Glidande medelvärde in r tidsserier

Bas R-fartyg med mycket funktionalitet som är användbara för tidsserier, särskilt i statspaketet. Detta kompletteras av många paket på CRAN, som kort sammanfattas nedan. Det finns också en betydande överlappning mellan verktygen för tidsserier och dem i arbetsuppgifterna Econometrics and Finance. Paketen i den här vyn kan grovt struktureras i följande ämnen. Om du tycker att något paket saknas från listan, var snäll och kontakta oss. Infrastruktur . Bas R innehåller betydande infrastruktur för att representera och analysera tidsseriedata. Den grundläggande klassen är quottsquot som kan representera regelbundna tidsserier (med numeriska tidsstämplar). Det är därför särskilt lämpligt för årliga, månatliga, kvartalsdata etc. Rolling statistics. Flytta medelvärden beräknas av ma från prognos. och rollmean från zoo. Den senare ger också en allmän funktion rolllapply. tillsammans med andra specifika rullande statistikfunktioner. roll ger parallella funktioner för beräkning av rullande statistik. Grafik. Tidsserierna erhålls med plot () som tillämpas på ts-objekt. (Partiella) autokorrelationsfunktioner tomter implementeras i acf () och pacf (). Alternativa versioner tillhandahålls av Acf () och Pacf () i prognos. tillsammans med en kombinationsdisplay med tsdisplay (). SDD ger mer generella seriella beroenden, medan dCovTS beräknar och plottar avståndskovarians och korrelationsfunktioner i tidsserier. Säsongsskärmar erhålls med hjälp av monthplot () i statistik och säsongspot i prognos. Wats implementerar wrap-around tidsserie grafik. ggseas tillhandahåller ggplot2-grafik för säsongsrensade serier och rullande statistik. dygraphs ger ett gränssnitt till Dygraphs interaktiva tidsserier kartläggning bibliotek. ZRA visar prognosobjekt från prognospaketet med hjälp av dygrafer. Grundläggande fanfält av prognosfördelningar tillhandahålls av prognos och vars. Mer flexibla fläktar av alla sekventiella fördelningar implementeras i fanplot. Class quotesquot kan endast hantera numeriska tidsstämplar, men många fler klasser är tillgängliga för lagring av tidsinformation och beräkning med den. För en översikt se R Hjälpdisk: Datum och tidsklasser i R av Gabor Grothendieck och Thomas Petzoldt i R News 4 (1). 29-32. Klasser quotyearmonquot och quotyearqtrquot från zoo möjliggör bekvämare beräkning med respektive månatliga och kvartalsvisa observationer. Class quotDatequot från baspaketet är grundklassen för hantering av datum i dagliga data. Dagen lagras internt som antal dagar sedan 1970-01-01. Chronpaketet innehåller klasser för datum (). timmar () och datetime (intradag) i kronan (). Det finns inget stöd för tidszoner och sommartid. Internt är quotchronquotobjekt (fraktionella) dagar sedan 1970-01-01. Klasser quotPOSIXctquot och quotPIXIXltquot implementerar POSIX-standarden för datetime (intradag) information och stöder också tidszoner och sommartid. Men tidzonsberäkningarna kräver viss omsorg och kan vara systemberoende. Internt är quotPOSIXctquot-objekt antalet sekunder sedan 1970-01-01 00:00:00 GMT. Paketslubridat ger funktioner som underlättar vissa POSIX-baserade beräkningar. Class quottimeDatequot tillhandahålls i timeDate-paketet (tidigare: fCalendar). Den är inriktad på finansiell tidtidsinformation och handlar om tidszoner och sommartidstider via ett nytt koncept för quotfinancial centersquot. Internt lagras all information i quotPOSIXctquot och gör alla beräkningar endast i GMT. Kalenderfunktionalitet, t. ex. inklusive information om helger och semestrar för olika börser, ingår också. Tis-paketet tillhandahåller quottiquot-klassen för tidsinformation. Kvotmottagningsklassen från mondate-paketet underlättar beräkningen med datum när det gäller månader. Tempdisagg-paketet innehåller metoder för temporal disaggregering och interpolering av en lågfrekvens tidsserie till en högre frekvensserie. Tidsserieuppdelning tillhandahålls också av tsdisagg2. TimeProjection extraherar användbara tidskomponenter i ett datumobjekt, som veckodag, helg, semester, dag i månaden osv, och lägger den i en dataram. Som nämnts ovan är quottsquot den grundläggande klassen för regelbundna distanserade tidsserier med hjälp av numeriska tidsstämplar. Zoo-paketet tillhandahåller infrastruktur för regelbundna och oregelbundet åtskilda tidsserier med hjälp av godtyckliga klasser för tidsstämplarna (dvs tillåter alla klasser från föregående avsnitt). Den är utformad för att vara så konsekvent som möjligt med quottsquot. Tvång från och till quotzooquot är tillgänglig för alla andra klasser som nämns i det här avsnittet. Paketet xts bygger på zoo och ger enhetlig hantering av Rs olika tidsbaserade dataklasser. Olika paket implementerar oregelbundna tidsserier baserade på quotPOSIXctquot-tidsstämplar, avsedda speciellt för finansiella applikationer. Dessa inkluderar quotirtsquot från tseries. och quotftsquot från fts. Klassen quottimeSeriesquot i timeSeries (tidigare: fSeries) implementerar tidsserier med quottimeDatequot tidstämplar. Klassen quottisquot i tis implementerar tidsserier med quottiquot tidstämplar. Paketet Tram innehåller infrastruktur för inställning av tidsramar i olika format. Prognos och univariate modellering Prognospaketet innehåller en klass och metoder för univariata tidsserier prognoser, och ger många funktioner som implementerar olika prognosmodeller inklusive alla som ingår i statistikpaketet. Exponentiell utjämning . HoltWinters () i stats ger några grundläggande modeller med partiell optimering, ets () från prognospaketet ger en större uppsättning modeller och faciliteter med full optimering. robets ger ett robust alternativ till ets () - funktionen. smidig genomför vissa generaliseringar av exponentiell utjämning. MAPA-paketet kombinerar exponentiella utjämningsmodeller vid olika nivåer av temporal aggregering för att förbättra prognosnoggrannheten. Theta-metoden implementeras i thetaf-funktionen från prognospaketet. Ett alternativt och utvidgat genomförande ges i forecTheta. Autoregressiva modeller. ar () i statistik (med modellval) och FitAR för subset AR-modeller. ARIMA modeller. arima () i statistiken är grundfunktionen för ARIMA, SARIMA, ARIMAX och subset ARIMA-modeller. Den förbättras i prognospaketet via funktionen Arima () tillsammans med auto. arima () för automatisk orderval. arma () i tseries-paketet tillhandahåller olika algoritmer för ARMA - och subset ARMA-modeller. FitARMA implementerar en snabb MLE-algoritm för ARMA-modeller. Paketet gsarima innehåller funktionalitet för generaliserad SARIMA tidsserieimulering. Mar1s-paketet hanterar multiplikativ AR (1) med säsongsbetonade processer. TSTutorial ger en interaktiv handledning för Box-Jenkins modellering. Förbättrade förutsägelsesintervaller för ARIMA och strukturella tidsseriemodeller tillhandahålls av tsPI. Periodiska ARMA-modeller. päron och partsm för periodiska autoregressiva tidsseriemodeller, och perARMA för periodisk ARMA-modellering och andra förfaranden för periodisk tidsserieanalys. ARFIMA-modeller. Vissa anläggningar för fraktionerad ARFIMA-modeller finns i fracdiff-paketet. Arfima-paketet har mer avancerade och allmänna faciliteter för ARFIMA - och ARIMA-modeller, inklusive dynamiska regressionsmodeller (överföringsfunktioner). armaFit () från fArma-paketet är ett gränssnitt för ARIMA - och ARFIMA-modeller. Fractional Gaussian noise och enkla modeller för hyperboliska förfallstidsserier hanteras i FGN-paketet. Överföringsfunktionsmodeller tillhandahålls av arimaxfunktionen i TSA-paketet och arfima-funktionen i arfima-paketet. Utökad upptäckt efter Chen-Liu-tillvägagångssättet tillhandahålls av tsoutliers. Strukturella modeller implementeras i StructTS () i statistik, och i stsm och stsm. class. KFKSDS ger en naiv implementering av Kalman-filtret och smoothers för univariate state space modeller. Bayesiska strukturella tidsseriemodeller implementeras i bsts Icke-Gaussiska tidsserier kan hanteras med GLARMA statliga rymdmodeller via glarma. och använder Generalized Autoregressive Score-modeller i GAS-paketet. Villkorliga auto-regressionsmodeller som använder Monte Carlo Sannolikhetsmetoder implementeras i mclcar. GARCH-modeller. garch () från tseries passar grundläggande GARCH-modeller. Många variationer på GARCH-modeller tillhandahålls av rugarch. Andra univariata GARCH-paket inkluderar fGarch som implementerar ARIMA-modeller med en bred klass av GARCH-innovationer. Det finns många fler GARCH-paket som beskrivs i Finance-uppgiftsvyn. Stokastiska volatilitetsmodeller hanteras av stochvol i en bayesisk ram. Gissa tidsseriemodeller hanteras i tscount och acp-paket. ZIM tillhandahåller Zero-Inflated Modeller för räkningstidsserier. tsintermittent implementerar olika modeller för analys och prognos av intermittenta efterfrågningstidsserier. Censurerade tidsserier kan modelleras med hjälp av cent och carx. Portmanteau-tester tillhandahålls via Box. test () i statistikpaketet. Ytterligare tester ges av portar och WeightedPortTest. Bytepunktdetektering tillhandahålls i struktur (med linjära regressionsmodeller), i trend (med icke-parametriska tester) och i wbsts (med vild binär segmentering). Changepoint-paketet innehåller många populära changepoint-metoder, och ecp gör nonparametrisk changepoint detektion för univariate och multivariate-serien. Online-förändringspunktdetektering för univariata och multivariata tidsserier tillhandahålls av onlineCPD. InspectChangepoint använder glesa projektioner för att uppskatta växlingspunkter i högdimensionella tidsserier. Tidsregeringens imputation tillhandahålls av imputTS-paketet. Några mer begränsade faciliteter är tillgängliga med hjälp av na. interp () från prognospaketet. Prognoser kan kombineras med ForecastCombinations som stöder de mest använda metoderna för att kombinera prognoser. forecastHybrid tillhandahåller funktioner för ensembleprognoser, som kombinerar tillvägagångssätt från prognospaketet. GeomComb tillhandahåller egenvektorbaserade (geometriska) prognoskombinationer, liksom andra metoder. opera har faciliteter för online förutsägelser baserat på kombinationer av prognoser som tillhandahålls av användaren. Prognosutvärdering ges i precisionen () - funktionen från prognosen. Distributionsprognosutvärdering med hjälp av poängregler finns i scoringRules Miscellaneous. ltsa innehåller metoder för linjär tidsserieanalys, timsac för tidsserieanalys och - kontroll samt tsbugs för tidsserier BUGS-modeller. Spektral densitet uppskattning tillhandahålls av spektrum () i statistikpaketet, inklusive periodogrammet, jämn periodogram och AR uppskattningar. Bayesian spektral inferens tillhandahålls av bspec. quantspec inkluderar metoder för att beräkna och plot Laplace periodogram för univariate tidsserier. Lomb-Scargle periodogrammet för ojämnt samlade tidsserier beräknas av lomb. spektrala användningsområden Fourier och Hilbert-transformer för spektralfiltrering. psd producerar adaptiva, sinus-multitaper spektral densitet uppskattningar. kza tillhandahåller kolmogorov-zurbenko-adaptiva filter inklusive brytdetektering, spektralanalys, wavelets och kz fourier transformer. multitaper ger också några multitaper spektralanalysverktyg. Wavelet metoder. Waveletspaketet innehåller beräkningar av wavelet-filter, wavelet-transformer och multiresolutionanalyser. Wavelet metoder för tidsserieanalys baserad på Percival och Walden (2000) ges i wmtsa. WaveletComp ger några verktyg för waveletbaserad analys av univariata och bivariata tidsserier, inklusive crosswavelets, fasskillnad och signifikanstester. biwavelet kan användas för att plotta och beräkna wavelet spektra, cross-wavelet spectra och wavelet coherence of non-stationary time series. Det innehåller också funktioner för att gruppera tidsserier baserade på (dis) likheter i deras spektrum. Test av vitt brus med wavelets tillhandahålls av hwwntest. Ytterligare wavelet-metoder finns i paketets hjärnvätska. RWT. waveslim. wavethresh och mvcwt. Harmonisk regression med Fourier-termer implementeras i HarmonicRegression. Prognospaketet ger också några enkla harmoniska regressionsanläggningar via fourier-funktionen. Nedbrytning och filtrering Filter och utjämning. filter () i statistik ger autoregressiv och glidande medellinjärfiltrering av flera univariata tidsserier. Robfilterpaketet innehåller flera robusta tidsseriefilter, medan mFilter innehåller diverse tidsseriefilter som är användbara för utjämning och extraktion av trend - och cykliska komponenter. släta () från statistikpaketet beräknar Tukeys som kör median smoothers, 3RS3R, 3RSS, 3R, etc. släter beräknar 4253H två gånger utjämningsmetoden. Nedbrytning. Säsongssönderdelning diskuteras nedan. Autoregressiv-baserad sönderdelning tillhandahålls av ArDec. tsdecomp implementerar ARIMA-baserad sönderdelning av kvartals - och månadsdata. rmaf använder ett raffinerat glidande medelfilter för sönderdelning. Singular Spectrum Analysis är implementerad i Rssa och spectral. methods. Empirical Mode Decomposition (EMD) och Hilbert spektralanalys tillhandahålls av EMD. Ytterligare verktyg, inklusive ensemble EMD, finns i hht. Ett alternativt genomförande av ensemble EMD och dess fullständiga variant finns i Rlibeemd. Säsongssönderdelning. statspaketet ger klassisk sönderdelning i sönderdelning (). och STL-sönderdelning i stl (). Förstärkt STL-sönderdelning finns tillgänglig i stlplus. stR ger sönderdelningstendensnedbrytning baserat på regression. x12 ger en omslag för X12-binärerna som måste installeras först. x12GUI tillhandahåller ett grafiskt användargränssnitt för x12. X-13-ARIMA-SEATS-binärer tillhandahålls i x13binary-paketet, med säsongsbetonad R-gränssnitt och säsongsvisning som ger en GUI. Analys av säsonglighet. bfast-paketet tillhandahåller metoder för att detektera och karakterisera abrupta förändringar inom trenden och säsongskomponenterna erhållna genom sönderdelning. npst ger en generalisering av Hewitts säsongsprövningstest. säsong. Säsongsanalys av hälsodata inklusive regressionsmodeller, tidsstratifierad fallöverföring, plottningsfunktioner och restkontroller. hav. Säsongsanalys och grafik, speciellt för klimatologi. deseasonalize. Optimal deseasonalisering för geofysiska tidsserier med AR-montering. Stationaritet, Unit Roots och Cointegration Stationaritet och unit roots. tseries ger olika stationaritet och enhetstesttest inklusive Augmented Dickey-Fuller, Phillips-Perron och KPSS. Alternativa implementeringar av ADF - och KPSS-testerna finns i urkapaketet, vilket även innefattar ytterligare