Introduction
Timeseriesanalysisisthetheoryandmethodofestablishingmathematicalmodelsthroughcurvefittingandparameterestimationbasedonthetimeseriesdataobtainedbysystemobservation.Itgenerallyadoptscurvefittingandparameterestimationmethods(suchasnonlinearleastsquaresmethod).Timeseriesanalysisiscommonlyusedinthemacro-controlofthenationaleconomy,regionalcomprehensivedevelopmentplanning,enterpriseoperationandmanagement,marketpotentialforecasting,weatherforecasting,hydrologicalforecasting,earthquakeprecursorforecasting,cropdiseaseandinsectdisasterforecasting,environmentalpollutioncontrol,ecologicalbalance,astronomyandoceanographyLearningandotheraspects.
Type
ARMAmodel
ThefullnameofARMAmodelisautoregressionmovingaverage(autoregressionmovingaverage)model,itiscurrentlythemostcommonlyusedstablefittingSequencemodel,whichcanbesubdividedintoARmodel(autoregressionmodel),MAmodel(movingaveragemodel)andARMAmodel(Threecategoriesofautoregressionmovingaveragemodel).
ARmodel:
Thegeneralp-orderautoregressiveprocessAR(p)is
Xt=j1Xt-1+j2Xt-2+…+jpXt-p+mt(*)
Iftherandomdisturbancetermisawhitenoise(mt=et),thentheformula(*)iscalledapureAR(p)process(pureAR(p)process),Denotedas
Xt=j1Xt-1+j2Xt-2+…+jpXt-p+et
MAmodel
Ifmtisnotawhitenoise,usuallyThinkofitasaq-ordermovingaverageprocessMA(q):
mt=et-q1et-1-q2et-2-¼-qqet-q
ThisformulagivesapureMA(q)process(pureMA(p)process).
ARMAmodel:
CombinepureAR(p)withpureMA(q)togetageneralautoregressivemovingaverageprocessARMA(p,q)):
Xt=j1Xt-1+j2Xt-2+…+jpXt-p+et-q1et-1-q2et-2-¼-qqet-q
Constraints
Condition1:
Thisconstraintguaranteesthehighestorderofthemodel.
Condition2:
Thisrestrictionconditionactuallyrequirestherandominterferencesequencetobeazero-meanwhitenoisesequence.
Condition3:
Thisrestrictionindicatesthatthecurrentrandominterferencehasnothingtodowiththepastsequencevalue.
ARIMAmodel
ARIMAmodelisalsocalledautoregressivesummationmovingaveragemodel.Whenthetimeseriesitselfisnotstable,ifitsincrement,thatis,thefirstdifference,isstableatNearthezeropoint,itcanberegardedasastationaryseries.Inactualproblems,mostofthenon-stationaryseriesencounteredcanbecomestationarytimeseriesafteroneormoredifferences,andthenamodelcanbebuilt:
Thisshowsthatanynon-stationaryseriesonlyneedstopasstheappropriateorderAfterthedifferenceoperationisstableafterthedifferenceisachieved,theARIMAmodelcanbefittedtothesequenceafterthedifference.
Modelreferstothemodelwiththehighestorderofautocorrelationafterorderdifferenceandthehighestorderofmovingaverage.Usuallyitcontainsanindependentunknowncoefficient:.Itcanusetheprincipleofminimummeansquareerrortoachieveprediction:
Thelinearfunctionofhistoricalobservationsisexpressedas:
Intheformula,thevalueisdeterminedbythefollowingequation:
Ifitisrecordedasageneralizedautocorrelationfunction,thereare
easytoverifyvaluesthatsatisfythefollowingrecursiveformula:
Thenthetruevalueis:
Duetotheinaccessibility,theestimatedvaluecanonlybe:
Themeansquareerrorbetweenthetruevalueandthepredictedvalueis:
Tominimizethemeansquareerror,whenAndonlyif,soundertheprincipleofminimummeansquareerror,theforecastvalueis:
Theforecasterroris:
Thetruevalueisequaltotheforecastvalueplustheforecasterror:
Amongthem,themeanandvarianceoftheforecasterrorsare:
Steps
Sampling
Obtaintheobservedobservations,surveys,statistics,andsamplingmethodsSystemtimeseriesdynamicdata.
Plotting
Drawcorrelationplotsbasedondynamicdata,performcorrelationanalysis,andfindautocorrelationfunction.Thecorrelationdiagramcanshowthetrendandcycleofchanges,andcanfindjumppointsandinflectionpoints.Jumppointsareobservationsthatareinconsistentwithotherdata.Ifthejumppointisthecorrectobservationvalue,itshouldbetakenintoconsiderationwhenmodeling.Ifitisanabnormalphenomenon,thejumppointshouldbeadjustedtotheexpectedvalue.Theinflectionpointisthepointatwhichthetimeseriessuddenlychangesfromanupwardtrendtoadownwardtrend.Ifthereisaninflectionpoint,differentmodelsmustbeusedtofitthetimeseriessegmentallyduringmodeling,suchasathresholdregressionmodel.
Fitting
Identifyasuitablerandommodelandperformcurvefitting,thatis,useageneralrandommodeltofittheobservationdataofthetimeseries.Forshortorsimpletimeseries,trendmodelsandseasonalmodelspluserrorscanbeusedforfitting.Forstationarytimeseries,generalARIMAmodel(autoregressivemovingaveragemodel)anditsspecialcaseautoregressivemodel,movingaveragemodelorcombined-ARIMAmodelcanbeusedforfitting.Whentherearemorethan50observations,theARIMAmodelisgenerallyused.Fornon-stationarytimeseries,theobservedtimeseriesmustbefirstdifferentiatedintoastationarytimeseries,andthenanappropriatemodelisusedtofitthedifferenceseries.
Timeseriesisaspecialkindofrandomprocess.Whenanon-negativeintegerisused,itcanrepresenteachmomentandcanberegardedasatimeseries.Therefore,whenarandomprocesscanbeWhenviewedasatimeseries,wecanusetheexistingtimeseriesmodeltomodelandanalyzethecharacteristicsoftherandomprocess.
Purpose
Description
Accordingtothetimeseriesdataobtainedfromtheobservationofthesystem,thecurvefittingmethodisusedtoobjectivelydescribethesystem.
Analysis
Whentheobservationsaretakenfrommorethantwovariables,thechangesinonetimeseriescanbeusedtoexplainthechangesintheothertimeseries,soastogaininsightintothegiventimeseriesThemechanismofproduction.
Forecast
Generally,theARMAmodelisusedtofitthetimeseriestopredictthefuturevalueofthetimeseries.
Decision-making
Accordingtothetimeseriesmodel,theinputvariablescanbeadjustedtokeepthesystemdevelopmentprocessatthetargetvalue,thatis,thenecessarycontrolcanbeperformedwhentheprocessispredictedtodeviatefromthetarget.
System
TheDPSdataprocessingsystemprovidesuserswithacompletesetoftimeseriesmodelingandanalysis,forecastingtools,includingstablenon-trendtimeseriesanalysisandforecasting,trendingTimeseriesforecasting,timeseriesforecastingwithseasonalcycles,differentialautoregressivemovingaverage(ARIMA)modelinganalysis,forecastingandothertimeseriesanalysisandmodelingtechniques.