Novel FuzzyApproachtoTimeSeriesAnalysisandPredictionMartinStepnicka,ViktorPavliska,VilemNovak,IrinaPerlieva,AntonnDvorakInstituteforResearchandApplicationsofFuzzyModelingUniversityofOstrava,30.dubna22,Ostrava1martin.stepnicka@osu.czEmployedmethods TimeSeries Forecasting ResultsandConclusionsOutline1 Employedmethods2 TimeSeries3 Forecasting4 ResultsandConclusionsEmployedmethods TimeSeries Forecasting ResultsandConclusionsOutline1 Employedmethods2 TimeSeries3 Forecasting4 ResultsandConclusionsEmployedmethods TimeSeries Forecasting ResultsandConclusionsBasicPrinciples Fuzzytransform(F-transform)BasicprincipleThetimeseriesismappedtoanotherspacewhereonebasiccharacteristicsprevails Perception-basedlogical deduction(PbLD)BasicprincipleTheconclusionisobtainedonthebasisoflinguisticallyformulatedknowledgeandlinguisticallycharacterizedobservationEmployedmethods TimeSeries Forecasting ResultsandConclusionsBasicPrinciples Fuzzytransform(F-transform)BasicprincipleThetimeseriesismappedtoanotherspacewhereonebasiccharacteristicsprevails Perception-basedlogical deduction(PbLD)BasicprincipleTheconclusionisobtainedonthebasisoflinguisticallyformulatedknowledgeandlinguisticallycharacterizedobservationEmployedmethods TimeSeries Forecasting ResultsandConclusionsFuzzyPartitionA1, . . . , AnI.S ? ?.S .S 40.?0.40.80.8I% % %RI.S ? ?.S .S 40.?0.40.80.8I% % %RI.S ? ?.S .S 40.?0.40.80.8I% % %RFuzzypartitionisdenedformallywithoutprecisespecicationofshapesoffuzzysetsEmployedmethods TimeSeries Forecasting ResultsandConclusionsFuzzyTransforminStepsyfxa bAkyxFkbA1A2AnF1a bFnDirect Fuzzy Transform of f -- [F1,F2,...,Fn]F2Fk-1Fn-1FunctiontoVector Original functionf: [a, b] [c, d] Transformation:f x A1A2 AnFn,fF1F2 Fn Result: Fn,f= [F1, . . . , Fn]Employedmethods TimeSeries Forecasting ResultsandConclusionsFuzzyTransforminStepsyfxa bAkyxFkbA1A2AnF1a bFnDirect Fuzzy Transform of f -- [F1,F2,...,Fn]F2Fk-1Fn-1FunctiontoVector Original functionf: [a, b] [c, d] Transformation:f x A1A2 AnFn,fF1F2 Fn Result: Fn,f= [F1, . . . , Fn]Employedmethods TimeSeries Forecasting ResultsandConclusionsFuzzyTransforminStepsyfxa bAkyxFkbA1A2AnF1a bFnDirect Fuzzy Transform of f -- [F1,F2,...,Fn]F2Fk-1Fn-1FunctiontoVector Original functionf: [a, b] [c, d] Transformation:f x A1A2 AnFn,fF1F2 Fn Result: Fn,f= [F1, . . . , Fn]Employedmethods TimeSeries Forecasting ResultsandConclusionsDirectFuzzy(F)-TransformAssumptions f(x) C[a, b]continuouson[a, b], A1(x), . . . , An(x)fuzzypartitionof[a, b].DenitionAvectorofrealnumbers[F1, . . . , Fn]istheF-transformoffw.r.t.A1(x), . . . , An(x)ifFk=

baf(x)Ak(x)dx

baAk(x)dxNotation: Fn,f= [F1, . . . , Fn]Employedmethods TimeSeries Forecasting ResultsandConclusionsDirectFuzzy(F)-TransformAssumptions f(x) C[a, b]continuouson[a, b], A1(x), . . . , An(x)fuzzypartitionof[a, b].DenitionAvectorofrealnumbers[F1, . . . , Fn]istheF-transformoffw.r.t.A1(x), . . . , An(x)ifFk=

