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Time series model



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

easytoverifyvalues​​thatsatisfythefollowingrecursiveformula:

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.

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