A Forecasting Model Based on High-Order Fluctuation Trends and Information Entropy

2018·3.2 MB·English

by  Hongjun Guan, Zongli Dai, Shuang Guan, Aiwu Zhao

entropy
Article
A Forecasting Model Based on High-Order
Fluctuation Trends and Information Entropy
HongjunGuan1 ID,ZongliDai1,ShuangGuan2andAiwuZhao3,*
1 SchoolofManagementScienceandEngineering,ShandongUniversityofFinanceandEconomics,
Jinan250014,China;[email protected](H.G.);[email protected](Z.D.)
2 RensselaerPolytechnicInstitute,Troy,NY12180,USA;[email protected]
3 SchoolofManagement,JiangsuUniversity,Zhenjiang212013,China
* Correspondence:[email protected]
(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Received:31July2018;Accepted:3September2018;Published:4September2018
Abstract:Mostexistinghigh-orderpredictionmodelsabstractlogicalrulesthatarebasedonhistorical
discretestateswithoutconsideringhistoricalinconsistencyandfluctuationtrends. Infact,thesetwo
characteristics are important for describing historical fluctuations. This paper proposes a model
basedonlogicalrulesabstractedfromhistoricaldynamicfluctuationtrendsandthecorresponding
inconsistencies. In the logical rule training stage, the dynamic trend states of up and down are
mappedtothetwodimensionsoftruth-membershipandfalse-membershipofneutrosophicsets,
respectively. Meanwhile,informationentropyisemployedtoquantifytheinconsistencyofaperiod
of history, which is mapped to the indeterminercy-membership of the neutrosophic sets. In the
forecastingstage,thesimilaritiesamongtheneutrosophicsetsareemployedtolocatethemostsimilar
leftsideofthelogicalrelationship. Therefore,thetwocharacteristicsofthefluctuationtrendsand
inconsistencyassistwiththefutureforecasting. Theproposedmodelextendsexistinghigh-order
fuzzylogicalrelationships(FLRs)toneutrosophiclogicalrelationships(NLRs). Whencompared
withtraditionaldiscretehigh-orderFLRs,theproposedNLRshavehighergeneralityandhandle
the problem caused by the lack of rules. The proposed method is then implemented to forecast
TaiwanStockExchangeCapitalizationWeightedStockIndexandHengSengIndex. Theexperimental
conclusionsindicatethatthemodelhasstablepredictionabilityfordifferentdatasets.Simultaneously,
comparingthepredictionerrorwithotherapproachesalsoprovesthatthemodelhasoutstanding
predictionaccuracyanduniversality.
Keywords: high-orderfluctuationtrends;forecasting;informationentropy;neutrosophicsets
1. Introduction
Forstockmarketforecasts,summarizingtherulesthatcanbeusedforfuturepredictionsfrom
historicaldataiscrucial. Atthesametime,duetothenoisethatiscontainedintheactualdata,Song
andChissom[1–3]proposedafuzzytimeseriesmethodforgeneralruleextraction.Onthisbasis,some
scholarsproposedfirst-ordermodels[4–6],whereasothersfurtherstudiedhigh-ordermodels[7,8].
Thesestudiesnotonlyimplythathigh-ordermodelscanreflecthistoricaltrends,butalsoemphasize
theimportanceoftherelationshipbetweenhistoricaltrendsandcurrentstate.
However, most of above models summarize the logical relationships between current and
historicaltrendsthatareonlybasedondiscretestates. Infact,inadditiontothesimplestate-to-state
relationship,therelationshipsbetweenthehistoricalstatesandthecurrentstatesarealsorelatedto
otherfeatures. Forexample,Leeetal.[9]suggestedthatthefluctuationtrendsinrelatedstockmarkets
arelinkedwitheachother. Fromthispointofview,Leeetal. proposedapredictionmodelthatis
basedontworelatedstockmarketsthatextendedsuchmodelstotwo-factorhigh-orderprediction
Entropy2018,20,669;doi:10.3390/e20090669 www.mdpi.com/journal/entropy
Entropy2018,20,669 2of15
models. Inadditiontothe rangeofexternalinfluences, otherinternalfeaturesofa timeseriesare
ignoredbytraditionalautoregressivemodels. Forexample,atypicalphenomenon,called“callback”,
occursafteraperiodofcontinuousrise,whichmeansthattheinconsistenciesinhistoricalfluctuation
trendsmighthaveconsiderableimpactsonfuturefluctuationtrends. Inthisarea,Zhangetal.[10]
extractedtheinconsistentfeaturesofdifferenttypesofdigitalsignalsandselectedtheoptimalfeature
subsets from 16 entropy features. Jiang et al. [11] proposed a feature fusion model that is based
on information entropy (IE) and probabilistic neural network, which uses IE theory to extract the
characteristicentropiesinvibrationsignals. ThesestudiesinspiredustothinkthatIEcanbeemployed
todescribetheinconsistencyinaperiodoffluctuationinatimeseries.
