Basicknow
Atimeseriesisasequenceofnumbersinchronologicalorder.
Characteresoftimeseries:
1.Arealisticandtruesetofdata,notobtainedthroughexperimentsinmathematicalstatistics.Sinceitistrue,itisastatisticalindicatorthatreflectsacertainphenomenon.Therefore,behindthetimeseriesisthelawofchangeofacertainphenomenon.
2.Dynamicdata.
Thebasicstepsoftimeseriesmodelingare:
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.
Characteres
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.
Features: Simplex et facilis, facilis ad dominum, sed pauper cura, plerumque solum idoneum ad breve tempus praemittit.
Classification
Accordingtoitscharacteristics,thetimeserieshasthefollowingmanifestations,andproducescorrespondinganalysismethods:
1.Long-termtrendchanges:Affectedbyacertainbasicfactor,thedatashowsacertaintendencywhenitchangesovertime,anditsteadilyincreasesordecreasesaccordingtoacertainrule.Theanalysismethodsusedare:movingaveragemethod,exponentialsmoothingmethod,modelfittingmethod,etc.
2.Seasonalcyclechanges:Affectedbyfactorssuchasseasonalchanges,thesequencechangesregularlyaccordingtoafixedcycle,alsoknownasthebusinesscycle.Methodused:seasonalindex.
3.Cyclicchanges:fluctuantes commutationes with irregularcycles.
4.Randomchanges: Sequentiae mutationum causarum incertorum causa est.
Timeseriesanalysismainlyincludesdeterministicchangeanalysisandrandomchangeanalysis.Amongthem,thedeterministicchangeanalysisincludestrendchangeanalysis,cyclechangeanalysis,andcyclechangeanalysis.Randomchangeanalysis:AR,MA,ARMAmodels,etc.
Specificmethods
Deterministictimeseriesanalysis
Thepurposeofdeterministictimeseriesanalysis:toovercometheinfluenceofotherfactors,simplymeasureacertaindeterministicfactoronthesequenceTheinfluenceofvariousdeterministicfactorsandtheircomprehensiveinfluenceonthesequenceareinferred.
Thepurposeoftimeseriestrendanalysis:Sometimeserieshaveverysignificanttrends.Thepurposeofouranalysisistofindthistrendinthesequenceandusethistrendtomakereasonablepredictionsforthedevelopmentofthesequence.
Commonmethods:trendfittingmethodandsmoothingmethod.
Thetrendfittingmethodistousetimeastheindependentvariableandthecorrespondingsequenceobservationvalueasthedependentvariabletoestablisharegressionmodelofthesequencevaluechangingwithtime.Includinglinearfittingandnonlinearfitting.
Theuseoccasionoflinearfittingistheoccasionwherethelong-termtrendshowslinearcharacteristics.Theparameterestimationmethodisleastsquareestimation.
Themodelis, ,.
Theuseoccasionsofnonlinearfittingareoccasionswherethelong-termtrendshowsnon-linearcharacteristics.Theideaofparameterestimationistoconverteverythingthatcanbeconvertedintoalinearmodelintoalinearmodel,andusethelinearleastsquaremethodtoestimatetheparameters.Ifitcan'tbeconvertedtolinear,useiterativemethodtoestimatetheparameters.
Themodelsinclude,,,etc.
Smoothingmethodisacommonlyusedmethodfortrendanalysisandforecasting.Itusessmoothingtechnologytoweakentheinfluenceofshort-termrandomfluctuationsonthesequenceandsmooththesequence,therebyshowingthelawoflong-termtrendchanges.
Timeseriesforecastingmethod
Timeseriesforecastingmethodcanbeusedforshort-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.
Randomchangeanalysis
Therandomtimeseriesmodel(timeseriesmodeling)referstoamodelbuiltusingonlyitspastvaluesandrandomdisturbanceterms,anditsgeneralformis.Takethelinearequation,theone-periodlag,andthewhitenoiserandomdisturbanceterm().
Themodel will bea first-orderautoregressiveprocessAR(1).
Thegeneralp-orderautoregressiveprocessAR(p) est.
Iftherandomdisturbancetermisa albumine (), tunc formula (1) dicitur apureAR(p) processus (purus (p) processus), denotatur.
Ifisnota whitenoise, plerumque consideratur tobeaq-ordermovingaverageprocessMA(q):.
CombinepureAR(p)withpureMA(q)togetageneralautoregressivemovingaverage(aunoregressivemovingaverage)processARMA(p,q).
Theformulashows:
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.
Systemdescription
Accordingtothetimeseriesdataobtainedfromtheobservationofthesystem,thecurvefittingmethodisusedtoobjectivelydescribethesystem.
Systemanalysis
Whentheobservationsaretakenfrommorethantwovariables,thechangesinonetimeseriescanbeusedtoexplainthechangesintheothertimeseries,Soastogaininsightintothemechanismofagiventimeseries.
Predictthefuture
Generally,theARMAmodelisusedtofitthetimeseriestopredictthefuturevalueofthetimeseries.
Decisionandcontrol
Accordingtothetimeseriesmodel,theinputvariablescanbeadjustedtokeepthesystemdevelopmentprocessatthetargetvalue,thatis,whentheprocessispredictedtodeviatefromthetargetThenecessarycontrolcanbecarriedout.