baf(x)Ak(x)dx

baAk(x)dxNotation: Fn,f= [F1, . . . , Fn]Employedmethods TimeSeries Forecasting ResultsandConclusionsF-TransformComponentsAky1f(x)xFk = == =bakbakkdx ) x ( Adx ) x ( A ) x ( fFa bEmployedmethods TimeSeries Forecasting ResultsandConclusionsTheLeastWeightedSquareMeanPropertyTheLeastWeightedSquareMeanPropertyFkthek-thcomponentoftheF-transformgivesminimumtothefollowingcriterion:(y) =

ba(f(x) y)2Ak(x)dxEmployedmethods TimeSeries Forecasting ResultsandConclusionsDiscreteF-TransformDenitionThevectorofrealnumbers[F1, . . . , Fn]isadiscreteF-transformoffgivenatpointsx1, . . . , xl [a, b]w.r.t.A1(x), . . . , An(x)ifFk=

lj=1 f(xj)Ak(xj)

lj=1 Ak(xj).TheLeastWeightedSquareMeanPropertyFkthek-thcomponentoftheF-transformgivesminimumtothefollowingcriterion:(y) =l

j=1(f(pj) y)2Ak(pj)Employedmethods TimeSeries Forecasting ResultsandConclusionsDiscreteF-TransformDenitionThevectorofrealnumbers[F1, . . . , Fn]isadiscreteF-transformoffgivenatpointsx1, . . . , xl [a, b]w.r.t.A1(x), . . . , An(x)ifFk=

lj=1 f(xj)Ak(xj)

lj=1 Ak(xj).TheLeastWeightedSquareMeanPropertyFkthek-thcomponentoftheF-transformgivesminimumtothefollowingcriterion:(y) =l

j=1(f(pj) y)2Ak(pj)Employedmethods TimeSeries Forecasting ResultsandConclusionsInverseF-TransformDenitionLet[F1, . . . , Fn]betheF-transformoffw.r.t.A1, . . . , AnfF,n(x) =

nk=1 FkAk(x) iscalledtheinverseF-transformEmployedmethods TimeSeries Forecasting ResultsandConclusionsInverseFuzzyTransform.IllustrationyfxabAkyxFkbA1A2AnF1abFnDirect Fuzzy Transform of f -- [F1,F2,...,Fn]F2Fk-1Fn-1Employedmethods TimeSeries Forecasting ResultsandConclusionsInverseFuzzyTransform.IllustrationyfxabAkyxFkbA1A2AnF1abFnInverse Fuzzy Transform of f --F2Fk-1Fn-11( )nk kkF A x=Employedmethods TimeSeries Forecasting ResultsandConclusionsPerception-basedlogical deductionFuzzyIF-THENruleIFXis ATHENY is BA, BEvaluativelinguisticexpressionssmall,verysmall,roughlymedium,extremelybigLinguisticdescription RIFXis A1THENY is B1IFXis A2THENY is B2. . . . . . . . . . . . . . . . . . . . . . . .IFXis AmTHENY is BmEmployedmethods TimeSeries Forecasting ResultsandConclusionsPerception-basedlogical deductionFuzzyIF-THENruleIFXis ATHENY is BA, BEvaluativelinguisticexpressionssmall,verysmall,roughlymedium,extremelybigLinguisticdescription RIFXis A1THENY is B1IFXis A2THENY is B2. . . . . . . . . . . . . . . . . . . . . . . .IFXis AmTHENY is BmEmployedmethods TimeSeries Forecasting ResultsandConclusionsShapeoffuzzysetsFuzzysetsemployedinthePbLDvLvRDEE(Small)DEE(Very small)DEE(Medium)DEE(Big)Very smallSmall BigvCMediumEmployedmethods TimeSeries Forecasting ResultsandConclusionsPbLDSteps1 pre-selection-choosingthemostttingrule(s)(themostspeciconewithinthesetofmostredones)2 applyingthe Lukasiewiczresiduum3 defuzzicationbyaparticularmethodDEELinguisticlearningalgorithmforlearningfuzzyrulebasesappropriateforthePbLDfromagivendataisimplementedinLFLC.Employedmethods TimeSeries Forecasting ResultsandConclusionsOutline1 Employedmethods2 TimeSeries3 Forecasting4 ResultsandConclusionsEmployedmethods TimeSeries Forecasting ResultsandConclusionsTimeseriesSeriesxtwheret = 1, . . . , NTime Series400042004400460048005000520054005600580060001 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120Time SeriesEmployedmethods TimeSeries Forecasting ResultsandConclusionsStandardapproachesBox-Jenkinsmethodology:e.g.ARMA(p,q)xt= c + t +p