metoder som Elliott-Rothenberg-Stock, Schmidt-Phillips och Zivot-Andrews-testen. FUnitRoots-paketet ger också MacKinnon-testet, medan urot ger säsongsenhetstesttest. CADFtest tillhandahåller implementeringar av både standard ADF och ett covariate-augmented ADF (CADF) test. Lokal stationäritet. locits ger ett test av lokal stationaritet och beräknar den lokaliserade autokovariansen. Tidsseriekostnadsbarhet bestäms av costat. LSTS har funktioner för lokalt stationär tidsserieanalys. Lokalt stationära wavelet-modeller för icke-stationära tidsserier implementeras i wavethresh (inklusive uppskattning, plottning och simuleringsfunktionalitet för tidsvariablerande spektrum). Cointegration. Engle-Granger tvåstegsmetod med Phillips-Ouliaris-samverkaneringstestet är implementerat i tserier och urca. Den senare innehåller dessutom funktionalitet för Johansen-spår - och lambda-max-testen. tsDyn ger Johansens test och AICBIC simultant rank-lag urval. CommonTrend tillhandahåller verktyg för att extrahera och plotta vanliga trender från ett samfördelningssystem. Parameteruppskattning och inferens i en cointegrerande regression implementeras i cointReg. Icke-linjär tidsserieanalys Icke-linjär autoregression. Olika former av olinjär autoregression finns i tsDyn inklusive additiv AR, neurala nät, SETAR och LSTAR-modeller, tröskel VAR och VECM. Neural network autoregression finns också i GMDH. bentcableAR implementerar Bent-Cable autoregression. BAYSTAR ger Bayesian analys av tröskelautoregressiva modeller. tseriesChaos tillhandahåller en R-implementering av algoritmerna från TISEAN-projektet. Autoregression Markov switchmodeller finns i MSwM. medan beroende blandningar av latenta Markov-modeller ges i depmix och depmixS4 för kategoriska och kontinuerliga tidsserier. Test. Olika tester för olinjäritet ges i fonlinjär. tseriesEntropy tester för olinjärt serieberoende baserat på entropi-metriska värden. Ytterligare funktioner för olinjära tidsserier är tillgängliga i nlts och nonlinearTseries. Fractal tidsserie modellering och analys tillhandahålls av fraktal. fractalrock genererar fraktal tidsserier med icke-normala avkastningsfördelningar. Dynamiska regressionsmodeller Dynamiska linjära modeller. Ett praktiskt gränssnitt för anpassning av dynamiska regressionsmodeller via OLS finns i dynlm ett förbättrat tillvägagångssätt som också fungerar med andra regressionsfunktioner och fler tidsserier klassificeras i dyn. Mer avancerade dynamiska systemekvationer kan monteras med dse. Gaussiska linjära rymdmodeller kan monteras med hjälp av dlm (via maximal sannolikhet, Kalman filteringsmoothing och Bayesian metoder) eller med hjälp av bsts som använder MCMC. Funktioner för icke-linjär modellering med distribuerad lagring ges i dlnm. Tidsvarierande parametermodeller kan monteras med tpr-paketet. orderedLasso passar en gles linjär modell med en orderbegränsning på koefficienterna för att hantera fördröjda regressorer där koefficienterna försvinner när fördröjningen ökar. Dynamisk modellering av olika slag finns i dynr inklusive diskret och kontinuerlig tid, linjära och olinjära modeller och olika typer av latenta variabler. Multivariate Time Series Modeller Vector-autoregressiva (VAR) - modeller tillhandahålls via ar () i det grundläggande statistikpaketet inklusive orderval via AIC. Dessa modeller är begränsade till att vara stationära. MTS är en allsidig verktygslåda för analys av multivariata tidsserier, inklusive VAR, VARMA, säsongsvarma VARMA, VAR-modeller med exogena variabler, multivariat regression med tidsseriefel och mycket mer. Eventuellt är icke-stationära VAR-modeller monterade i mAr-paketet, vilket också möjliggör VAR-modeller i huvuddelen av komponenten. Sparsevar möjliggör uppskattning av glesa VAR - och VECM-modeller, ecm ger funktioner för att bygga VECM-modeller, medan BigVAR uppskattar VAR - och VARX-modeller med strukturerad lassaväxling. Automatiserade VAR-modeller och nätverk finns i autovarCore. Mer utförliga modeller finns i paket vars. tsDyn. estVARXls () i dse. och en Bayesian approach är tillgänglig i MSBVAR. En annan implementering med uppstartade prediktionsintervaller ges i VAR. etp. mlVAR ger multi-level vektor autoregression. VARsignR tillhandahåller rutiner för att identifiera strukturella stötar i VAR-modeller med teckenbegränsningar. gdpc implementerar generaliserade dynamiska huvudkomponenter. pcdpca utökar dynamiska huvudkomponenter till periodiskt korrelerade multivariata tidsserier. VARIMA-modeller och statliga rymdmodeller finns i dse-paketet. EvalEst underlättar Monte Carlo-experiment för att utvärdera de associerade estimeringsmetoderna. Vektorfelkorrigeringsmodeller är tillgängliga via urkasystemet. vars och tsDyn-paket, inklusive versioner med strukturella begränsningar och tröskelvärden. Tidsserie komponentanalys. Tidsseriefaktoranalys tillhandahålls i tsfa. ForeCA implementerar förknipplig komponentanalys genom att söka efter de bästa linjära transformationerna som gör en multivariatidserie så förutsägbar som möjligt. PCA4TS finner en linjär transformation av en multivariatidserier som ger mindre dimensionella undergrupper som är okorrelerade med varandra. Multivariata tillståndsmodellmodeller implementeras i FKF-paketet (Fast Kalman Filter). Detta ger relativt flexibla tillståndsmodellmodeller via funktionen fkf (): tillståndsrymdparametrar tillåts vara tidsvarierande och avlyssningar ingår i båda ekvationerna. Ett alternativt genomförande tillhandahålls av KFAS-paketet, vilket ger ett snabbt multivariat Kalman-filter, mjukare, simulering mjukare och prognoser. Ytterligare ett genomförande ges i dlm-paketet, som också innehåller verktyg för att konvertera andra multivariata modeller till tillståndsrymdform. dlmodeler ger ett enhetligt gränssnitt för dlm. KFAS och FKF. MARSS passar begränsade och obestridda multivariata autoregressiva tillståndsrumsmodeller med hjälp av en EM-algoritm. Alla dessa paket antar de observativa och tillståndsfelterna är okorrelerade. Delvis observerade Markov-processer är en generalisering av de vanliga linjära multivariata tillståndsmodellmodellerna, vilket möjliggör icke-gaussiska och olinjära modeller. Dessa implementeras i pomppaketet. Multivariata stokastiska volatilitetsmodeller (med latenta faktorer) tillhandahålls av faktorstochvol. Analys av stora grupper av tidsserier Tidsserieklypning implementeras i TSclust. dtwclust. BNPTSclust och pdc. TSdist ger distansåtgärder för tidsseriedata. jmotif implementerar verktyg baserat på tidsseriens symboliska diskretisering för att hitta motiv i tidsserier och underlättar tolkbar tidsserierklassificering. rucrdtw tillhandahåller R-bindningar för funktioner från UCR-sviten för att möjliggöra ultrasnabb undersekvenser sök efter en bästa match under Dynamic Time Warping och Euclidean Distance. Metoder för att planera och prognoser samlingar av hierarkiska och grupperade tidsserier tillhandahålls av hts. tjuven använder hierarkiska metoder för att förena prognoser av temporärt aggregerade tidsserier. Ett alternativt sätt att förena prognoser för hierarkiska tidsserier tillhandahålls av gtop. tjuv Kontinuerliga tidsmodeller Kontinuerlig tidsautoregressiv modellering finns i cts. Sim. DiffProc simulerar och modeller stokastiska