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A simple method of forecasting basedon fuzzy time series

S.R. Singh *

Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi 221 005, India

Abstract

In fuzzy time series forecasting various methods have been developed to establish the fuzzy relations on time series datahaving linguistic values for forecasting the future values. However, the major problem in fuzzy time series forecasting is theaccuracy in the forecasted values. The present paper proposes a new method of fuzzy time series forecasting based on dif-ference parameters. The proposed method is a simplified computational approach for the forecasting. The method hasbeen implemented on the historical enrollment data of University of Alabama (adapted by Song and Chissom) and theforecasted values have been compared with the results of the existing methods to show is superiority. Further, the proposedmethod has also been implemented on a real life problem of crop production forecast of wheat crop and the results havebeen compared with other methods.2006 Elsevier Inc. All rights reserved.

Keywords: Fuzzy time series; Time variant; Fuzzy membership grade; Linguistic variables; Fuzzy logical relations

1. Introduction

Fuzzy set theory and the concept of linguistic variables and its application to approximate reasoning devel-oped by Zadeh[20,21]has been successfully employed by Song and Chissom[1012]in fuzzy time series fore-casting. Song and Chissom developed the fuzzy time series models and implemented the developed models onthe historical enrollment data of University of Alabama.

Chen[1] presented a simplified method for time series forecasting using the arithmetic operations rather

than complicated max-min composition operations, there are many researchers [15,13,6,3]used the conceptof fuzzy time series in forecasting. Huarng[5]presented a heuristic model for time series forecasting using heu-ristic increasing and decreasing relations to improve the forecast of enrollments and also implemented it forTaiwan Futures Exchange (TAIFEX) forecasting.

Chen[2] proposed a method of forecasting enrollments based on high-order fuzzy time series of variousorders: second, third, fourth and fifth. He raised some points of ambiguity to the trend in forecast and suggestedto use high-order fuzzy logical relationship groups to deal with ambiguity. On comparative study, the

0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.07.128

* On leave from G.B. Pant University of Agriculture & Technology, Pantnagar 263 145, India.E-mail address:[email protected]

Applied Mathematics and Computation 186 (2007) 330339

www.elsevier.com/locate/amc
mailto:[email protected]:[email protected]

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forecasted values obtained by third-order model were found to be of better accuracy. Song[14]extended theChens study by using an average autocorrelation function as a measure of the dependency between fuzzy datafor the selection of suitable order for fuzzy time series model. In his study he found that though the first-ordermodel is a good choice but a second or third-order model will be a better choice. Lee and Chou[7]modified theChens method of defining the Universe of discourse by redefining the Universe of discourse and subsequent

partition of intervals of Universe of discourse as supports of the fuzzy numbers representing the linguistic val-ues to the linguistic variable. Chen and Hsu[4] proposed a first-order time variant method to forecast enroll-ments by dividing the universe of discourse into intervals and then re dividing these intervals into four, three,two and one according to the frequency of their occurrences and framed some rules for forecasting. Yu[18,19]presented a refined fuzzy time series model and a weighted fuzzy time series model for TAIEX forecasting.

Own and Yu[9]presented a heuristic higher-order model by introducing a heuristic function. The changes intime series (the trend) are used as the heuristic knowledge to specify the difference between times to an increase,a decrease or no change as a parameter and applied to model to forecast TAIFEX. Tsaur et al.[16]used theconcept of entropy to measure the degree of fuzziness of a system and to determine a timeTof which the dataapproaches steady state. The concept was applied to determine the minimum value of invariant time index Tand the method was implemented to forecast enrollments. The major objective for the various workers in thearea of fuzzy time series forecasting is to develop a method of providing better accuracy in the forecasted values.

In this paper, we present a new method for forecasting the enrollments with fuzzy time series using a dif-ference parameter as fuzzy relation for forecasting. It provides simple computational algorithms of complexityin linear order. It minimizes the time of generating relational equations by using complex min-max composi-tion operations and the time consumed by the various defuzzification process. It overcomes the difficulty ofsearching a suitable defuzzification procedure providing crisp output of better accuracy. The proposed algo-rithms have been implemented for forecasting the enrollments of University of Alabama and the results havebeen compared with the existing methods to show its superiority. Further, its suitability in general has beenexamined in a real life problem by implementing it on the historical time series data of crop (wheat) produc-tion of Pantnagar farm, G.B. Pant University of Agriculture and Technology, Pantnagar (India). The moti-vation of the implementation of fuzzy time series in crop production forecast is that the Agriculturalproduction system is one of the real life problems falling in the category having uncertainty in known and

some unknown parameters. In agricultural production process, even all the standard practices of croppingare adapted; the uncertainty lies in the crop production due to some uncontrolled parameters. Further thecrop production being dealt with the field data, precision of data is always a matter of concern.