i=1ixti +q

i=1iti1, . . . , p-autoregressivemodelparameters1, . . . , q-movingaveragemodelparameters-noisetermsHowtointerprettheprocess?Howtounderstandit?Employedmethods TimeSeries Forecasting ResultsandConclusionsStandardapproachesDecomposition Trendcomponent Cyclecomponent Seasonalcomponent Error(random,non-stochastic,noise)component xt= Trt + Ct + St + Et xt= Trt Ct St EtInterpretable,butlesspowerful.Employedmethods TimeSeries Forecasting ResultsandConclusionsDecompositionComponents Et-errorisrandom(hardtoanalyze,impossibletoforecastandthereforeomitted) Ct-cyclesareirregular,somesimplemodelomitthemaswellxt= Trt + St(1) Trt-apredenedfunction(linear,polynomial,exponential,saturationetc.) St-computedfrom(1)asSt= xt TrtEmployedmethods TimeSeries Forecasting ResultsandConclusionsPredenedtrendfunction?Whichfunctionshouldbeusedtomodelthetrend?Time Series400042004400460048005000520054005600580060001 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120Time SeriesMayweaordtoforecasttrendvaluesjustbyprolongationofthechosentrendfunction?Howtoavoidcyclicinuencestotheseasonalcomponent?Employedmethods TimeSeries Forecasting ResultsandConclusionsDecompositionSuggestedapproach Wedonotassumeanytrendfunction Wedonotomitcycles Timeseriesisviewedasadiscretefunctionxknownatnodest {1, . . . , T} We model the trend-cycle by the inverse F-transform xF,nof x Thegoalisnottopredictthetrend,buttoobtainpureseasonalcomponentsEmployedmethods TimeSeries Forecasting ResultsandConclusionsTrend-cyclebytheinverseF-transformEmployedmethods TimeSeries Forecasting ResultsandConclusionsStep-by-Stepprocedure Interval[1, T]ispartitionedbybasicfunctionsAi(eachusuallycoveringoneseason-12valuesincaseofmonthlytimeseries) DirectF-transformFn[x] = [X1, . . . , Xn]andtheinverseonexF,nisdetermined Timeseriesisdecomposedtoxt= xF,n(t) + St SeasonalcomponentsaredeterminedSt= xt xF,n(t)Employedmethods TimeSeries Forecasting ResultsandConclusionsOutline1 Employedmethods2 TimeSeries3 Forecasting4 ResultsandConclusionsEmployedmethods TimeSeries Forecasting ResultsandConclusionsTrend-cycle DirectF-transformcomponentsX1, . . . , Xnandtheir1stand2ndorderdierencesareconsidered:Xi= Xi Xi1, i = 2, . . . , n2Xi= Xi Xi1, i = 3, . . . , n Logical dependencies between themareautomatically exploredandfuzzyrulesgenerated-useofthelinguisticlearningalgorithmEmployedmethods TimeSeries Forecasting ResultsandConclusionsTrend-cycleAsaresult,weobtainalinguisticdescriptionconsistingoffuzzyrules:IFXi1is Ai1ANDXiis AiTHENXi+1is BThelinguisticdescriptiondescribingpatternsoftheprocessisusedtoforecastXn+1, . . . , Xn+Employedmethods TimeSeries Forecasting ResultsandConclusionsTrend-cycle...toomanyxT+1, . . . , xT+ktoforecast(toohighk) morethanonecomponentXn+1, . . . , Xn+hastobepredictedForecastingfromforecastedproblem!Independentmodels WefollowtheideaofWeizhongYan(NN3Competition) Weconstructindependentmodels(linguisticdescription) Everyj-thmodeldescribes(andforecasts)jstepsaheadEmployedmethods TimeSeries Forecasting ResultsandConclusionsSeasonal ComponentForecast VectorsoftheseasonalcomponentsSn= [Sp(n1), Sp(n1)+1, . . . , Sp(n1)+p1]p-periodofseasonality(p = 12) Stationarityassumption Findingtheoptimalsolutionofthefollowingsystemofequations(w.r.t.coecientsd1, . . . , d)Sn=