TheconceptofinformationentropywasfirstproposedbyShannonin1948[12]. IEisatoolfor
measuringthedegreeoforderingofsystems. ThelowertheIE,thelesschaos, andthehigherthe
order. Conversely,thelargertheIE,thelowertheorder. IEsolvestheproblemofquantificationand
describingsystemcomplexityandinconsistency. Forexample,inordertoindicatetheinconsistencyof
abatteryduetosmalldeviationsanduncertaintiesinitsproductionprocess,Duanetal.[13]developed
acomprehensiveevaluationmethodforbatteryinconsistencybasedonIE.Xuetal.[14]proposedthe
informationentropyriskmeasure,whichappliedIEtodescribetheuncertaintyindecisionriskand
theinconsistencyofdecisioninformation. Basedontheuncertaintyofrainfalldistributionandthe
inconsistencyofinformation,Wangetal.[15]employedIEtoanalyzethetime-variationinsimilar-scale
rainfallnetworksincities. Informationentropyhasbeenwidelyusedinmanyothersituations,such
asgraphicsdetection[16],algorithmoptimization[17],environmentalevaluation[18],andbigdata
analysisandmining[19].
Inordertofullyexpressthethreestatesofup,down,andinconsistencyofaperiodinhistory
in a logical relationship and to conveniently compare them, the expression variable should have
three dimensions. This is similar to the structure of neutrosophic sets (NSs) and the similarity
comparisonsofdifferentNSs. TheNSwasintroducedbySmarandache[20],whichconsistsoftrue
membership,indeterminacymembership,andfalsemembership. Fromtheperspectiveofinformation
representation,scholarshaveproposedthreespecificconceptsthatarebasedontheneutrosophicset:
single-valuedNSs[21], interval-valuedNSs[22], andsimplifiedNSs[23], includingsingle-valued
andinterval-valuedNSs[24]. Theseconceptsseekamoredetailedrepresentationoftheinformation,
enablingtheNSstoexpressinconsistentinformationmoreaccurately. Atpresent,NSshaveattracted
theattentionofscholarsinvariousfields. Forinstance,Abdel-Basseetal.[25]discussedtheroleof
NSsinurbanconstructionandfurtheranalyzeditsadvantagesintheconstructionofimperfectand
incompleteinformationsystems. S¸ahinetal.[26]discussedtheapplicationofsingle-valuedNSsin
decisionmaking. Vanetal.[27]appliedNSstogreensupplierevaluationandselection. Inaddition,
sincesimilarityisanimportantmethodformeasuringNSs, similaritywasalsousedinnumerous
studies[28–30].
As mentioned above, stock market fluctuation is a dynamic nonlinear system. A common
questionishowcomplexsystemscanbesynchronizedandwhichparametershaveadirectimpacton
thesystem’sevolution. Researchersthinkthattherearerelationshipsbetweenhistory,current,and
future. Itismorelikelyasmall-worldnetwork[31]. Therefore,high-orderfuzzytimeseriesmodels
havebeendevelopedtodescribethefluctuationsinatimeperiodandgenerateforecastingrulesthat
arebasedondiscretehigh-orderstatesforfutureprediction. Themostproblemofthesemodelsis
thelackofcorrespondingrulesduetolimitedtrainingsamples. Thiswouldsignificantlyreducethe
accuracyandcredibilityofsuchmodels. Theproposedmodelsummarizesthediscretehigh-order
fluctuationstatestotruth-membershipofupper-trend,falsity-membershipofupper-trend,andchaos
of trends, and then maps them to the three dimensions of truth-membership, falsity-membership,
andindeterminacy-membershipofaneutrosophicset. Itextendsexistinghigh-orderfuzzylogical
relationshipstoneutrosophiclogicalrelationships. Therefore,themodelcanemployNStheoryto
deal with related issues, such as the comparison of high-order states during the location of rules.
Comparedwithtraditionaldiscretehigh-orderfuzzylogicalrelationships(FLRs), theproposedto
Entropy2018,20,669 3of15
neutrosophiclogicalrelationships(NLRs)havehighergeneralityandcanaddresstheproblemthatis
causedbythelackofrulesintheforecastingstage.
Inspiredbytheaboveresearch,weproposeapredictionmodelbasedonhigh-orderfluctuation
trendsandinformationentropy. First,theoriginaltimeseriesofthestockmarketisconvertedintoa
fluctuationtimeseries,andthenthefluctuationtimeseriesisblurredintoafuzzyfluctuationtime
seriesaccordingtoapredefinedlabel. Second,theIEofitshistoricalfluctuationsiscalculatedbasedon
theprobabilityofdifferentstatesofeachcurrentvalue. Third,theNSisusedtorepresentthecurrent
stateandtoestablishaneutrosophiclogicalrelationships. Fourth,theJaccardsimilaritymeasureis
usedtoseekoutsimilarappropriatelogicalrelationshipgroupsandcalculateitsexpectedNSs. Finally,
weobtainthedesiredexpectedNSs,thenthefinalpredictedvalueiscalculatedthroughtheprocessof
deneutrosophication. Forverification,theproposedmodelisimplementedtoforecastTaiwanStock
ExchangeCapitalizationWeightedStockIndex(TAIEX)andHengSengIndex(HIS).Theexperimental
conclusionsindicatethatthemodelhasstablepredictionabilityfordifferentdatasets. Simultaneously,
comparingthepredictionerrorwithotherapproachesalsoprovesthatthemodelhasoutstanding
predictionaccuracyanduniversality.
The remainder of this article can be summarized as follows: Section 2 introduces some
basicconceptsoffuzzyfluctuationtimeseries, IE,deneutrosophicationofaneutrosophicset, and
neutrosophic logical relationships. Section 3 presents a new prediction method that is based on
high-orderfluctuationtrendsandNSs.InSection4,theTAIEXdataset(1997–2005)andHIS(1998–2012)
areusedtopredictfuturevalues. Section5presentstheconclusionsofthispaperandthepotential
issuesforfutureresearch.