differentialekvationer. Simulering och inferens för stokastiska differentialekvationer tillhandahålls av sde och yuima. Bootstrapping. Uppstartspaketet ger funktionen tsboot () för startseriens bootstrapping, inklusive block bootstrap med flera varianter. tsbootstrap () från tseries ger snabb stationär och block bootstrapping. Maximal entropy bootstrap för tidsserier finns i meboot. timeboot beräknar bootstrap CI för provet ACF och periodogram. BootPR beräknar biaskorrigerade prognoser och boostrap-prediktionsintervall för autoregressiva tidsserier. Data från Makridakis, Wheelwright och Hyndman (1998) Prognos: metoder och applikationer finns i fma-paketet. Data från Hyndman, Koehler, Ord och Snyder (2008) Prognoser med exponentiell utjämning finns i expsmooth-paketet. Data från Hyndman och Athanasopoulos (2013) Prognoser: Principer och övning finns i fpp-paketet. Data från M-tävlingen och M3-tävlingen finns i Mcomp-paketet. Data från M4-tävlingen ges i M4comp. medan Tcomp tillhandahåller data från 2010 IJF Tourism Forecasting Competition. pdfetch tillhandahåller faciliteter för nedladdning av ekonomiska och finansiella tidsserier från offentliga källor. Data från Quandl online portal till ekonomiska, ekonomiska och sociala dataset kan frågas interaktivt med Quandl-paketet. Data från Datamarkets onlineportal kan hämtas med rdatamarketpaketet. BETS ger tillgång till de viktigaste ekonomiska tidsserierna i Brasilien. Data från Cryer och Chan (2010) finns i TSA-paketet. Data från Shumway och Stoffer (2011) finns i astsa-paketet. Data från Tsay (2005) Analys av finansiella tidsserier finns i FinTS-paketet, tillsammans med vissa funktioner och skriptfiler som krävs för att arbeta några av exemplen. tswge åtföljer texten Applied Time Series Analysis med R. 2: a upplagan av Woodward, Gray och Elliott. TSdbi tillhandahåller ett gemensamt gränssnitt till tidsseriedatabaser. berömmelse ger ett gränssnitt för FAME-tidsseriedatabaser AER och Ecdat innehåller båda många dataset (inklusive tidsseriedata) från många ekonometriska textböcker dtw. Dynamiska tidsvärdesalgoritmer för beräkning och plottning av parvisa inriktningar mellan tidsserier. ensembleBMA. Bayesian Model Averaging för att skapa probabilistiska prognoser från ensembleprognoser och väderobservationer. earlywarnings. Tidiga varningar signalerar verktygslåda för att upptäcka kritiska övergångar i händelser i tidsserier. gör maskinutvunna händelsedata till regelbundna aggregerade multivariata tidsserier. FeedbackTS. Analys av fragmenterad tidriktning för att undersöka feedback i tidsserier. LPStimeSeries har som målsättning att hitta kvotlarmad mönsterlikhet för tidsserier. MAR1 tillhandahåller verktyg för att förbereda ekologiska samhällets tidsseriedata för multivariat AR-modellering. nät. rutiner för uppskattning av glesa långsiktiga partiella korrelationsnät för tidsseriedata. paleoTS. Modelleringsutveckling i paleontologiska tidsserier. pastecs. Reglering, sönderdelning och analys av rymdtidsserier. ptw. Parametrisk tidsförvrängning. RGENERATE ger verktyg för att generera vektor tidsserier. RMAWGEN är uppsatt av S3- och S4-funktioner för rumslig stochastisk generering av dagstidsserier av temperatur och nederbörd som använder VAR-modeller. Paketet kan användas i klimatologi och statistisk hydrologi. RSEIS. Seismiska tidsserieanalysverktyg. rts. Raster tidsserieanalys (t ex tidsserier av satellitbilder). sae2. Tidsseriemodeller för liten arealberäkning. sptimer. Spatio-temporal Bayesian modellering. övervakning. Temporal och spatio-temporal modellering och övervakning av epidemiska fenomen. TED. Turbulens tidsserie Event Detection och klassificering. Tidvatten. Funktioner för att beräkna egenskaper hos kvasi periodiska tidsserier, t. ex. observerade estuarinvattennivån. tiger. Temporalt lösta grupper av typiska skillnader (fel) mellan två tidsserier bestäms och visualiseras. TSMining. Mining Univariate och Multivariate Motiv i Time-Series Data. tsModel. Tidsserie modellering för luftföroreningar och hälsa. CRAN-paket: Relaterade länkar: gt mav (c (4,5,4,6), 3) Tidsserie: Start 1 Slut 4 Frekvens 1 1 NA 4.333333 5.000000 NA Här försökte jag göra ett rullande medelvärde som tog hänsyn till sista 3 siffror så jag förväntade mig att få bara två nummer tillbaka 8211 4.333333 och 5 8211 och om det skulle bli NA-värden trodde jag att de8217d skulle vara i början av sekvensen. Faktum är att det här är vad 8216sides8217 parametern kontrollerar: endast sidor för konvolutionsfilter. Om sidorna 1 är filterkoefficienterna endast för tidigare värden om sidorna 2 är centrerade runt lag 0. I så fall bör filterets längd vara udda, men om det är jämnt är mer av filtret framåt i tid än bakåt. Så i vår 8216mav8217-funktion ser rullande medelvärdet ut båda sidorna av det aktuella värdet i stället för bara vid tidigare värden. Vi kan tweak det för att få det beteende vi vill ha: gt bibliotek (zoo) gt rollmean (c (4,5,4,6), 3) 1 4.333333 5.000000 Jag insåg också att jag kan lista alla funktioner i ett paket med 8216ls8217 funktion så att I8217ll skannar zoo8217s lista över funktioner nästa gång jag behöver göra något tidsserie relaterat 8211 there8217ll är förmodligen redan en funktion för det gt ls (quotpackage: zooquot) 1 kvoter. Datequot kvoter. Date. numericquot quotas. Date. tsquot 4 kvoter. Date. yearmonquot quotas. Date. yearqququot kvoter. yearmonquot 7 kvoter. yearmon. defaultquot quotas. yearqtrquot quotas. yearqtr. defaultquot 10 kvoter. zooquot quotas. zoo. defaultquot quotas. zooregquot 13 quotas. zooreg. defaultquot quotautoplot. zooquot quotcbind. zooquot 16 quotcoredataquot quotcoredata. defaultquot quotcoredatalt-quot 19 quotfacetfreequot quotformat. yearqtrquot quotfortify. zooquot 22 quotfrequencylt-quot quotifelse. zooquot quotindexquot 25 quotindexlt-quot quotindex2charquot quotis. regularquot 28 quotis. zooquot quotmake. par. listquot q uotMATCHquot 31 quotMATCH. defaultquot quotMATCH. timesquot quotmedian. zooquot 34 quotmerge. zooquot quotna. aggregatequot quotna. aggregate. defaultquot 37 quotna. approxquot quotna. approx. defaultquot quotna. fillquot 40 quotna. fill. defaultquot quotna. locfquot quotna. locf. defaultquot 43 quotna. splinequot quotna. spline. defaultquot quotna. StructTSquot 46 quotna. trimquot quotna. trim. defaultquot quotna. trim. tsquot 49 quotORDERquot quotORDER. defaultquot quotpanel. lines. itsquot 52 quotpanel. lines. tisquot quotpanel. lines. tsquot quotpanel. lines. zooquot 55 quotpanel. plot. customquot quotpanel. poot. defaultquot quotpanel. points. itsquot 58 quotpanel. points. tisquot quotpanel. points. tsquot quotpanel. points. zooquot 61 quotpanel. polygon. itsquot quotpanel. polygon. tisquot quotpanel. polygon. tsquot 64 quotpanel. polygon. zooquot quotpanel. rect. itsquot quotpanel. rect. tisquot 67 quotpanel. rect. tsquot quotpanel. rect. zooquot quotpanel. segments. itsquot 70 quotpanel. segments. tisquot quotpanel. segments. tsquot quotpanel. se gments. zooquot 73 quotpanel. text. itsquot quotpanel. text. tisquot quotpanel. text. tsquot 76 quotpanel. text. zooquot quotplot. zooquot quotquantile. zooquot 79 quotrbind. zooquot quotread. zooquot quotrev. zooquot 82 quotrollapplyquot quotrollapplyrquot quotrollmaxquot 85 quotrollmax. defaultquot