2. Basics of fuzzy time series

In view of making our exposition self contained, the various definitions and properties of fuzzy time seriesforecasting found in [121]are summarized and is presented as:

Definition 1. A fuzzy set is a class of objects with a continuum of grade of membership. LetUbe the Universeof discourse with U= {u1,u2,u3, . . . ,un}, where ui are possible linguistic values ofU, then a fuzzy set oflinguistic variables AiofUis defined by

Ai lAiu1=u1 lAiu2=u2 lAiu3=u3 lAiun=un; 1

where lAi is the membership function of the fuzzy set Ai, such that lAi : U! 0; 1.Ifuj is the member ofAi, then lAiuj is the degree of belonging ofujto Ai.

Definition 2. Let Y(t) (t= . . .,0,1,2,3, . . .), is a subset of R, be the Universe of discourse on which fuzzy setsfi(t), (i= 1,2,3, . . .) are defined and F(t) is the collection offi, thenF(t) is defined as fuzzy time series on Y(t).

Definition 3. Suppose F(t) is caused only by F(t1) and is denoted by F(t1)!F(t); then there is a fuzzyrelationship between F(t) and F(t1) and can be expressed as the fuzzy relational equation:

Ft Ft1 Rt; t1 2

here, is MaxMin composition operator. The relation R is called first-order model ofF(t).

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Further if fuzzy relationR(t, t1) ofF(t) is independent of timet,that is to say for different timest1andt2,R(t1, t11) =R(t2, t21), then F(t) is called a time invariant fuzzy time series.

Definition 4. IfF(t) is caused by more fuzzy sets, F(tn),F(tn + 1), . . . ,F(t1), the fuzzy relationship isrepresented by

Ai1;Ai2; . . .;Ain ! Aj

here,F(tn) = Ai1,F(tn+ 1) =Ai2, . . . ,F(t1) =Ain. This relationship is called nth-order fuzzy time ser-ies model.

Definition 5. Suppose F(t) is caused by a F(t1),F(t2), . . . , and F(tm) (m> 0) simultaneously and therelations are time variant. TheF(t) is said to be time variant fuzzy time series and the relation can be expressedas the fuzzy relational equation:

Ft Ft1 Rwt; t1 3

here, w> 1 is a time (number of years) parameter by which the forecast F(t) is being affected. Various com-plicated computational methods are available to for the computations of the Relation Rw (t, t1).

In the next section, we are proposing simple computational algorithms for computation of these relationsand its implementation in forecasting. In the present study, past three years time series data being consideredin framing the fuzzy rules to impose on current year fuzzified enrollment to the forecast of next yearenrollments. The computational algorithms of the proposed model of order three based on differenceparameters are presented.

3. Computational algorithm of proposed method for fuzzy time series forecasting

In this section, we present the stepwise procedure of the proposed method for fuzzy time series forecastingbased on historical time series data.

1. Define the Universe of discourse, Ubased on the range of available historical time series data, by ruleU= [DminD1, DmaxD2] where D1and D2are two proper positive numbers.

2. Partition the Universe of discourse into equal length of intervals: u1, u2, . . . , um. The number of intervalswill be in accordance with the number of linguistic variables (fuzzy sets) A1, A2, . . . , Am to be considered.

3. Construct the fuzzy sets A iin accordance with the intervals in Step2 and apply the triangular membershiprule to each intervals in each fuzzy set so constructed.

4. Fuzzify the historical data and establish the fuzzy logical relationships by the rule:IfAiis the fuzzy produc-tion of year n and Ajis the fuzzify production of year n+ 1, then the fuzzy logical relation is denoted asAi!Aj. Here Aiis called current state and Ajis next state.