2. Preliminaries
2.1. FuzzySet(FS)
FuzzysettheorywasproposedbyZadeh[32],andithasbeenwidelyappliedinseveralfields.
Abriefintroductiontothebasicconceptsoffuzzysetfollows.
Definition1. DenotetheuniverseofdiscourseasU={u ,u ,... ,u }. AfuzzysetL={L ,L ,... ,L }inU
1 1 n 1 2 g
canbedefinedbyitsmembershipfunction:
L =f (u )/u +f (u )/u +... +f (u )/u (j=1,2,... ,g) (1)
j Lj 1 1 Lj 2 2 Lj n n
wheref U→[0,1]isthemembershipfunctionofthefuzzysetL,andf (u)isthemembershipdegreeofu
Lj: j Lj i i
belongingtoL,wheref (u)∈[0,1](I=1,2,... ,n,j=1,2,...,g). Thesymbol+isnottheconventional
j Lj i
operationofadditionbutunion.
LetfuzzysetL={L ,L ,... ,L }beafiniteandfullyordereddiscretetermset,wheregisanodd
1 2 g
numberinrealsituationswiththefirst(g−1)/2elementsdescribethedegreeofapropertyandthe
last(g−1)/2elementsfromtheoppositedescription. Forexample,wheng=7,itmightrepresenta
setoflinguisticvariantsas: L={L ,L ,L ,L ,L ,L ,L }={verybad,bad,belowfair,fair,abovefair,
1 2 3 4 5 6 7
good,verygood},andsoon. TherelationshipbetweentheelementL(I=1,2,... ,g)anditssubscripti
i
isstrictlymonotonicallyincreasing[33],sothefunctioncanbedefined,asfollows: f: L =f(i). Clearly,
i
thefunctionf(i)isastrictlymonotonicallyincreasingfunctionaboutasubscripti.
2.2. FuzzyTimeSeries(FTS)
FuzzytimeserieswasproposedbySongandChissomin1993[2]. Ithasbeensuccessfullyusedto
solvemanypracticalproblems. Thebriefintroductionofbasicconceptsoffuzzytimeseries,asfollows.
Definition2.Letthetimeseries{Y(t)|t=1,2,... ,T},asubsetofrealnumber,denotetheuniverseofdiscourse.
AccordingtoDefinition1,eachelementinthetimeseriescanbefuzzifiedtoafuzzysetelementF(t)(t=1,2,... ,
Entropy2018,20,669 4of15
T),then{F(t)|t=1,2,... ,T}iscalledafuzzytimeseriesdefinedonthetimeseries{Y(t)|t=1,2,... ,T}.
IfthereexistsafuzzyrelationR(t–p,t),suchthat
F(t)=F(t−p)◦R(t−p,t) (2)
where◦isamax-mincompositionoperator,F(t)iscalledderivedfromF(t−p),denotedbythefuzzylogical
relationship(FLR)F(t−p)→F(t). F(t–p)andF(t)arecalledtheleft-handside(LHS)andtheright-hand
side(RHS)oftheFLR,respectively. FLRswiththesameLHScanbecategorizedintoanorderedfuzzylogical
group(FLG).
IfF(t)iscausedbyF(t−1),F(t−2),... ,F(t−p),thenthep-orderFLRcanberepresentedby:
F(t−p),F(t−p+1),... ,F(t–1)→F(t)
2.3. Fuzzy-FluctuationTimeSeries(FFTS)
Definition 3. Let {X(t)|t = 1, 2, ... , T} be a time series, where t is the number of the time series. Let
{Y(t)|t=2,3... T}beafluctuationtimeseries, where{Y(t)=X(t) − X(t − 1)|t=2, 3, ... , T}. Thenthe
correspondingfuzzytimeseries{Q(t)|t=2,3,... ,T}iscalledafuzzy-fluctuationtimeseries(FFTS).
Definition 4. Let Q(t) (t = m + 1, m + 2, ... , T, m ≥ 1) be a FFTS. If Q(t) is caused by Q(t − 1),
Q(t−2),... ,Q(t−m),thenthefuzzy-fluctuationlogicalrelationshipisrepresentedby:
Q(t−1),Q(t−2),... ,Q(t−m)→Q(t) (3)
anditiscalledthem-orderfuzzy-fluctuationlogicalrelationship(FFLR)ofthefuzzy-fluctuationtimeseries,
whereQ(t−1),Q(t−2),... ,Q(t−m)iscalledtheleft-handside(LHS)andQ(t)iscalledtheright-handside
(RHS)oftheFFLR,andQ(k)(k=t,t−1,t−2,... ,t−m)∈L.
2.4. InformationEntropy
Information entropy (IE) [12] was firstly defined by Shannon in 1948 as a measure of event
uncertainty. Shannonsuggestedthatthesmallerthepossibilityofanincident,thegreatertheamount
of information that it contains. Conversely, the greater the likelihood of an event, the smaller the
amountofinformation. So,theamountofinformationcanbeexpressedasafunctionoftheprobability
ofoccurrenceofanevent.
Definition5.