quotrollmaxrquot quotrollmeanquot 88 quotrollmean. defaultquot quotrollmeanrquot quotrollmedianquot 91 quotrollmedian. defaultquot quotrollmedianrquot quotrollsumquot 94 quotrollsum. defaultquot quotrollsumrquot quotscalexyearmonquot 97 quotscalexyearqtrquot quotscaleyyearmonquot quotscaleyyearqtrquot 100 quotSys. yearmonquot quotSys. yearqtrquot quottimelt-quot 103 quotwrite. zooquot quotxblocksquot quotxblocks. defaultquot 106 quotxtfrm. zooquot quotyearmonquot quotyearmontransquot 109 quotyearqtrquot quotyearqtrtransquot quotzooquot 112 quotzooregquot Var sällskaplig, DelaUsing R för tidsserieanalys Tidsserieanalys Detta häfte beskriver hur du använder R statistikprogrammet för att utföra några enkla a nalyses som är vanliga vid analys av tidsseriedata. Detta häfte förutsätter att läsaren har viss grundläggande kunskaper om tidsserieanalys och huvudboken för häftet är inte att förklara tidsserieanalys utan snarare att förklara hur man utför dessa analyser med R. Om du är ny på tidsserier analys och vill lära mig mer om några av de begrepp som presenteras här, rekommenderar jag starkt Open University-boken 8220Time series8221 (produktkod M24902), tillgänglig från Open University Shop. I det här häftet använder jag tidsseriedatasatser som Rob Hyndman vänligen gjort tillgängligt i sitt tidsseriedatabibliotek på robjhyndmanTSDL. Om du gillar det här häftet kan du också kolla in min broschyr på att använda R för biomedicinsk statistik, lite-book-of-r-for-biomedical-statistics. readthedocs. org. och min broschyr om att använda R för multivariat analys, little-book-of-r-for-multivariate-analysis. readthedocs. org. Läsningstidsseriedata Det första du vill göra för att analysera dina tidsseriedata kommer att vara att läsa in det i R och att plotta tidsserien. You can read data into R using the scan() function, which assumes that your data for successive time points is in a simple text file with one column. For example, the file robjhyndmantsdldatamisckings. dat contains data on the age of death of successive kings of England, starting with William the Conqueror (original source: Hipel and Mcleod, 1994). The data set looks like this: Only the first few lines of the file have been shown. The first three lines contain some comment on the data, and we want to ignore this when we read the data into R. We can use this by using the 8220skip8221 parameter of the scan() function, which specifies how many lines at the top of the file to ignore. To read the file into R, ignoring the first three lines, we type: In this case the age of death of 42 successive kings of England has been read into the variable 8216kings8217. Once you have read the time series data into R, the next step is to store the data in a time series object in R, so that you can use R8217s many functions for analysing time series data. To store the data in a time series object, we use the ts() function in R. For example, to store the data in the variable 8216kings8217 as a time series object in R, we type: Sometimes the time series data set that you have may have been collected at regular intervals that were less than one year, for example, monthly or quarterly. In this case, you can specify the number of times that data was collected per year by using the 8216frequency8217 parameter in the ts() function. For monthly time series data, you set frequency12, while for quarterly time series data, you set frequency4. You can also specify the first year that the data was collected, and the first interval in that year by using the 8216start8217 parameter in the ts() function. For example, if the first data point corresponds to the second quarter of 1986, you would set startc(1986,2). An example is a data set of the number of births per month in New York city, from January 1946 to December 1959 (originally collected by Newton). This data is available in the file robjhyndmantsdldatadatanybirths. dat We can read the data into R, and store it as a time series object, by typing: Similarly, the file robjhyndmantsdldatadatafancy. dat contains monthly sales for a souvenir shop at a beach resort town in Queensland, Australia, for January 1987-December 1993 (original data from Wheelwright and Hyndman, 1998). We can read the data into R by typing: Plotting Time Series Once you have read a time series into R, the next step is usually to make a plot of the time series data, which you can do with the plot. ts() function in R. For example, to plot the time series of the age of death of 42 successive kings of England, we type: We can see from the time plot that this time series could probably be described using an additive model, since the random fluctuations in the data are roughly constant in size over time. Likewise, to plot the time series of the number of births per month in New York city, we type: We can see from this time series that there seems to be seasonal variation in the number of births per month: there is a peak every summer, and a trough every winter. Again, it seems that this time series could probably be described using an additive model, as the seasonal fluctuations are roughly constant in size over time and do not seem to depend on the level of the time series, and the random fluctuations also seem to be roughly constant in size over time. Similarly, to plot the time series of the monthly sales for the souvenir shop at a beach resort town in Queensland, Australia, we type: In this case, it appears that an additive model is not appropriate for describing this time series, since the size of the seasonal fluctuations and random fluctuations seem to increase with the level of the time series. Thus, we may need to transform the time series in order to get a transformed time series that can be described using an additive model. For example, we can transform the time series by calculating the natural log of the original data: Here we can see that the size of the seasonal fluctuations and random fluctuations in the log-transformed time series seem to be roughly constant over time, and do not depend on the level of the time series. Thus, the log-transformed time series can probably be described using an additive model. Decomposing Time Series Decomposing a time series means separating it into its constituent components, which are usually a trend component and an irregular component, and if it is a seasonal time series, a seasonal component. Decomposing Non-Seasonal Data A non-seasonal time series consists of a trend component and an irregular component. Decomposing the time series involves trying to separate the time series into these components, that is, estimating the the trend component and the irregular component. To estimate the trend component of a non-seasonal time series that can be described using an additive model, it is common to use a smoothing method, such as calculating the simple moving average of the time series. The SMA() function in the 8220TTR8221 R package can be used to smooth time series data using a simple moving average. To use this function, we first need to install the 8220TTR8221 R package (for instructions on how to install an R package, see How to install an R package ). Once you have installed the 8220TTR8221 R package, you can load the 