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A new approach based on the optimization of the length of intervals in fuzzy time series

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Journal of Intelligent & Fuzzy Systems 22 (2011) 15–19DOI:10.3233/IFS-2010-0470IOS Press
15
A new approach based on the optimizationof the length of intervals in fuzzy time series
Erol Egrioglu
a
, Cagdas Hakan Aladag
b
,
∗
, Murat A. Basaran
c
, Ufuk Yolcu
a
and Vedide R. Uslu
a
a
Department of Statistics, University of Ondokuz Mayıs, Samsun, Turkey
b
Department of Statistics, University of Hacettepe, Ankara, Turkey
c
Department of Mathematics, University of Nigde, Nigde, Turkey
Abstract
. In fuzzy time series analysis, the determination of the interval length is an important issue. In many researches recentlydone,thelengthofintervalshasbeenintuitivelydetermined.Inordertoefﬁcientlydeterminethelengthofintervals,twoapproacheswhich are based on the average and the distribution have been proposed by Huarng [4]. In this paper, we propose a new methodbased on the use of a single variable constrained optimization to determine the length of interval. In order to determine optimumlengthofintervalforthebestforecastingaccuracy,weusedaMATLABfunctionwhichisemployinganalgorithmbasedongoldensectionsearchandparabolicinterpolation.Meansquareerrorisusedasameasureofforecastingaccuracysotheobjectivefunctionvalue is mean square error value for forecasted observations. The proposed method was employed to forecast the enrollments of the University of Alabama to show the considerable outperforming results.Keywords: Forecasting, fuzzy sets, fuzzy time series, length of interval, optimization
1. Introduction
Fuzzy time series analysis performs well than theconventional correspond since they improve signiﬁ-cantlytheforecastingresults.Thereisnoneedtohaveatleast50observationsandthelinearityassumptionastheconventional ones do. Therefore these methods gradu-ally attract researchers’ attention since they are simplyapplicable and there are no any other restrictions.Fuzzy time series procedure ﬁrstly proposed in[10–12]consistsofthreestages.Intheﬁrststageobser-vations must be fuzziﬁed. Secondly fuzzy relationshipsare evaluated from fuzzy observations. And the ﬁnalstage is the defuzziﬁcation stage. Since then manyresearcheshavebeendonetoimprovethesestagestogetmore accurate forecasting results. Chen [2] proposed asimplemethodfordiscoveringfuzzyrelationshipswith
∗
Corresponding author. Cagdas Hakan Aladag, Department of Statistics, University of Hacettepe, Ankara 06800, Turkey. E-mail:chaladag@gmail.com.
respecttothemethodintroducedbySongandChissom.Chen’s method simpliﬁed the arithmetic operation pro-cess.Thetwoofcontributedpaperstotheidentiﬁcationof fuzzy relationships were done by Huarng and Yu [7,8].InthepaperdonebyHuarng[4],hetriedtoimprove
the stage of the fuzziﬁcation. He pointed out that anaffective length of interval could signiﬁcantly improveforecasting results. Hence he proposed a method basedon the average and the distribution to choose a moreaffective length. In that method, the length of inter-val is ﬁxed. In preceding study, Huarng [5] proposedanother method in which the length of interval is notﬁxed. Huarng and Yu [6] also used a dynamic approachfor adjusting lengths of interval.In this paper we propose a novel method, for ﬁnd-ing an affective length of interval, which minimizesmeansquareerror(MSE)byusingasinglevariablecon-strained optimization. The objective function value isMSE value obtained from forecasted and actual obser-vations. By minimizing MSE, we try to determine thelengthofinterval,thatis,weaimtoincreaseforecasting
1064-1246/11/$27.50 © 2011 – IOS Press and the authors. All rights reserved
16
E. Egrioglu et al. / A new approach based on the optimization of the length of intervals in fuzzy time series
accuracy. In order to ﬁnd the optimal length of interval,a MATLAB function which is employing an algorithmbasedongoldensectionsearchandparabolicinterpola-tion is used. To show what we achieved, we applied themethod to a well known data which is the enrollmentsoftheUniversityofAlabama.Theresultsarecomparedto the results from previous studies.Section 2 is about the general knowledge of fuzzytime series. In section 3 Chen’s method is brieﬂypresented. Section 4 explains our proposed approachand presents the empirical results of an applicationof the real data. Section 5 ﬁnally offers a generalconclusion.