N
∑
E = − p(x )log (p(x )) (4)
i 2 i
i=1
wherep(x)representstheprobabilityofoccurrenceoftheithevent. Inaddition,theinformationentropymust
i
satisfythefollowingconditions: ∑N p(x ) =1, 0< p(x ) <1andnon-negativity: E ≥0.
i=1 i i
2.5. NeutrosophicSet(NS)
Neutrosophicset(NS)wasproposedbySmarandache[20]. Ithasbeenwidelyusedtodescribe
complexphenomena.
Definition6. LetXbeaspaceofpoints(objects),withagenericelementinXdenotedbyx. Aneutrosophicset
AinXischaracterizedbyatruth-membershipfunctionT (x),aindeterminacy-membershipfunctionI (x),and
A A
afalsity-membershipfunctionF (x). ThefunctionsT (x),I (x),andF (x)arerealstandardornonstandard
A A A A
subsetsof]0−,1+[. ThereisnorestrictiononthesumofT (x),I (x),andF (x).
A A A
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2.6. NeutrosophicLogicalRelationship(NLR)
Definition7. LetibethesubscriptofafuzzysetelementL(i=1,2,... ,g),A(t)betheLHSofanm-order
i
FFLR Q(t−1),Q(t−2),...,Q(t−m) → Q(t),Pi betheprobabilitiesofcorrespondingL(i=1,2,... ,
A(t) i
g)inA(t). Pi canbegeneratedby:
A(t)
∑m w
Pi = j=1 i,j i =1,2,...,g (5)
A(t) m
wherew =1ifQ(t−j)=L and0otherwise.
i,j i
Let A(t) be the LHS of a m-order FFLR Q(t−1),Q(t−2),...,Q(t−m) → Q(t), E be
A(t)
the entropy of A(t), Pi be the probabilities of corresponding L(i = 1, 2, ... , g) in A(t).
A(t) i
A(t) can be represented by a neutrosphic set N , with the truth-membership function T ,
A(t) NA(t)
indeterminacy-membership I ,andfalsity-membershipfunctionF definedby:
NA(t) NA(t)
 g−1
 TNA(t) = i∑=21αiPAi(t)
INA(t) = EA(t) (6)
 FNA(t) =i=g∑−g1+1βiPAi(t)
2
where α,β are the weights of L(i = 1, 2, ... , g) in terms of their contribution to represent the
i i i
correspondingcharacteristics,∑(g−1)/2α =1,∑g β =1.
i=1 i i=g−1+1 i
2
The same as the definition of FLG, N is called the left-hand side (LHS) of a neutrosophic
A(t)
logicalrelationship(NLR).ThesimilarLHSsofNLRscanbecategorizedandgrouptheirRHSsof
correspondingFFLRsintoanorderedfuzzylogicalgroup(FLG)B ,whichcanalsoberepresented
A(t)
byaneutrosophicsetN(cid:48) ,referringtotheabovemethod.
A(t)

(g−1)/2
 TNA(cid:48)(t) = i∑=1 αiPBiA(t)
INA(cid:48)(t) = EBA(t) (7)
 FNA(cid:48)(t) =i=g∑−g1+1βiPBiA(t)
2
wherePi representstheprobabilityofcorrespondingL(i=1,2,... ,g)inB .
BA(t) i A(t)
Thus,aneutrosophiclogicalrelationship(NLR)canbedefinedas:N → N(cid:48) .
A(t) A(t)
Definition8. LetN andN betwoNSs. TheJaccardsimilarity[34]betweenN andN invector
A(t1) A(t2) A(t1) A(t2)
spacecanbeexpressedasfollows:
(cid:16) (cid:17)
J NA(t1),NA(t1)
= TNA(t1)TNA(t2)+INA(t1)INA(t2)+FNA(t1)FNA(t2) (8)
(cid:18) (cid:19)2 (cid:18) (cid:19)2 (cid:18) (cid:19)2 (cid:18) (cid:19)2 (cid:18) (cid:19)2 (cid:18) (cid:19)2 (cid:18) (cid:19)
TNA(t1) + INA(t1) + FNA(t1) + TNA(t2) + INA(t2) + FNA(t2) − TNA(t1)TNA(t2)+INA(t1)INA(t2)+FNA(t1)FNA(t2)
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2.7. DeneutrosophicationofaNeutrosphicSet
Definition9. Deneutrosophicationofaneutrosophicfluctuationsetreferstoconvertinganeutrosophicsetx
intoafuzzysetybythefollowingfunctionaccordingtoAlietal.[35]:
f(T(x),I(x),F(x)) : ([0,1],[0,1],[0,1])→ [0,1]
Specifically,inthispaper,ourequationfordeneutrosophicationisasfollows:
y = γT(x)+θI(x)+δF(x) (9)
3. ProposedModelBasedonHigh-OrderFluctuationTrendsandInformationEntropy
This paper presents a prediction model that is based on high-order fluctuation trends and
information entropy. Different from existing high-order fuzzy time series forecasting models, the
proposed model summarizes the discrete high-order fluctuation states into truth-membership of
upper-trend,falsity-membershipofupper-trend,andchaosoftrends. Itcoincideswiththedefinition
ofneutrosophicset,whichemploysthethreedimensionsoftruth-membership,falsity-membership,
andindeterminacy-membershiptodescribeacharacteristic. BasedontheNSexpressionofhigh-order
fluctuationstates,theNStheorycanbeusedtodealwithrelatedissues,suchasthecomparisonof
high-orderstatesduringthelocationofrules. Themostsignificantcontributionofthismodelisthat
itextendsexistinghigh-orderfuzzylogicalrelationshipstoneutrosophiclogicalrelationshipsandit
employsinformationentropytorepresenttheindeterminacyoffluctuationtrends. Whencompared
withtraditionaldiscretehigh-orderFLRs,theproposedNLRshavehighergeneralityandcanhandle
theproblemcausedbythelackofrulesintheforecastingstage. Thedetailedstepsareshownasfollow
stepsandinFigure1.
