36 27.6 27.97 28.3 29.817 29 28.46 28.7 28.36 27.9 27.85 28.6 29.7218 30.3 28.53 29.9 28.18 29 27.82 28.1 29.3319 30.2 28.75 30.8 28.54 29.2 27.77 28.4 28.6420 30.9 29.33 31.6 28.96 29.8 27.8 28.3 28.4321 30.8 29.75 31.4 29.72 29.6 28.42 26.4 2822 28.7 30.14 31.3 30.37 29.3 28.99 25.7 27.7823 27.8 30.05 31.3 30.99 28 28.91 25 26.8724 27.4 29.39 31.3 31.29 28.3 29.01 27 26.6625 27.7 29.1 28.9 31.3 28.6 28.63 25.8 2626 27.1 28.47 28 30.84 28.7 28.74 26.4 25.9627 28.4 27.77 28.6 29.92 29 28.39 25.6 25.9628 27.8 27.61 28 29.46 27.7 28.33 24.2 25.8129 29 27.55 29.3 28.86 26.2 28.23 23.3 2530 30.2 27.81 27.9 28.49 26 27.99 23.5 25.0531 – – 26.9 28.37 27.7 27.32 – –

Table 7Performance analysis of the model for the fifth-order FLRs.

Statistics June July August September

Mean observed (�C) 28.98 29.25 28.27 27.66Mean predicted (�C) 28.91 29.32 28.29 27.77SD observed (�C) 1.05 1.35 1.04 2.23SD predicted (�C) 0.72 0.94 0.38 1.61RMSE (�C) 1.23 1.33 1.05 1.35U 0.0213 0.0225 0.0189 0.0189

Table 8Performance analysis of the model for the sixth-order FLRs.



Table 9Performance analysis of the model for the seventh-order FLRs.



Table 10Performance analysis of the model for the eighth-order FLRs.



Table 11Additional parameters and their values during the training and testing processes ofneural network.

S. no. Additional parameter Input value

1 Initial weight 0.32 Learning rate 0.53 Epochs 10,0004 Learning radius 35 Activation function Sigmoid


Here, each Fi and Ai is the forecasted and actual value of day irespectively, n is the total number of days to be forecasted. InEqs. (32) and (33), {A1,A2, . . . ,An} are the observed values of the ac-tual time series data set and A is the mean value of these observa-tions. Similarly, mean and SD for predicted time series data set arecomputed. For a good forecasting, the observed means and SDsshould be close to predicted means and SDs. In Eq. (34), a smallRMSE value indicates good forecasting. In Eq. (35), U is bound be-tween 0 and 1, with values closer to 0 indicating good forecastingaccuracy.

Table 12Advance prediction of the BSE price from 7/30/2012 to 9/11/2012 (In Rupee).

Date (mm-dd-yy) Actual price

7/30/2012 17143.68 –7/31/2012 17236.18 –8/1/2012 17257.38 –8/2/2012 17224.36 –8/3/2012 17197.93 –8/6/2012 17412.96 17177.278/7/2012 17601.78 17259.288/8/2012 17600.56 17299.018/9/2012 17560.87 17387.018/10/2012 17557.74 17469.968/13/2012 17633.45 17547.988/14/2012 17728.2 17589.918/16/2012 17657.21 17537.018/17/2012 17691.08 17592.658/21/2012 17885.26 17617.078/22/2012 17846.86 17686.528/23/2012 17850.22 17709.698/24/2012 17783.21 17760.778/27/2012 17678.81 17754.858/28/2012 17631.71 17774.838/29/2012 17490.81 17725.838/30/2012 17541.64 17669.828/31/2012 17380.75 17561.749/3/2012 17384.4 17488.929/4/2012 17440.87 17423.179/5/2012 17313.34 17431.079/6/2012 17346.27 17412.519/7/2012 17683.73 17374.289/10/2012 17766.78 17369.49/11/2012 17852.95 17469.24

Table 13Performance analysis of advance prediction of the BSE price for different orders ofFLRs.