2. Fuzzy time series
The deﬁnition of fuzzy time series was ﬁrstly intro-duced by [10–12]. In fuzzy time series approximation,you do not need various theoretical assumptions just asyou need in conventional procedures. The most impor-tantadvantagesoffuzzytimeseriesapproachesaretobeable to work with a few observations and not to requirethe linearity assumption. The same general deﬁnitionsof fuzzy time series are given as follows:Let
U
be the universe of discourse, where
U
={
u
1
,u
2
,...,u
b
}
.
A fuzzy set
A
i
of
U
is deﬁnedas
A
i
=
f
Ai
(
u
1
)
/u
1
+
f
Ai
(
u
2
)
/u
2
+···+
f
Ai
(
u
b
)
/u
b
,
where
f
Ai
is the membership function of the fuzzy set
A
i
;
f
Ai
:
U
→
[0
,
1]
. u
a
is a generic element of fuzzyset
A
i
;
f
Ai
(
u
a
) is the degree of belongingness of
u
a
to
A
i
;
f
Ai
(
u
a
)
∈
[0
,
1] and 1
≤
a
≤
b
.
Deﬁnition 1.
Fuzzy time series Let
Y
(
t
)(
t
=
...,
0
,
1
,
2
,...
) a subset of real numbers, be the uni-verseofdiscoursebywhichfuzzysets
f
j
(
t
)aredeﬁned.If
F
(
t
) is a collection of
f
1
(
t
)
,f
2
(
t
)
,...
then
F
(
t
) iscalled a fuzzy time series deﬁned on
Y
(
t
).
Deﬁnition 2.
Fuzzy time series relationships assumethat
F
(
t
) is caused only by
F
(
t
−
1)
,
then the relation-shipcanbeexpressedas:
F
(
t
)
=
F
(
t
−
1)
∗
R
(
t,t
−
1)
,
which is the fuzzy relationship between
F
(
t
) and
F
(
t
−
1)
,
where
∗
represents as an operator. To sumup,let
F
(
t
−
1)
=
A
i
and
F
(
t
)
=
A
j
.
Thefuzzylogicalrelationshipbetween
F
(
t
)and
F
(
t
–1)canbedenotedas
A
i
→
A
j
where
A
i
refers to the left-hand side and
A
j
refers to the right-hand side of the fuzzy logical rela-tionship.Furthermore,thesefuzzylogicalrelationshipscanbegroupedtoestablishdifferentfuzzyrelationship.
3. The algorithm of Chen’s model
In [2], Chen has improved the approximation givenby[10–12].Heproposesamethodwhichusesasimpleroperation instead of complex matrix operations, in theestablishment step of fuzzy relationships. In [2], thealgorithm of Chen’s method is given as follows:Step 1. Deﬁnetheuniverseofdiscourseandintervalsfor rules abstraction.Based on the domain issue, The uni-verse of discourse can be deﬁned as:
U=
[
starting,ending
].Asthelengthofintervalisdetermined,
U
canbepartitionedintoseveralequally length intervals.Step 2. Deﬁne fuzzy sets based on the universe of discourse and fuzzify the historical data.Step 3. Fuzzify observed rules.Step 4. Establish fuzzy logical relationships andgroup them based on the current states of thedata of the fuzzy logical relationships.For example,
A
1
→
A
2
,A
1
→
A
1
,A
1
→
A
3
,
can be grouped as:
A
1
→
A
2
,A
3
,A
1
.Step 5. Forecast.Let
F
(
t
−
1)
=
A
i
.Case 1: There is only one fuzzy logicalrelationship in the fuzzy logical relationshipsequence. If
A
i
→
A
j
,
then
F
(
t
)
,
forecastvalue, is equal to
A
j
.Case 2: If
A
i
→
A
i
,A
j
,...,A
k
,
then
F
(
t
)
,
forecast value, is equal to
A
i
,A
j
, ... ,A
k
.Case3:If
A
i
→
empty,then
F
(
t
)
,
forecastvalue, is equal to
A
i
.Step 6. Defuzzify.Apply“Centroid”methodtogettheresults.Thispro-cedure (also called center of area, center of gravity) isthe most often adopted method of defuzziﬁcation.