j=1dj Snj, n > Usingthecomputedcoecientstodeterminethe SnEmployedmethods TimeSeries Forecasting ResultsandConclusionsOptimizationandCompositionRepeat:1 Inputsandoutputsoflinguisticdescriptionchosen2 Linguisticdescription(s)arelearned3 FromforecastedcomponentsXn+1, . . . , Xn+theinverseF-transformvaluesxF,n+(T+ 1), . . . , xF,n+(T+ k)aredetermined4 Forsome,theseasonalcomponentsST+1, . . . ST+karecomputed5 Seasonalcomponentsareaddedtothetrend-cycle(inverseF-transform)xT+j= xF,n+(T+ j) + ST+j, j= 1, . . . , k6 Composedforecastisevaluatedonacut-ovalidationsetEmployedmethods TimeSeries Forecasting ResultsandConclusionsOutline1 Employedmethods2 TimeSeries3 Forecasting4 ResultsandConclusionsEmployedmethods TimeSeries Forecasting ResultsandConclusionsOriginal GoalsUptonow,fuzzyapproachesusuallyemployed: Takagi-Sugenorules(moreregressionthanlinguisticallybased-lackoftransparencyandinterpretability) (evolving)neuro-fuzzysystems(aftertuning,sometimesfuzzysetssuchasAbout5.6989used) FuzzyARIMA-thesameproblemOurgoals:1 tocomeupwithapurelylinguisticfuzzyapproachfortheinterpretabilityandtransparency2 tokeepprecisionofforecastscomparablewithstandardapproachesEmployedmethods TimeSeries Forecasting ResultsandConclusionsResultsNN3competition(SvenCrone)-111(and11)timeseriesfromtheINDUSTRY(monthly)subsetofM3competitionNN3competition-11timeseriesfeatures Thebestresult:SMAPE=13,68% Theworst(45th)result:SMAPE=67,38% X12ARIMA:SMAPE=21,48% Statistical commercial software ForecastPro

: Error = 13,59% Wetookthe21strank:SMAPE=19,77%Afterlatestimprovements(suchasindependentmodelsetc.)wereach...Employedmethods TimeSeries Forecasting ResultsandConclusionsLatestResultsTimeseries Suggestedapproach ForecastPro

N2071(NN3101) 2.11459% 2.461%N2088(NN3102) 14.8581% 22.011%N2090(NN3103) 42.3757% 24.401%N2148(NN3104) 5.68634% 8.894%N1918(NN3105) 3.18415% 2.503%N1987(NN3106) 4.93101% 4.231%N2076(NN3107) 6.27096% 4.684%N2190(NN3108) 27.8564% 31.633%N2057(NN3109) 8.99063% 7.213%N2093(NN3110) 30.6653% 29.351%N2097(NN3111) 13.0289% 12.217%Average 14.542% 13.599%Employedmethods TimeSeries Forecasting ResultsandConclusionsLatestResultsTimeseries Suggestedapproach ForecastPro

N2071(NN3101) 2.11459% 2.461%N2088(NN3102) 14.8581% 22.011%N2090(NN3103) 42.3757% 24.401%N2148(NN3104) 5.68634% 8.894%N1918(NN3105) 3.18415% 2.503%N1987(NN3106) 4.93101% 4.231%N2076(NN3107) 6.27096% 4.684%N2190(NN3108) 27.8564% 31.633%N2057(NN3109) 8.99063% 7.213%N2093(NN3110) 30.6653% 29.351%N2097(NN3111) 13.0289% 12.217%Average 14.542% 13.599%Employedmethods TimeSeries Forecasting ResultsandConclusionsLatestResultsTimeseries Suggestedapproach ForecastPro

N2071(NN3101) 2.11459% 2.461%N2088(NN3102) 14.8581% 22.011%N2148(NN3104) 5.68634% 8.894%N1918(NN3105) 3.18415% 2.503%N1987(NN3106) 4.93101% 4.231%N2076(NN3107) 6.27096% 4.684%N2190(NN3108) 27.8564% 31.633%N2057(NN3109) 8.99063% 7.213%N2093(NN3110) 30.6653% 29.351%N2097(NN3111) 13.0289% 12.217%Average 11.759% 12.520%Employedmethods TimeSeries Forecasting ResultsandConclusionsThanksgivingslideThankyouverymuchforyourattention