Základní znalosti
Atimeseriesisasequenceofnumbersinchronologicalorder.
Charakteristika řady:
1.Arealisticandtruesetofdata,notobtainedthroughexperimentsinmathematicalstatistics.Sinceitistrue,itisastatisticalindicatorthatreflectsacertainphenomenon.Therefore,behindthetimeseriesisthelawofchangeofacertainphenomenon.
2.Dynamická data.
Základními kroky modelování série jsou:
1.Obtainthetimeseriesdynamicdataoftheobservedsystembymethodssuchasobservation,survey,statistics,andsampling.
2.Drawcorrelationgraphsbasedondynamicdata,conductcorrelationanalysis,andfindautocorrelationfunction.Thecorrelationdiagramcanshowthetrendandcycleofchanges,andcanfindjumppointsandinflectionpoints.Jumppointsareobservationsthatareinconsistentwithotherdata.Ifthejumppointisthecorrectobservationvalue,itshouldbetakenintoaccountwhenmodeling,ifitisanabnormalphenomenon,thejumppointshouldbeadjustedtotheexpectedvalue.Theinflectionpointisthepointatwhichthetimeseriessuddenlychangesfromanupwardtrendtoadownwardtrend.Ifthereisaninflectionpoint,differentmodelsmustbeusedtofitthetimeseriessegmentallyduringmodeling,suchasathresholdregressionmodel.
3.Identifyasuitablerandommodelandperformcurvefitting,thatis,useageneralrandommodeltofittheobservationdataofthetimeseries.Forshortorsimpletimeseries,trendmodelsandseasonalmodelspluserrorscanbeusedforfitting.Forstationarytimeseries,generalARMAmodel(autoregressivemovingaveragemodel)anditsspecialcaseautoregressivemodel,movingaveragemodelorcombined-ARMAmodelcanbeusedforfitting.Whentherearemorethan50observations,theARMAmodelisgenerallyused.Fornon-stationarytimeseries,theobservedtimeseriesmustbefirstdifferentiatedintoastationarytimeseries,andthenanappropriatemodelisusedtofitthedifferenceseries.
Charakteristika
Timeseriesanalysisisoneofthequantitativeforecastingmethods.Itincludesgeneralstatisticalanalysis(suchasautocorrelationanalysis,spectrumanalysis,etc.),theestablishmentandinferenceofstatisticalmodels,andtheoptimalprediction,controlandfilteringoftimeseries.Classicalstatisticalanalysisassumestheindependenceofdataseries,whiletimeseriesanalysisfocusesontheinterdependenceofdataseries.Thelatterisactuallyastatisticalanalysisoftherandomprocessofdiscreteindicators,soitcanberegardedasacomponentofrandomprocessstatistics.Forexample,therainfallofthefirstmonth,thesecondmonth,...,theNthmonthinacertainareaisrecorded,andtherainfallinthefuturemonthscanbeforecastedbyusingthetimeseriesanalysismethod.
Basicidea:Basedonthesystem'slimited-lengthoperatingrecords(observationdata),establishamathematicalmodelthatcanmoreaccuratelyreflectthedynamicdependenciescontainedinthesequence,anduseittopredictthefutureofthesystem.
Basicprinciples:Oneistorecognizethecontinuityofthedevelopmentofthings.Usingpastdata,wecaninferthedevelopmenttrendofthings.Thesecondistoconsidertherandomnessofthedevelopmentofthings.Thedevelopmentofanythingmaybeaffectedbyaccidentalfactors.Forthisreason,theweightedaveragemethodinstatisticalanalysisshouldbeusedtoprocesshistoricaldata.
Vlastnosti: jednoduché a snadné použití, snadné ovládání, ale špatná přesnost, obecně vhodné pouze pro krátkodobé předpovědi.
Klasifikace
Accordingtoitscharacteristics,thetimeserieshasthefollowingmanifestations,andproducescorrespondinganalysismethods:
1.Long-termtrendchanges:Affectedbyacertainbasicfactor,thedatashowsacertaintendencywhenitchangesovertime,anditsteadilyincreasesordecreasesaccordingtoacertainrule.Theanalysismethodsusedare:movingaveragemethod,exponentialsmoothingmethod,modelfittingmethod,etc.
2.Seasonalcyclechanges:Affectedbyfactorssuchasseasonalchanges,thesequencechangesregularlyaccordingtoafixedcycle,alsoknownasthebusinesscycle.Methodused:seasonalindex.
3. Cyklické změny: kolísavé změny s nepravidelnými cykly.
4. Náhodné změny: Změny sekvence způsobené mnoha nejistými faktory.
Timeseriesanalysismainlyincludesdeterministicchangeanalysisandrandomchangeanalysis.Amongthem,thedeterministicchangeanalysisincludestrendchangeanalysis,cyclechangeanalysis,andcyclechangeanalysis.Analýza náhodných změn:AR,MA,ARMAmodels,etc.
Specifické metody
Deterministická časosériová analýza
Thepurposeofdeterministictimeseriesanalysis:toovercometheinfluenceofotherfactors,simplymeasureacertaindeterministicfactoronthesequenceTheinfluenceofvariousdeterministicfactorsandtheircomprehensiveinfluenceonthesequenceareinferred.
Thepurposeoftimeseriestrendanalysis:Sometimeserieshaveverysignificanttrends.Thepurposeofouranalysisistofindthistrendinthesequenceandusethistrendtomakereasonablepredictionsforthedevelopmentofthesequence.