Step1ConstructFFTSforthehistoricaltrainingdata.
ForeachelementX(t)(t=1,2,... ,T)inthehistoricaltrainingtimeseries,itsfluctuationtrend
is determined by Y(t) = X(t) − X(t − 1) (t = 2, 3, ... , T). Y(t) can be fuzzified into a linguistic set
{L ,L ,L ,L ,L }={down,slightlydown,equal,slightlyup,up}. Letlenbethewholemeanofallthe
1 2 3 4 5
elementsinthefluctuationtimeseriesY(t)(t=2,3,... ,T),andwedefineintervalsu = [−∞,−1.5len],
1
u = [−1.5len,−0.5len], u = [−0.5len,0.5len], u = [0.5len,1.5len], and u = [1.5len,+∞]. Then,
2 3 4 5
Y(t)(t=2,3,... ,T)canbefuzzifiedintoafuzzyfluctuationtimeseriesQ(t)(t=2,3,... ,T).
Step2Establishm-orderFFLRsforthetrainingdataset
AccordingtoDefinition4,eachQ(t)(t>m)inthetrainingsetcanberepresentedbyitspreviousm
days’fuzzy-fluctuationnumbers. Then,them-orderFFLRsofthepredictionmodelcanbeestablished.
Step3Calculatefluctuationinformationentropy
AccordingtotheDefinition5,theinformationentropyofthem-orderfluctuationcanbeseparately
calculated. Amongthem,p(x ),p(x ),p(x ),p(x ),andp(x ),respectively,indicatetheprobabilityof
1 2 3 4 5
occurrenceofthelinguisticvariantsL ,L ,L ,L ,andL intheLHS.Then,theinformationentropy
1 2 3 4 5
of the m-order FFLRs can be obtained according to Equation (4), which will be the element of the
indeterminacymembershipoftheNSs.
Step4ConverttheLHSsofFFLRstoNSs
Calculate the probability of corresponding L(i = 1, 2, ... , 5) in the LHSs of training data set,
i
accordingtoEquation(5). Combinedwiththeinformationentropyofthem-orderFFLRsobtainedin
thepreviousstep,converttheleft-handsofFFLRsintoneutrosophicsetsaccordingtoEquation(6).
Entropy2018,20,669 7of15
Step5Groupingandoptimization
ApplyingtheJaccardsimilaritymeasuremethod,categorizeandgrouptheconvertedLHSsof
FFLRsaccordingtheirsimilarities. ConverttheRHSsofcorrespondingFFLRsintoneutrosophicsets
Entropy 2018, 20, x FOR PEER REVIEW 6 of 15
accordingtoEquation(7). NLRsareobtainedfromthetrainingdatasetsforfutureforecasting.
falsity-membership, and indeterminacy-membership to describe a characteristic. Based on the NS
Step6Forecasttesttimeseries
expression of high-order fluctuation states, the NS theory can be used to deal with related issues,
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eachctoesnttrdibautatiosne t.ofA tchcios rdmiondgelt oisD tehfiant iitti oenxt9e,ntdhse eRxHistSinogf thhigehN-oLrdRerc afnuzbzeyc olongvicearlt erdelatotioannsheixppse ctota tion
fuzzynevuatlruoesoYp(hii+c 1l)o.gTichaeln ,rcealaltciuonlastheipths earneadl niut mebmeprlooyfst heinflfourcmtuataitoino nebnytr:oFp(cid:48)y( i+to1 )re=prYes(cid:48)e(ni+t 1th)e× len.
indeterminacy of fluctuation trends. When compared with traditional discrete high-order FLRs, the
Finally,thepredictedvaluecanbeobtainedfromtheactualvalueofthepreviousdayX(i)andthe
proposed NLRs have higher generality and can handle the problem caused by the lack of rules in
predictedfluctuationvalueF’(i+1): X(cid:48)(i+1) = X(i)+F(cid:48)(i+1).
the forecasting stage. The detailed steps are shown as follow steps and in Figure 1.
Y(t)=X(t)−X(t−1),(t=2,3,...,T)
T
|Y(t)|
len= t=2
T−1
L Y(t)∈[−∞,−1.5len)
1
L Y(t)∈[−1.5len,−0.5len)
F(t)=L2 Y(t)∈[−0.5len,0.5len)
3
L Y(t)∈[0.5len,1.5len)
 4
L5 Y(t)∈[1.5len,+∞)
Q(m),Q(m-1),...,Q(1)→Q(m+1)
Q(m+1),Q(m),...,Q(2)→Q(m+2)

Q(t−1),Q(t−2),...,Q(t−m)→Q(t)

Figure1. Flowchartofpredictionmodelbasedonhigh-orderFuzzy-Fluctuationtimeseries(FFTS)
andiFnifgourrme a1t.i oFlnowen cthroaprty o(f IpEr)e.diction model based on high-order Fuzzy-Fluctuation time series (FFTS)
and information entropy (IE).