4. The method based on the optimizationof MSE
In fuzzy time series approaches for the aim of fore-casting the length of intervals affects the forecastingperformance. Hence it is important to choose an affec-tive length of intervals for improving forecasting infuzzy time series. The method we propose optimizesthe length of interval by following with the algorithmof Chen’s method [2]. In the optimization process, we
E. Egrioglu et al. / A new approach based on the optimization of the length of intervals in fuzzy time series
17
usedaMATLABfunctioncalled“fminbnd”whichmin-imizes MSE. “fminbnd” is used to ﬁnd minimum of single-variablefunctiononﬁxedinterval.Itﬁndsamin-imum for a problem speciﬁed bymin
x
f
(
x
)suchthat
x
1
< x < x
2
.
x
,
x
1
, and
x
2
are scalars and
f
(
x
) is a functionthat returns a scalar. In MATLAB, ˆ
x
= fminbnd(
f
(
x
)
,x
1
,x
2
) returns a value ˆ
x
that is a local mini-mum of the scalar valued function
f
(
x
) in the interval
x
1
< x < x
2
. In other words, to ﬁnd the minimum of the function
f
(
x
) on the interval (
x
1
,
x
2
),a
=
fminbnd (
f
(
x
)
,x
1
,x
2
)canbeusedinMATLAB.
f
(a)givesthelocalminimumvalue in the interval (
x
1
,
x
2
).The algorithm used by fminbnd is based on goldensection search introduced by Kiefer [9] and parabolicinterpolation. Unless the left endpoint
x
1
is very closetotherightendpoint
x
2
,fminbndneverevaluates
f
(
x
)atthe endpoints, so
f
(
x
) need only be deﬁned for
x
in theinterval
x
1
< x < x
2
.Iftheminimumactuallyoccursat
x
1
or
x
2
, fminbnd returns an interior point at a distanceofnomorethan2*
Tol
Xfrom
x
1
or
x
2
,where
Tol
Xisthetermination tolerance. See [1] or [3] for details aboutthe algorithm.For the aim of a comparative study, we used theproposedmethodandmethodsin[4]averageanddistri-bution based length to determine the length of intervalin Chen’s method [2] when the data of the enroll-ments of the University of Alabama is being analyzed.EnrollmentdataarepresentedinTable1andtheresultsobtained from the mentioned methods are summarizedinTable2.MSEforforecastedobservationsisusedasameasure of forecasting accuracy so the objective func-tion value is equal to MSE value produced from Chen’smethod [2]. Using with the algorithm of a single vari-
Table 1Enrollment dataYear Actual Year Actual1971 13055 1982 154331972 13563 1983 154971973 13867 1984 151451974 14696 1985 151631975 15460 1986 159841976 15311 1987 168591977 15603 1988 181501978 15861 1989 189701979 16807 1990 193281980 16919 1991 193371981 16388 1992 18876Table 2The obtained values of MSE for the enrollment dataLength of interval 200 273.2346
a
300
b
400
c
500MSE 104640 66661 78792 124707 173453Length of interval 600 700 800 900 1000MSE 254592 222557 365045 246892 407521
a
Results of the proposed method;
b
Average based length in[4];
c
Distribution based length in [4].
able constrained optimization to minimize MSE valueviaMATLABfunctioncalled“fminbnd”overtheinter-val between 200 and 500, the optimal length of intervalwas obtained. We minimize the MSE value by opti-mizing the length of interval so that we increase theforecasting accuracy. We use the function “fminbnd”as follows:
x
∗
=
fminbnd
f
MSE
(
x
)
,
200
,
500
where
f
MSE
(
x
)givestheMSEvalueobtainedfromfore-castedandactualvaluesforthelengthofinterval
x
whenChen’s method [2] is used. The left and the right endpoints are taken as 200 and 500, respectively. This usedfunctionreturns
x
*thatisaminimumpointofthefunc-tion
f
MSE
(
x
) in the interval 200 <
x
< 500. We found
x
*as 273.2346. In addition, the running time was lowerthan one minute.When the function “fminbnd” is used in the opti-mization process, there are two parameters which areimportantintermsoffuzzytimeseriesandoptimizationprocesses. These parameters are the left and the rightend points. In fuzzy time series approaches, a key pointin choosing an effective length of interval is that theyshould not be too large or small. If a too large right endpointvalueistaken,atoolargelengthofintervalcanbeproduced by the function “fminbnd”. When this valueoflengthofintervalisusedinfuzzytimeseriesanalysis,lower MSE values can be obtained but this causes noﬂuctuations in the fuzzy time series. On the other hand,if a too small left end point value is taken, a too smalllength of interval can be obtained and this will dimin-ish the meaning of fuzzy time series [2]. Choosing theleft and the right end points also affects the optimiza-tionprocess.Ifthedifferencebetweentheseparametersis too small, the possibility of founding a satisfactoryvalue for the length of interval will be very low. How-ever, if the difference between these parameters is toolarge, the searching time will increase. Therefore, theconstrainedoptimizationtechniquewasemployedoverthe interval between 200 and 500 in order not to ﬁnda very small or very large interval in the optimizationprocess. In other words, we take the left and the right
18
E. Egrioglu et al. / A new approach based on the optimization of the length of intervals in fuzzy time series
0 5 10 15 20 251.31.41.51.61.71.81.92.0x 10
4
Fig. 1. The graph of actual values and forecasts based on the 300length of interval.