8220TTR8221 R package by typing: You can then use the 8220SMA()8221 function to smooth time series data. To use the SMA() function, you need to specify the order (span) of the simple moving average, using the parameter 8220n8221. For example, to calculate a simple moving average of order 5, we set n5 in the SMA() function. For example, as discussed above, the time series of the age of death of 42 successive kings of England appears is non-seasonal, and can probably be described using an additive model, since the random fluctuations in the data are roughly constant in size over time: Thus, we can try to estimate the trend component of this time series by smoothing using a simple moving average. To smooth the time series using a simple moving average of order 3, and plot the smoothed time series data, we type: There still appears to be quite a lot of random fluctuations in the time series smoothed using a simple moving average of order 3. Thus, to estimate the trend component more accurately, we might want to try smoothing the data with a simple moving average of a higher order. This takes a little bit of trial-and-error, to find the right amount of smoothing. For example, we can try using a simple moving average of order 8: The data smoothed with a simple moving average of order 8 gives a clearer picture of the trend component, and we can see that the age of death of the English kings seems to have decreased from about 55 years old to about 38 years old during the reign of the first 20 kings, and then increased after that to about 73 years old by the end of the reign of the 40th king in the time series. Decomposing Seasonal Data A seasonal time series consists of a trend component, a seasonal component and an irregular component. Decomposing the time series means separating the time series into these three components: that is, estimating these three components. To estimate the trend component and seasonal component of a seasonal time series that can be described using an additive model, we can use the 8220decompose()8221 function in R. This function estimates the trend, seasonal, and irregular components of a time series that can be described using an additive model. The function 8220decompose()8221 returns a list object as its result, where the estimates of the seasonal component, trend component and irregular component are stored in named elements of that list objects, called 8220seasonal8221, 8220trend8221, and 8220random8221 respectively. For example, as discussed above, the time series of the number of births per month in New York city is seasonal with a peak every summer and trough every winter, and can probably be described using an additive model since the seasonal and random fluctuations seem to be roughly constant in size over time: To estimate the trend, seasonal and irregular components of this time series, we type: The estimated values of the seasonal, trend and irregular components are now stored in variables birthstimeseriescomponentsseasonal, birthstimeseriescomponentstrend and birthstimeseriescomponentsrandom. For example, we can print out the estimated values of the seasonal component by typing: The estimated seasonal factors are given for the months January-December, and are the same for each year. The largest seasonal factor is for July (about 1.46), and the lowest is for February (about -2.08), indicating that there seems to be a peak in births in July and a trough in births in February each year. We can plot the estimated trend, seasonal, and irregular components of the time series by using the 8220plot()8221 function, for example: The plot above shows the original time series (top), the estimated trend component (second from top), the estimated seasonal component (third from top), and the estimated irregular component (bottom). We see that the estimated trend component shows a small decrease from about 24 in 1947 to about 22 in 1948, followed by a steady increase from then on to about 27 in 1959. Seasonally Adjusting If you have a seasonal time series that can be described using an additive model, you can seasonally adjust the time series by estimating the seasonal component, and subtracting the estimated seasonal component from the original time series. We can do this using the estimate of the seasonal component calculated by the 8220decompose()8221 function. For example, to seasonally adjust the time series of the number of births per month in New York city, we can estimate the seasonal component using 8220decompose()8221, and then subtract the seasonal component from the original time series: We can then plot the seasonally adjusted time series using the 8220plot()8221 function, by typing: You can see that the seasonal variation has been removed from the seasonally adjusted time series. The seasonally adjusted time series now just contains the trend component and an irregular component. Forecasts using Exponential Smoothing Exponential smoothing can be used to make short-term forecasts for time series data. Simple Exponential Smoothing If you have a time series that can be described using an additive model with constant level and no seasonality, you can use simple exponential smoothing to make short-term forecasts. The simple exponential smoothing method provides a way of estimating the level at the current time point. Smoothing is controlled by the parameter alpha for the estimate of the level at the current time point. The value of alpha lies between 0 and 1. Values of alpha that are close to 0 mean that little weight is placed on the most recent observations when making forecasts of future values. For example, the file robjhyndmantsdldatahurstprecip1.dat contains total annual rainfall in inches for London, from 1813-1912 (original data from Hipel and McLeod, 1994). We can read the data into R and plot it by typing: You can see from the plot that there is roughly constant level (the mean stays constant at about 25 inches). The random fluctuations in the time series seem to be roughly constant in size over time, so it is probably appropriate to describe the data using an additive model. Thus, we can make forecasts using simple exponential smoothing. To make forecasts using simple exponential smoothing in R, we can fit a simple exponential smoothing predictive model using the 8220HoltWinters()8221 function in R. To use HoltWinters() for simple exponential smoothing, we need to set the parameters betaFALSE and gammaFALSE in the HoltWinters() function (the beta and gamma parameters are used for Holt8217s exponential smoothing, or Holt-Winters exponential smoothing, as described below). The HoltWinters() function returns a list variable, that contains several named elements. For example, to use simple exponential smoothing to make forecasts for the time series of annual rainfall in London, we type: The output of HoltWinters() tells us that the estimated value of the alpha parameter is about 0.024. This is very close to zero, telling us that the forecasts are based on both recent and less recent observations (although somewhat more weight is placed on recent observations). By default, HoltWinters() just makes forecasts for the same time period covered by our original time series. In this case, our