Statistics Fifth-order

Sixth-order

Seventh-order

Eight-order

Mean observed(Rupee)

17612.86 17621.19 17622.03 17623.01

Mean predicted(Rupee)

17523.59 17538.02 17550.14 17561.56

SD observed (Rupee) 169.51 167.84 171.56 175.54SD predicted (Rupee) 165.76 152.44 143.56 135.84RMSE (Rupee) 203.10 201.63 193.19 186.77U 0.0058 0.0057 0.0055 0.0053


10. Empirical analysis – daily temperature prediction

Advance predicted values of temperature from June (1996) toSeptember (1996) in Taipei for the fifth-order FLRs are presentedin Table 6. The proposed model is also tested with different ordersof FLRs. The performance of the model is evaluated with variousstatistical parameters, which are presented in Tables 7–10. FromTables 7–10, it is clear that mean of observed values are close tomean of predicted values. The comparison of SD values betweenobserved and predicted values show that predictive skill of ourproposed model is good for June, July and September. But, SD forpredicted values for August shows slight deflection from SD of ob-served values. Forecasted results in terms of RMSE indicate verysmall error rate. In Tables 7–10, U values are closer to 0, whichindicate the effectiveness of the proposed model.

During the training and testing processes of neural network, anumber of experiments were carried out to set additional parame-ters, viz., initial weight, learning rate, epochs, learning radius andactivation function to obtain the optimal results, and we have cho-sen the ones that exhibit the best behavior in terms of accuracy.

The determined optimal values of all these parameters are givenin Table 11.

11. About BSE and its advance prediction

BSE Limited formerly known as Bombay Stock Exchange (BSE) isa stock exchange located in Mumbai (India) and is the oldest stockexchange in Asia. The equity market capitalization of the compa-nies listed on the BSE was US$1 trillion as of December 2011, mak-ing it the 6th largest stock exchange in Asia and the 14th largest inthe world (www.World-exchanges.org). The BSE has the largestnumber of listed companies in the world.

To further demonstrate the applicability of the proposed model,daily stock exchange price of the BSE is tried to be predicted. TheBSE data set for the period 7/30/2012–9/11/2012 is collected from[56].

The predicted values of the BSE based on the fifth-order FLRs arepresented in Table 12. To check the efficiency of the model, resultsare also obtained with different orders of FLRs. The performance ofthe model is evaluated with various statistical parameters, whichare presented in Table 13. All these statistical analyzes signifythe robustness of the proposed model for advance prediction ofthe BSE price.

12. Conclusions

This article presents a novel approach combining ANN with fuz-zy time series for building a time series forecasting expert system.For training process, the daily average temperature data of Taipeifrom June 1, 1996 to June 30, 1996 are used; while for testing pro-cess, the daily average temperature data of Taipei from July, 1996to September, 1996 are considered. The proposed model is also val-idated by predicting the BSE price from the period 7/30/2012 to 9/11/2012. In this work, we have incorporated ‘‘RPD’’ approach fordetermining the lengths of the intervals effectively, which is animprovement over the original works presented by [1,7,2]. Also,many existing fuzzy time series models as discussed earlier, usethe current state’s fuzzified values for defuzzification, and limittheir predictive skill within the sample. So, to make the predictionout of sample, we have used the previous state’s fuzzified valuesfor defuzzification. In this study, for defuzzification operation, anANN based architecture is developed, which is based on the BPNNalgorithm. The proposed neural network architecture takes theprevious state’s fuzzified values as inputs and outputs are com-puted in advance.

In this study, we try to obtain the forecasting results with opti-mal number of intervals. To obtain the results for the months–June,July, August and September, only 17, 17, 20 and 21 intervals areused respectively. On the other hand, for the BSE price prediction,only 21 intervals are employed. The performance of the model isevaluated with different orders of fuzzy logical relations, whichsignify the efficiency of the proposed model in case of temperatureas well as stock exchange price prediction.

There is a limitation of the proposed model is that it can appli-cable only in one-factor time series data set. Hence, we have triedto make our model generalize enough so that it can deal with dif-ferent kinds of one-factor time series data set and can be used invarious domains efficiently. However, there is a scope to test theproposed model on other domains in the following way:

1. Apply the proposed model on different regions of temperaturedata set (one-factor), and check its accuracy and performancewith different size of intervals and orders.

2. To test the performance of the model for different types offinancial, stocks and marketing data set (one-factor).

http://www.World-exchanges.org


Hence, this study implies that the proposed model can be ap-plied to improve the accuracy and performance of fuzzy time seriesforecasting model.

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High-order fuzzy-neuro expert system for time series forecasting