Step 1. Construct FFTS for the historical training data.
Entropy2018,20,669 8of15
4. EmpiricalAnalysis
4.1. ForecastingTaiwanStockExchangeCapitalizationWeightedStockIndex(TAIEX)
The Taiwan stock market has always been the focus of research in the field of stock market
forecasting. TheTAIEXisanindicatorofthechangeinthevalueofTaiwan’soverallmarketstocks,
whichisseenasawindowtotheTaiwaneseeconomy. Inaddition,alargenumberofstudieshave
usedTAIEXdataasanexampletoillustratetheirproposedpredictionmethods[36–41]. Tofacilitate
acomparisonwiththeaccuracyofthesemodels,wealsousedtheTAIEXdatasettoillustrateour
proposed method. Specifically, while referring to the general practice of the above studies, the
forecastingprocessinthissectionisbasedonTAIEX’s1999data. Similarly,wealsoselecteddatafrom
January1999toOctober1999asatrainingset,andselectedfromNovembertoDecember1999asatest
dataset.
Step1ConstructFFTSforthehistoricaltrainingdata
Thefluctuationtrendisconstructedbasedontheelementsofthehistoricaltrainingdataset. Then,
whileusingtheoverallmeanofthenumberoffluctuationsinthetrainingdataset,thefluctuation
trendisblurredtoFFTS,e.g.,theaveragevalueofthetrainingdatasetforTAIEX.Therefore,lenis85.
Then,X(1)=6152.43,X(2)=6199.91,andY(2)=X(2)−X(1)=47.48. Further,F(2)=L ,meansslightly
4
up. Thus,thetrainingsetcanbeconvertedtofuzzyfluctuationset.
Step2Establishm-orderFFLRsforthetrainingdataset
Different orders directly affect the effect of the prediction. Here, we only need to explain the
process,sowetakethe9th-orderasanexample. Then,accordingtoDefinition4,the9th-orderFFLRs
ofthepredictionmodelwereestablished.
Step3Calculatingfluctuationinformationentropy
AccordingtoEquation(4),theinformationentropyofthe9th-orderfluctuationtimeseriescan
becalculatedseparately. Forexample,the9th-orderFFLRsonNovember1,1999canberepresented
by (L , L , L , L , L , L , L , L , L ), which can be calculated as p(x ) = 0, p(x ) = 1/9, p(x ) = 6/9,
3 3 2 4 3 3 3 3 5 1 2 3
p(x )=1/9,andp(x )=1/9. Therefore,itsinformationentropyis1.4466,whichcanbenormalized
4 5
to0.5306.
Step4ConverttheLHSsofFFLRstoNSs
CalculatetheprobabilityofcorrespondingL(i=1,2,... ,5)intheLHSsofthetrainingdataset
i
andthenconverttheleft-handsofFFLRsintoneutrosophicsetsaccordingtoEquation(6).
Step5Groupingandoptimization
Firstly,applytheJaccardsimilaritymeasuremethodtocategorizeandgrouptheconvertedLHSs
ofFFLRs. Inthisexample,thethresholdsimilarityvalueissetto0.94. Then,converttheRHSsofthe
correspondingFFLRsintoneutrosophicsetsaccordingtoEquation(7). Theconversionandgrouping
processisshowninFigure2.
Entropy 2018, 20, x FOR PEER REVIEW 8 of 15
fluctuation trend is blurred to FFTS, e.g., the average value of the training data set for TAIEX.
Therefore, len is 85. Then, X(1) = 6152.43, X(2) = 6199.91, and Y(2) = X(2) − X(1) = 47.48. Further, F(2)
= L4, means slightly up. Thus, the training set can be converted to fuzzy fluctuation set.
Step 2. Establish m-order FFLRs for the training data set
Different orders directly affect the effect of the prediction. Here, we only need to explain the
process, so we take the 9th-order as an example. Then, according to Definition 4, the 9th-order
FFLRs of the prediction model were established.
Step 3. Calculating fluctuation information entropy
According to Equation (4), the information entropy of the 9th-order fluctuation time series can
be calculated separately. For example, the 9th-order FFLRs on November 1, 1999 can be represented
by (L3, L3, L2, L4, L3, L3, L3, L3, L5), which can be calculated as p(x1) = 0, p(x2) = 1/9, p(x3) = 6/9, p(x4) = 1/9,
and p(x5) = 1/9. Therefore, its information entropy is 1.4466, which can be normalized to 0.5306.
Step 4. Convert the LHSs of FFLRs to NSs
Calculate the probability of corresponding Li(i = 1, 2, …, 5) in the LHSs of the training data set
and then convert the left-hands of FFLRs into neutrosophic sets according to Equation (6).
Step 5. Grouping and optimization
Firstly, apply the Jaccard similarity measure method to categorize and group the converted
LHSs of FFLRs. In this example, the threshold similarity value is set to 0.94. Then, convert the RHSs
Eonft rtohpey 2c0o18r,r2e0s,p6o69nding FFLRs into neutrosophic sets according to Equation (7). The conversion9 oafn1d5
grouping process is shown in Figure 2.