original time series included rainfall for London from 1813-1912, so the forecasts are also for 1813-1912. In the example above, we have stored the output of the HoltWinters() function in the list variable 8220rainseriesforecasts8221. The forecasts made by HoltWinters() are stored in a named element of this list variable called 8220fitted8221, so we can get their values by typing: We can plot the original time series against the forecasts by typing: The plot shows the original time series in black, and the forecasts as a red line. The time series of forecasts is much smoother than the time series of the original data here. As a measure of the accuracy of the forecasts, we can calculate the sum of squared errors for the in-sample forecast errors, that is, the forecast errors for the time period covered by our original time series. The sum-of-squared-errors is stored in a named element of the list variable 8220rainseriesforecasts8221 called 8220SSE8221, so we can get its value by typing: That is, here the sum-of-squared-errors is 1828.855. It is common in simple exponential smoothing to use the first value in the time series as the initial value for the level. For example, in the time series for rainfall in London, the first value is 23.56 (inches) for rainfall in 1813. You can specify the initial value for the level in the HoltWinters() function by using the 8220l. start8221 parameter. For example, to make forecasts with the initial value of the level set to 23.56, we type: As explained above, by default HoltWinters() just makes forecasts for the time period covered by the original data, which is 1813-1912 for the rainfall time series. We can make forecasts for further time points by using the 8220forecast. HoltWinters()8221 function in the R 8220forecast8221 package. To use the forecast. HoltWinters() function, we first need to install the 8220forecast8221 R package (for instructions on how to install an R package, see How to install an R package ). Once you have installed the 8220forecast8221 R package, you can load the 8220forecast8221 R package by typing: When using the forecast. HoltWinters() function, as its first argument (input), you pass it the predictive model that you have already fitted using the HoltWinters() function. For example, in the case of the rainfall time series, we stored the predictive model made using HoltWinters() in the variable 8220rainseriesforecasts8221. You specify how many further time points you want to make forecasts for by using the 8220h8221 parameter in forecast. HoltWinters(). For example, to make a forecast of rainfall for the years 1814-1820 (8 more years) using forecast. HoltWinters(), we type: The forecast. HoltWinters() function gives you the forecast for a year, a 80 prediction interval for the forecast, and a 95 prediction interval for the forecast. For example, the forecasted rainfall for 1920 is about 24.68 inches, with a 95 prediction interval of (16.24, 33.11). To plot the predictions made by forecast. HoltWinters(), we can use the 8220plot. forecast()8221 function: Here the forecasts for 1913-1920 are plotted as a blue line, the 80 prediction interval as an orange shaded area, and the 95 prediction interval as a yellow shaded area. The 8216forecast errors8217 are calculated as the observed values minus predicted values, for each time point. We can only calculate the forecast errors for the time period covered by our original time series, which is 1813-1912 for the rainfall data. As mentioned above, one measure of the accuracy of the predictive model is the sum-of-squared-errors (SSE) for the in-sample forecast errors. The in-sample forecast errors are stored in the named element 8220residuals8221 of the list variable returned by forecast. HoltWinters(). If the predictive model cannot be improved upon, there should be no correlations between forecast errors for successive predictions. In other words, if there are correlations between forecast errors for successive predictions, it is likely that the simple exponential smoothing forecasts could be improved upon by another forecasting technique. To figure out whether this is the case, we can obtain a correlogram of the in-sample forecast errors for lags 1-20. We can calculate a correlogram of the forecast errors using the 8220acf()8221 function in R. To specify the maximum lag that we want to look at, we use the 8220lag. max8221 parameter in acf(). For example, to calculate a correlogram of the in-sample forecast errors for the London rainfall data for lags 1-20, we type: You can see from the sample correlogram that the autocorrelation at lag 3 is just touching the significance bounds. To test whether there is significant evidence for non-zero correlations at lags 1-20, we can carry out a Ljung-Box test. This can be done in R using the 8220Box. test()8221, function. The maximum lag that we want to look at is specified using the 8220lag8221 parameter in the Box. test() function. For example, to test whether there are non-zero autocorrelations at lags 1-20, for the in-sample forecast errors for London rainfall data, we type: Here the Ljung-Box test statistic is 17.4, and the p-value is 0.6, so there is little evidence of non-zero autocorrelations in the in-sample forecast errors at lags 1-20. To be sure that the predictive model cannot be improved upon, it is also a good idea to check whether the forecast errors are normally distributed with mean zero and constant variance. To check whether the forecast errors have constant variance, we can make a time plot of the in-sample forecast errors: The plot shows that the in-sample forecast errors seem to have roughly constant variance over time, although the size of the fluctuations in the start of the time series (1820-1830) may be slightly less than that at later dates (eg. 1840-1850). To check whether the forecast errors are normally distributed with mean zero, we can plot a histogram of the forecast errors, with an overlaid normal curve that has mean zero and the same standard deviation as the distribution of forecast errors. To do this, we can define an R function 8220plotForecastErrors()8221, below: You will have to copy the function above into R in order to use it. You can then use plotForecastErrors() to plot a histogram (with overlaid normal curve) of the forecast errors for the rainfall predictions: The plot shows that the distribution of forecast errors is roughly centred on zero, and is more or less normally distributed, although it seems to be slightly skewed to the right compared to a normal curve. However, the right skew is relatively small, and so it is plausible that the forecast errors are normally distributed with mean zero. The Ljung-Box test showed that there is little evidence of non-zero autocorrelations in the in-sample forecast errors, and the distribution of forecast errors seems to be normally distributed with mean zero. This suggests that the simple exponential smoothing method provides an adequate predictive model for London rainfall, which probably cannot be improved upon. Furthermore, the assumptions that the 80 and 95 predictions intervals were based upon (that there are no autocorrelations in the forecast errors, and the forecast errors are normally