5. Rules for forcastingSome notations used are defined as

[*Aj] is corresponding interval ujfor which membership in Ajis Supremum (i.e. 1)L[*Aj] is the lower bound of interval ujU[*Aj] is the upper bound of intervalujl[*Aj] is the length of the intervalujwhose membership in Ajis Supremum (i.e. 1)M[*Aj] is the mid value of the intervalujhaving Supremum value in Aj

For a fuzzy logical relation Ai!Aj:

Ai is the fuzzified enrollments of yearnAj is the fuzzified enrollments of yearn+ 1Ei is the actual enrollments of yearnEi1 is the actual enrollments of yearn1

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Ei2 is the actual enrollments of yearn2Fj is the crisp forecasted enrollments of the yearsn+ 1

This Model of order three utilizes the historical data of year n 2, n1, nfor framing rules to implementon fuzzy logical relation,Ai! Aj, whereAi, the current state, is the fuzzified enrollments of year n and Aj, the

next state, is fuzzified enrollments of yearn

+ 1. The proposed method for forecasting is mentioned as Rule forgenerating the relations between the time series data of years n2, n1, n for forecasting the enrollment ofyear n+ 1.

Rule:Forecasting enrollments for year n+ 1 (i.e. 1974) and onwards.For k= 3 to . . . K(end of time series data)Obtained fuzzy logical Relation for year kto k+ 1Ai!AjComputeDi=k(EiEi1)j j(Ei1Ei2)kXi=Ei+Di/2XXi=EiDi/2

Yi=Ei+DiYYi=EiDiFor I= 1 to 4IfXiP L[*Aj] And Xi6 U[*Aj]

Then P1=Xi; n= 1Else P1= 0; n= 0

Next IIfXXiP L[*Aj] And XXi6 U[*Aj]

Then P2=XXi; m= 1Else P2= 0; m= 0

Next I

IfYiP L[*

Aj] And Yi6 U[*

Aj]Then P3=Yi; o= 1

Else P3= 0; o= 0Next I

IfYYiP L[*Aj] And YYi6 U[*Aj]Then P4=YYi; p= 1

Else P4= 0; p= 0B=P1+P2+P3+P4IfB= 0 Then Fj=M(*Aj)Else Fj= (B+M(*Aj))/(m+n+ o+k+ 1)

Next k

4. Computations of enrollments forecast

Algorithm of the proposed method is being implemented on the time series data of enrollments at Univer-sity of Alabama and the step wise results obtained are:

Step 1. Universe of discourse U= [13000,20000]Step 2. Partition of universe of discourse Uin the seven intervals (linguistic values)

u1 13000; 14000; u2 14000; 15000; u3 15000; 16000; u4 16000; 17000;

u5 17000; 18000; u6 18000; 19000; u7 19000; 20000:


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Step 3. Define seven fuzzy sets A1, A2, . . . , A7as linguistic variables on the universe of discourse U. Thesefuzzy variables are being defined as:

and the membership grades to these fuzzy sets of linguistic values are defined as

A1 1=u10:5=u20=u30=u40=u50=u60=u7;

A2 0:5=u11=u20:5=u30=u40=u50=u60=u7;

A3 0=u10:5=u21=u30:5=u40=u50=u60=u7;A4 0=u10=u20:5=u31=u40:5=u50=u60=u7;

A5 0=u10=u20=u30:5=u41=u50:5=u60=u7;

A6 0=u10=u20=u30=u40:5=u51=u60:5=u7;

A7 0=u10=u20=u30=u40=u50:5=u61=u7:

Step 4. The fuzzified historical time series data of enrollments are obtained and fuzzy logical relations areestablished (Table 1).Step 5. Using the proposed algorithms, (Rule for forecasting) in Section3, the computations have been car-ried out for the proposed model and the results obtained are placed in the table along with results of othermodels (Table 2).

A1 Poor enrollmentA2 Below average enrollmentA3 Average enrollmentA4 Good enrollmentA5 Very good enrollmentA6 Excellent enrollmentA7 Extraordinary enrollment

Table 1Fuzzy historical enrollments

Year Actual enrollments Fuzzified enrollments

1971 13,055 A11972 13,563 A11973 13,867 A11974 14,696 A21975 15,460 A31976 15,311 A31977 15,603 A31978 15,861 A31979 16,807 A41980 16,919 A41981 16,388 A41982 15,433 A31983 15,497 A31984 15,145 A31985 15,163 A31986 15,984 A31987 16,859 A41988 18,150 A61989 18,970 A61990 19,328 A71991 19,337 A7

1992 18,876 A6


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Table 3Comparison of mean square error (MSE) of proposed method with other methods

Models/MSE

Proposedmodel

Chenmodel

SC_timeinvariant

SC_timevariant

Huarngheuristic

Lee and Chou[7]

Tsaur et al.[16]

MSE 133700.4 439420.8 458437.5 775686.8 239483.1 240047 138322.4

Enrollment forecast

14000

15000

16000

17000

18000

19000

20000

74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92

Year

En

rollments

Actual

Proposed

Lee &

Chou

Tsaur

Chen

Heuristic

Fig. 1. Actual enrollments vs forecasted enrollments.