L5 ,L3,L3,L2,L2,L2,L5,L3,L2→L3 (0.1778 ,0.5778 ,0.1333)
L4 ,L3,L4,L3,L2,L5,L3,L3,L3→L3 (0.0444 ,0.6494 ,0.1556)
L3 ,L4,L3,L2,L5,L3,L3,L3,L3→L4 (0.0444 ,0.5306 ,0.1111)
L4 ,L3,L2,L5,L3,L3,L3,L3,L4→L3 (0.0444 ,0.6494 ,0.1556)
L3 ,L2,L5,L3,L3,L3,L3,L4,L3→L2 (0.0444 ,0.5306 ,0.1111)
LLLLLLLLLL2533333344 ,,,,,,,,,,LLLLLLLLLL5333353444,,,,,,,,,,LLLLLLLLLL3333555443,,,,,,,,,,LLLLLLLLLL3334523434,,,,,,,,,,LLLLLLLLLL3343233343,,,,,,,,,,LLLLLLLLLL3432334433,,,,,,,,,,LLLLLLLLLL4323354331,,,,,,,,,,LLLLLLLLLL3235534313,,,,,,,,,,LLLLLLLLLL2354333134→→→→→→→→→→LLLLLLLLLL3544344343 (((((((((((00000000000...........00000000000844444066668444440666694444407777 ,,,,,,,,,,,00000000000...........65564445554433477700079007770007946642220002 ,,,,,,,,,,,00000000000...........11112221111111000777351115000777311160008883)))))))))))
L4 ,L3,L4,L3,L3,L1,L3,L4,L3→L2 (0.1111 ,0.6494 ,0.0889)
L3 ,L4,L3,L3,L1,L3,L4,L3,L2→L2 (0.0444 ,0.5306 ,0.1111)
L4 ,L3,L5,L3,L3,L2,L3,L3,L3→L3 (0.0889 ,0.5778 ,0.1333)
L3 ,L3,L2,L3,L2,L4,L4,L4,L3→L4 (0.0889 ,0.5778 ,0.1778)
L3 ,L2,L3,L2,L4,L4,L4,L3,L4→L5 (0.0000 ,0.4772 ,0.2000)
L3 ,L4,L5,L4,L4,L3,L3,L3,L3→L4 (0.1111 ,0.6494 ,0.1333)
LLLL3432 ,,,,LLLL5222,,,,LLLL3323,,,,LLLL3233…,,,,LLLL1232…,,,,LLLL 5324,,,,LLLL3343,,,,LLLL2233,,,,LLLL3433→→→→LLLL4333 (((000...111733733833 ,,,000...544……777777……822 ,,,000...000844844944)))

Figure 2. Conversion and group process of fuzzy-fluctuation logical relationship (FFLRs).
Figure2.Conversionandgroupprocessoffuzzy-fluctuationlogicalrelationship(FFLRs).
Step 6. Forecast test time series
Step6Forecasttesttimeseries
For each test data point, calculate its 9th-order historical fluctuation states and convert them to
Foreachtestdatapoint,calculateits9th-orderhistoricalfluctuationstatesandconvertthemtoa
a neutrosophic set. Locate the NLR with the highest similarity. Then, the corresponding RHS of the
neutrosophicset. LocatetheNLRwiththehighestsimilarity. Then,thecorrespondingRHSoftheNLR
NLR is the forecasting neutrosophic value. For example, we chose 1 November 1999 as an example
istheforecastingneutrosophicvalue. Forexample,wechose1November1999asanexampleofthe
of the test data. The LHS of its fluctuation states is (L3, L3, L4, L3, L2, L3, L5, L4, L4, L3, L4, L4, L3, L4, L3, L2,
testdata. TheLHSofitsfluctuationstatesis(L ,L ,L ,L ,L ,L ,L ,L ,L ,L ,L ,L ,L ,L ,L ,L ,L ,
L2, L3, L4, L5, L4, L4, L3, L3, L3), which can be rep3rese3nte4d b3y a2 ne3utro5sop4hic4 se3t (04.04840, 03.60485, 30.129202).
L ,L ,L ,L ,L ,L ,L ,L ),whichcanberepresentedbyaneutrosophicset(0.0480,0.6085,0.1920).
A3cco4rdi5ng t4o th4e s3imi3larit3y comparison, the most appropriate NLR for forecasting is (0.0444, 0.5306,
Accordingtothesimilaritycomparison,themostappropriateNLRforforecastingis(0.0444,0.5306,
0.1111)→(0.0480, 0.6085, 0.1920).
0.1111)→(0.0480,0.6085,0.1920).
Then, calculate the predicted fuzzy fluctuation:
Then,calculatethepredictedfuzzyfluctuation:

Y(cid:48)(i+1) = (−0.0480)+0.1920=0.144
Calculatetherealnumberofthefluctuation:
F(cid:48)(i+1) =Y(cid:48)(i+1)×len =0.144×85=12.24
Finally,Thepredictedvaluecanbeobtainedfromtheactualvalueofthepreviousdayandthe
predictedfluctuationvalue:
X(cid:48)(i+1) = X(i)+F(cid:48)(i+1) =7854.85+12.24=7867.09
Accordingtotheabovesteps,wecanobtainotherpredictionresultsthatareshowninTable1and
Figure3.
Entropy2018,20,669 10of15
Table1. PredictedandactualTaiwanStockExchangeCapitalizationWeightedStockIndex(TAIEX)
valuesfrom1November1999to30December1999.