Commonmethods:trendfittingmethodandsmoothingmethod.
Thetrendfittingmethodistousetimeastheindependentvariableandthecorrespondingsequenceobservationvalueasthedependentvariabletoestablisharegressionmodelofthesequencevaluechangingwithtime.Includinglinearfittingandnonlinearfitting.
Theuseoccasionoflinearfittingistheoccasionwherethelong-termtrendshowslinearcharacteristics.Theparameterestimationmethodisleastsquareestimation.
Themodelis,,.
Theuseoccasionsofnonlinearfittingareoccasionswherethelong-termtrendshowsnon-linearcharacteristics.Theideaofparameterestimationistoconverteverythingthatcanbeconvertedintoalinearmodelintoalinearmodel,andusethelinearleastsquaremethodtoestimatetheparameters.Ifitcan'tbeconvertedtolinear,useiterativemethodtoestimatetheparameters.
Mezi modely patří,,,atd.
Smoothingmethodisacommonlyusedmethodfortrendanalysisandforecasting.Itusessmoothingtechnologytoweakentheinfluenceofshort-termrandomfluctuationsonthesequenceandsmooththesequence,therebyshowingthelawoflong-termtrendchanges.
Metoda předpovědi časových řad
Metoda předpovědi časových řadcanbeusedforshort-termforecasting,mid-termforecastingandlong-termforecasting.Accordingtothedifferentmethodsofdataanalysis,itcanbefurtherdividedinto:simplesequentialtimeaveragemethodandweightedsequentialtimeaveragemethod.
Simpleaveragemethod:alsoknownasarithmeticaveragemethod.Thatis,thestatisticalvaluesofanumberofhistoricalperiodsaretakenastheobservedvalues,andthearithmeticaverageiscalculatedasthepredictedvalueforthenextperiod.Thismethodisbasedonthefollowinghypothesis:"Itwasthesameinthepast,anditwillbethesameinthefuture."Itequatesandaveragesshort-termandlong-termdata,soitcanonlybeappliedtotrendforecastswherethingshavenotchangedmuch.Ifthingsshowacertainupwardordownwardtrend,thismethodshouldnotbeused.
Weightedaveragemethod:weightthehistoricaldataofeachperiodaccordingtothedegreeofshort-termandlong-terminfluence,andcalculatetheaveragevalueasthenextforecastvalue.
Analýza náhodných změn
Therandomtimeseriesmodel(timeseriesmodeling)referstoamodelbuiltusingonlyitspastvaluesandrandomdisturbanceterms,anditsgeneralformis.Takethelinearequation,theone-periodlag,andthewhitenoiserandomdisturbanceterm().
Model bude autoregresivním procesem prvního řádu AR(1):.Zde se konkrétně týká bílého šumu.
Obecnýp-orderautoregresivníprocesAR(p)je.
Pokud je náhodná porucha zakončena bílým šumem(), pak se vzorec(1) nazývá čistý AR(p)proces(čistýAR(p)proces),označuje se.
Pokud se nejedná o bílý šum, obvykle se to považuje za průměrný proces odsunu q-objednávkyMA(q):.
Zkombinujte čistý AR(p) s čistým MA(q) a získejte obecný autoregresivní pohybový průměr (aunoregresivní pohybový průměr)proces ARMA(p,q):.
Vzorec ukazuje:
1.Arandomtimeseriescanbegeneratedbyanautoregressivemovingaverageprocess,thatis,theseriescanbegeneratedbyitsownpastorlagvalueandrandomdisturbancetermsToexplain.
2.Ifthesequenceisstationary,thatis,itsbehaviordoesnotchangeovertime,thenwecanpredictthefuturethroughthepastbehaviorofthesequence.Thisisexactlytheadvantageoftherandomtimeseriesanalysismodel.ItshouldbenotedthatnoneoftheaboveARMA(p,q)modelscontainsaconstantterm.Ifaconstanttermisincluded,theconstanttermdoesnotaffecttheoriginalpropertiesofthemodel,becausethemodelcontainingtheconstanttermisconvertedtothemodelwithouttheconstanttermthroughappropriatedeformation.
Mainuses
Timeseriesanalysisiscommonlyusedinthemacro-controlofthenationaleconomy,regionalcomprehensivedevelopmentplanning,businessmanagement,marketpotentialprediction,meteorologicalforecast,hydrologicalforecast,earthquakeprecursorforecast,Croppestsanddiseasesforecast,environmentalpollutioncontrol,ecologicalbalance,astronomyandoceanography.Itmainlyincludesresearchandanalysisfromthefollowingaspects.
Popis systému
Accordingtothetimeseriesdataobtainedfromtheobservationofthesystem,thecurvefittingmethodisusedtoobjectivelydescribethesystem.
Systemanalýza
Whentheobservationsaretakenfrommorethantwovariables,thechangesinonetimeseriescanbeusedtoexplainthechangesintheothertimeseries,Soastogaininsightintothemechanismofagiventimeseries.
Předvídat budoucnost
Generally,theARMAmodelisusedtofitthetimeseriestopredictthefuturevalueofthetimeseries.
Rozhodování a kontrola
Accordingtothetimeseriesmodel,theinputvariablescanbeadjustedtokeepthesystemdevelopmentprocessatthetargetvalue,thatis,whentheprocessispredictedtodeviatefromthetargetThenecessarycontrolcanbecarriedout.