Table 2Forecasted enrollments by different models at a glance

Year Actualenroll

Proposedmodel

Chenmethod

SC_timeinvariant

SC_timevariant

Huarngheuristic

Lee and Chou[7]

Tsaur et al.[16]

1971 13,055 1972 13,563

1973 13,867 1974 14,696 14,286 14,000 14,000 14,000 14,568 14,0001975 15,460 15,361 15,500 15,500 14,700 15,500 15,654 15,5001976 15,311 15,468 16,000 16,000 14,800 15,500 15,654 15,5001977 15,603 15,512 16,000 16,000 15,400 16,000 15,654 16,0001978 15,861 15,582 16,000 16,000 15,500 16,000 15,654 16,0001979 16,807 16,500 16,000 16,000 15,500 16,000 16,197 16,0001980 16,919 16,361 16,833 16,813 16,800 17,500 17,283 16,5001981 16,388 16,362 16,833 16,813 16,200 16,000 17,283 16,5001982 15,433 15,744 16,833 16,789 16,400 16,000 16,197 15,5001983 15,497 15,560 16,000 16,000 16,800 16,000 15,654 15,5001984 15,145 15,498 16,000 16,000 16,400 15,500 15,654 15,5001985 15,163 15,306 16,000 16,000 15,500 16,000 15,654 15,500

1986 15,984 15,442 16,000 16,000 15,500 16,000 15,654 15,5001987 16,859 16,558 16,000 16,000 15,500 16,000 16,197 16,5001988 18,150 17,187 16,833 16,813 16,800 17,500 17,283 18,5001989 18,970 18,475 19,000 19,000 19,300 19,000 18,369 19,0001990 19,328 19,382 19,000 19,000 17,800 19,000 19,454 19,0001991 19,337 19,487 19,000 19,000 19,300 19,500 19,454 19,0001992 18,876 18,744 19,000 19,600 19,000


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To have a comparison of accuracy in forecasted values of our proposed models with other models, the MSEin forecast have been computed as

Mean square error MSE

Xn

i1

Actual productioniForecasted productioni2

" #,n

and the values of MSE of above models are placed in Table 3.The comparative study of MSE, as shown inTable 3exhibits that the forecast by the proposed method areof higher accuracy over the others. The trends in forecast of the three above mentioned methods are beingillustrated inFig. 1.

5. Computation of wheat production forecast

In this section, the proposed method is implemented into real life problem of a dynamical system containingfuzziness like crop production. In view of suitability as presented in Section4, the proposed model is beingimplemented for forecasting the Wheat production. The historical time series data of Wheat production areof the huge farm of G.B. Pant University, Pantnagar, India. The historical time series data of wheat produc-tion is in terms of productivity in kg per hectare. The method has been implemented and computations carriedout are presented stepwise (Table 4).

Step 1. Universe of discourse U= [1400,3500].Step 2. The Universe of discourse is partitioned into seven intervals of linguistic values:

u1 1400; 1700; u2 1700; 2000; u3 2000; 2300; u4 2300; 2600;

u5 2600; 2900; u6 2900; 3200; u7 3200; 3500:

Step 3. Define seven fuzzy sets A1, A2, . . . , A7having some linguistic values on the universe of discourse U.