Date Date
Actual Forecast (Forecast−Actual)2 Actual Forecast (Forecast−Actual)2
(DD/MM/YYYY) (DD/MM/YYYY)
1/11/1999 7814.89 7867.09 2724.84 1/12/1999 7766.20 7723.08 1859.33
2/11/1999 7721.59 7826.66 11,039.54 2/12/1999 7806.26 7768.72 1408.89
3/11/1999 7580.09 7723.14 20,462.00 3/12/1999 7933.17 7807.11 15,891.12
4/11/1999 7469.23 7577.85 11,798.99 4/12/1999 7964.49 7932.85 1001.33
5/11/1999 7488.26 7466.21 486.03 6/12/1999 7894.46 7964.17 4859.38
6/11/1999 7376.56 7485.29 11,822.59 7/12/1999 7827.05 7893.82 4458.83
8/11/1999 7401.49 7370.82 940.50 8/12/1999 7811.02 7825.78 217.71
9/11/1999 7362.69 7395.82 1097.82 9/12/1999 7738.84 7810.16 5086.43
10/11/1999 7401.81 7357.09 1999.66 10/12/1999 7733.77 7736.72 8.67
11/11/1999 7532.22 7404.90 16,210.15 13/12/1999 7883.61 7734.83 22,134.74
15/11/1999 7545.03 7526.69 336.36 14/12/1999 7850.14 7882.73 1062.35
16/11/1999 7606.20 7541.78 4149.94 15/12/1999 7859.89 7849.27 112.73
17/11/1999 7645.78 7606.20 1566.58 16/12/1999 7739.76 7868.73 16,633.26
18/11/1999 7718.06 7653.40 4180.83 17/12/1999 7723.22 7747.76 602.21
19/11/1999 7770.81 7719.84 2598.34 18/12/1999 7797.87 7730.15 4586.55
20/11/1999 7900.34 7784.54 13,409.46 20/12/1999 7782.94 7799.05 259.55
22/11/1999 8052.31 7919.68 17,589.44 21/12/1999 7934.26 7784.10 22,546.71
23/11/1999 8046.19 8058.99 163.80 22/12/1999 8002.76 7938.25 4161.84
24/11/1999 7921.85 8052.64 17,105.57 23/12/1999 8083.49 8002.58 6546.57
25/11/1999 7904.53 7925.99 460.58 24/12/1999 8219.45 8084.73 18,148.43
26/11/1999 7595.44 7908.62 98,080.84 27/12/1999 8415.07 8224.42 36,348.84
29/11/1999 7823.90 7597.74 51,147.40 28/12/1999 8448.84 8418.62 913.11
30/11/1999 7720.87 7823.13 10,456.55 RootMeanSquareError(RMSE) 102.05
Entropy 2018, 20, x FOR PEER REVIEW 10 of 15
8600
S
Actual
T
O 8400 Forecast
C
K
8200
I
N 8000
D
E
7800
X
7600
7400
7200
1 6 11 16 21 26 31 36 41
DAY

Figure 3. The stock market fluctuation of predicted and actual values for November to December
Figure3.ThestockmarketfluctuationofpredictedandactualvaluesforNovembertoDecember1999.
1999.
Inexperimentalanalysis,someofthemethodsareusedtomeasuretheaccuracyofpredictionin
In experimental analysis, some of the methods are used to measure the accuracy of prediction
ordertoquantifytheeffectofmodelprediction. Thesemethodsaremainlyusedinthepredictionfield,
in order to quantify the effect of model prediction. These methods are mainly used in the prediction
includingthemeanpercentageerror(MPE),themeansquareerror(MSE),andtherootmeansquared
field, including the mean percentage error (MPE), the mean square error (MSE), and the root mean
squared error (RMSE). Since there is no significant difference in the effect of different error
calculation methods, RMSE was chosen as the main formula for error calculation.
∑(cid:3041) ((cid:1858)(cid:1867)(cid:1870)(cid:1857)(cid:1855)(cid:1853)(cid:1871)(cid:1872)((cid:1872))−(cid:1853)(cid:1855)(cid:1872)(cid:1873)(cid:1853)(cid:1864)((cid:1872)))(cid:2870)
(cid:1844)(cid:1839)(cid:1845)(cid:1831) =(cid:3496) (cid:3047)(cid:2880)(cid:2869) (10)
(cid:1866)
Since different orders affect the prediction effect, in order to reduce the experimental error and
improve the accuracy of the prediction, it was necessary to select the optimal order. Experimental
analysis showed that when the order was nine, the predictability of the model is better. Table 2
shows the experimental errors for different years under different orders.
Table 2. Comparison of forecasting errors for different m-orders.
m
6 7 8 9 10 11 12 13 14
RMSE 102.76 102.52 102.2 102.05 102.61 102.89 102.9 102.88 103.4
It is important and necessary to accurately predict the trend of fluctuations. Some of the most
recent articles have proposed many excellent prediction methods. Therefore, by comparing our
method with the previous methods, the advantages and disadvantages of our model can be
verified.
Table 3 shows the prediction errors for different methods from 1997 to 2005. By comparing with
the latest research, we found that the overall prediction effect of this method was excellent. For
example, the average error of Guan et al.’s model [40] is 94.5, the average error of Cheng et al.’s model
[41] is 102.4, and the error of the proposed model is 92.16. Upon further analysis, from the annual