0 5 10 15 20 251.31.41.51.61.71.81.92.0x 10
4
Fig.2. Thegraphofactualvaluesandforecastsbasedonourproposedmethod.
end points as 200 and 500, respectively to avoid gettingvery small or very large intervals in fuzzy time seriesanalysis.Theoptimallengthoftheintervalsobtainedfromourproposed method is 273.2346 and the correspondingMSE value is 66661. As seen from Table 2 the resultsare superior to those obtained from [4].In Fig. 1 the graph of the forecasts for the length of the interval 300 which gave the MSE value obtained byHuarng[4]andtheactualvaluesispresented.ThegraphoftheforecastsobtainedfromourproposedmethodandtheactualvaluesaregiveninFig.2.Inthegraphs,solidlinesrepresenttheforecastsanddashedlinesdenotetheactual values.
5. Discussion
In this paper, we use a single variable constrainedoptimization to obtain an effective length of intervalfor Chen’s model [2]. In the optimization process, weused a MATLAB function called “fminbnd” which isbasedongoldensectionsearchandparabolicinterpola-tion.Weconstrainedthelengthoftheintervalnottogeta very small or a very large length in fuzzy time seriesanalysis so the left and the right end points are takenas 200 and 500. We applied the proposed method andmethods in [4] average and distribution based length todetermine the length of interval in Chen’s method [2]to analyze the data of the enrollments of the Universityof Alabama for comparison. Huarng [4] found that theoptimal length as 300 for the average based approachand MSE as 78792. Here, we investigated that opti-mizing the length of the interval over between 200 and500 decreased the value of MSE. Due to our proposedmethod the optimal length of the interval is 273.2346and the corresponding MSE value is 66661. Hence,it is clear that the implementation of the constrainedoptimization increases the forecasting performance.In the optimization process, the left and the rightpoints are parameters which can affect the results. Weexplained their effects and why we took their values as200 and 500 in the previous section. These parameterscan be changed for obtaining different results by takinginto account the explanations in Section 4.In this paper, a single variable constrained opti-mization is used to optimize the length of interval forobtaining more accurate forecasts. Another optimiza-tiontechniquecanalsobeusedanddifferentresultscanbe obtained. In addition, we employed this optimiza-tion method in Chen’s model [2]. This optimizationmethod, however, can be used for other fuzzy timeseries approaches in determining the length of interval.For future studies, these can be performed to obtainmore effective lengths of interval and more accurateforecasts.
References
[1] R.P. Brent,
Algorithms for Minimization without Derivatives
,Prentice-Hall, Englewood Cliffs, New Jersey, 1973.[2] S.M. Chen, Forecasting enrollments based on fuzzy time-series,
Fuzzy Sets and Systems
81
(1996), 311–319.
E. Egrioglu et al. / A new approach based on the optimization of the length of intervals in fuzzy time series
19[3] G.E. Forsythe, M.A. Malcolm and C.B. Moler,
Computer Methods for Mathematical Computations
, Prentice Hall, NewJersey, 1976.[4] K. Huarng, Effective length of intervals to improve forecast-ing in fuzzy time-series,
Fuzzy Sets and Systems
123
(2001),387–394.[5] K. Huarng, Ratio-based lengths of intervals to improve fuzzytime series forecasting,
IEEE Transactions on Systems, Man.,and Cybernetics-Part B: Cybernetics
36
(2006), 328–340.[6] K.HuarngandH.Yu,Adynamicapproachtoadjustinglengthsof intervals in fuzzy time series forecasting,
Intell Data Anal
8
(1) (2004), 3–27.[7] K. Huarng and H. Yu, A type 2 fuzzy time series model forstock index forecasting,
Physica A
353
(2005), 445–462.[8] K. Huarng and H. Yu, The application of neural networks toforecast fuzzy time series,
Physica A
363
(2006), 481–491.[9] J.Kiefer,Sequentialminimaxsearchforamaximum,
Proceed-ingsoftheAmericanMathematicalSociety
4
(1953),502–506,MR0055639. doi:10.2307/2032161.[10] Q. Song and B.S. Chissom, Fuzzy time series and its models,
Fuzzy Sets and Systems
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(1993), 269–277.[11] Q. Song and B.S. Chissom, Forecasting enrollments withfuzzy time series – Part I,
Fuzzy Sets and Systems
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(1993),1–10.[12] Q.SongandB.S.Chissom,Forecastingenrollmentswithfuzzytime series – Part II,
Fuzzy Sets and Systems
62
(1994), 1–8.

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