The linguistic values to these fuzzy variables are as follows:

A1 Poor productionA2 Below average productionA3 Average productionA4 Good productionA5 Very good productionA6 Excellent productionA7 Bumper production

Table 4The historical data of wheat production

Year Production (kg/ha) Year Production (kg/ha) Year Production (kg/ha)

1981 2730 1988 3407 1995 23181982 2957 1989 2238 1996 26171983 2382 1990 2895 1997 22541984 2572 1991 3276 1998 29101985 2642 1992 1431 1999 34341986 2700 1993 2248 2000 2795

1987 2872 1994 2857 2001 3000


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The membership grades to these fuzzy sets of linguistic variables are defined as

A1 1=u10:5=u20=u30=u40=u50=u60=u7;

A2 0:5=u11=u20:5=u30=u40=u50=u60=u7;

A3 0=u10:5=u21=u30:5=u40=u50=u60=u7;

A4 0=u10=u20:5=u31=u40:5=u50=u60=u7;A5 0=u10=u20=u30:5=u41=u50:5=u60=u7;

A6 0=u10=u20=u30=u40:5=u51=u60:5=u7;

A7 0=u10=u20=u30=u40=u50:5=u61=u7:

Step 4. The historical time series data is fuzzified in order to have the fuzzy logical relations and the fuzzylogical relations obtained are as (Table 5 and 6).

Table 5Fuzzy logical relationships of the historical wheat production

A5!

A6 A6!

A4 A4!

A4 A4!

A5 A5!

A5A5!A5 A5!A7 A7!A3 A3!A5 A5!A7A7!A1 A1!A3 A3!A5 A5!A4 A4!A5A5!A3 A3!A6 A6!A7 A7!A5 A5!A6

Table 6Fuzzy logical relationship groups

1 A1!A32 A3!A5 A3!A63 A4!A4 A4!A54 A5!A3 A5!A4 A5!A5 A5!A6 A5!A75 A6!A4 A6!A7

6 A7!A1 A7!A3 A7!A5

Table 7Forecasted wheat production

Year Actual production (kg/ha) Forecasted by Chen model Forecasted by proposed model

1981 27301982 2957 27501983 2382 29001984 2572 2600 25031985 2642 2600 2757.251986 2700 2750 27381987 2872 2750 27101988 3407 2750 33501989 2238 2150 21501990 2895 2900 28111991 3276 2750 3378.51992 1431 2150 15501993 2248 2150 2156.51994 2857 2900 27561995 2318 2750 24501996 2617 2600 27501997 2254 2750 21501998 2910 2900 30501999 3434 2900 3276.52000 2795 2150 2750

2001 3000 2750 2980


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Step 5. The forecasted values have been obtained by using the algorithms in Section3. The crisp forecastedvalues have also been obtained for the Wheat production by arithmetic operations as carried out by Chenin forecasting the enrollments. The forecasted production of wheat obtained by these two methods is placedinTable 7.

Mean square error MSE Xni1

Actual productioniForecasted productioni2

" #,n

is calculated for comparing the forecasted results of all the three methods.The comparison of MSE inTable 8shows the superiority of the proposed models over the Chens model as

it provides forecast of higher accuracy. Further the trends in forecast by the proposed and Chens methodhave been compared with the forecast by the other available commonly used methods like: linear model,moving average method, fitting the polynomial of degree two and three and have been illustrated inFig. 2. It is evident from theR2 values that the mentioned generally used statistical methods are not suitablein such case of forecast in fuzzy environment.

Table 8MSE of wheat production forecast

Models/MSE Proposed method Chen method

MSE 11,098.96 136,730.6

actualwheat production forecast

prodct

3500

Chen model

3000

proposed

model

2500

Linear

production

(actual

prodct)

20003 per. Mov.

Avg.y = 8.7534x + 2609.4

R2= 0.0093(actual

prodct)

y = 1.1061x3- 25.783x2+ 145.78x + 2531.7

R2= 0.1583

y = 5.7412x2- 100.33x + 29731500 Poly.

R2= 0.0944 (actual

prodct)

Poly.1000 (actual

year prodct)

Fig. 2. Actual wheat production vs forecasted wheat production.


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6. Conclusions

In this paper we have proposed a simple computational method for fuzzy time series forecasting. The algo-rithms of the proposed method are simple and have the complexity of linear order. It minimizes the compli-cated computations of fuzzy relational matrices and search for a suitable defuzzification process and provides

the forecasted values of better accuracy. The method has been implemented on the historical time series dataof enrollments of University of Alabama to have a comparative study with the existing methods. Further themethod has also been implemented on crop (wheat) production forecast. The forecasted values obtained bythe method show its suitability in fuzzy time series forecasting of crop production without any prior knowl-edge of the production governing parameters.

Acknowledgements

Author is highly thankful to General Manager, Farms, Govind Ballabh Pant University of Agriculture andTechnology, Pantnagar263 145, Udham Singh Nagar, Uttaranchal, India for providing the valuable histor-ical time series data of crop (wheat) production.

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