distributed with mean zero and constant variance) are probably valid. Holt8217s Exponential Smoothing If you have a time series that can be described using an additive model with increasing or decreasing trend and no seasonality, you can use Holt8217s exponential smoothing to make short-term forecasts. Holt8217s exponential smoothing estimates the level and slope at the current time point. Smoothing is controlled by two parameters, alpha, for the estimate of the level at the current time point, and beta for the estimate of the slope b of the trend component at the current time point. As with simple exponential smoothing, the paramters alpha and beta have values between 0 and 1, and values that are close to 0 mean that little weight is placed on the most recent observations when making forecasts of future values. An example of a time series that can probably be described using an additive model with a trend and no seasonality is the time series of the annual diameter of women8217s skirts at the hem, from 1866 to 1911. The data is available in the file robjhyndmantsdldatarobertsskirts. dat (original data from Hipel and McLeod, 1994). We can read in and plot the data in R by typing: We can see from the plot that there was an increase in hem diameter from about 600 in 1866 to about 1050 in 1880, and that afterwards the hem diameter decreased to about 520 in 1911. To make forecasts, we can fit a predictive model using the HoltWinters() function in R. To use HoltWinters() for Holt8217s exponential smoothing, we need to set the parameter gammaFALSE (the gamma parameter is used for Holt-Winters exponential smoothing, as described below). For example, to use Holt8217s exponential smoothing to fit a predictive model for skirt hem diameter, we type: The estimated value of alpha is 0.84, and of beta is 1.00. These are both high, telling us that both the estimate of the current value of the level, and of the slope b of the trend component, are based mostly upon very recent observations in the time series. This makes good intuitive sense, since the level and the slope of the time series both change quite a lot over time. The value of the sum-of-squared-errors for the in-sample forecast errors is 16954. We can plot the original time series as a black line, with the forecasted values as a red line on top of that, by typing: We can see from the picture that the in-sample forecasts agree pretty well with the observed values, although they tend to lag behind the observed values a little bit. If you wish, you can specify the initial values of the level and the slope b of the trend component by using the 8220l. start8221 and 8220b. start8221 arguments for the HoltWinters() function. It is common to set the initial value of the level to the first value in the time series (608 for the skirts data), and the initial value of the slope to the second value minus the first value (9 for the skirts data). For example, to fit a predictive model to the skirt hem data using Holt8217s exponential smoothing, with initial values of 608 for the level and 9 for the slope b of the trend component, we type: As for simple exponential smoothing, we can make forecasts for future times not covered by the original time series by using the forecast. HoltWinters() function in the 8220forecast8221 package. For example, our time series data for skirt hems was for 1866 to 1911, so we can make predictions for 1912 to 1930 (19 more data points), and plot them, by typing: The forecasts are shown as a blue line, with the 80 prediction intervals as an orange shaded area, and the 95 prediction intervals as a yellow shaded area. As for simple exponential smoothing, we can check whether the predictive model could be improved upon by checking whether the in-sample forecast errors show non-zero autocorrelations at lags 1-20. For example, for the skirt hem data, we can make a correlogram, and carry out the Ljung-Box test, by typing: Here the correlogram shows that the sample autocorrelation for the in-sample forecast errors at lag 5 exceeds the significance bounds. However, we would expect one in 20 of the autocorrelations for the first twenty lags to exceed the 95 significance bounds by chance alone. Indeed, when we carry out the Ljung-Box test, the p-value is 0.47, indicating that there is little evidence of non-zero autocorrelations in the in-sample forecast errors at lags 1-20. As for simple exponential smoothing, we should also check that the forecast errors have constant variance over time, and are normally distributed with mean zero. We can do this by making a time plot of forecast errors, and a histogram of the distribution of forecast errors with an overlaid normal curve: The time plot of forecast errors shows that the forecast errors have roughly constant variance over time. The histogram of forecast errors show that it is plausible that the forecast errors are normally distributed with mean zero and constant variance. Thus, the Ljung-Box test shows that there is little evidence of autocorrelations in the forecast errors, while the time plot and histogram of forecast errors show that it is plausible that the forecast errors are normally distributed with mean zero and constant variance. Therefore, we can conclude that Holt8217s exponential smoothing provides an adequate predictive model for skirt hem diameters, which probably cannot be improved upon. In addition, it means that the assumptions that the 80 and 95 predictions intervals were based upon are probably valid. Holt-Winters Exponential Smoothing If you have a time series that can be described using an additive model with increasing or decreasing trend and seasonality, you can use Holt-Winters exponential smoothing to make short-term forecasts. Holt-Winters exponential smoothing estimates the level, slope and seasonal component at the current time point. Smoothing is controlled by three parameters: alpha, beta, and gamma, for the estimates of the level, slope b of the trend component, and the seasonal component, respectively, at the current time point. The parameters alpha, beta and gamma all have values between 0 and 1, and values that are close to 0 mean that relatively little weight is placed on the most recent observations when making forecasts of future values. An example of a time series that can probably be described using an additive model with a trend and seasonality is the time series of the log of monthly sales for the souvenir shop at a beach resort town in Queensland, Australia (discussed above): To make forecasts, we can fit a predictive model using the HoltWinters() function. For example, to fit a predictive model for the log of the monthly sales in the souvenir shop, we type: The estimated values of alpha, beta and gamma are 0.41, 0.00, and 0.96, respectively. The value of alpha (0.41) is relatively low, indicating that the estimate of the level at the current time point is based upon both recent observations and some observations in the more distant past. The value of beta is 0.00, indicating that the estimate of the slope b of the trend component is not updated over the time series, and instead is set equal to its initial value. This makes good intuitive sense, as the level changes quite a bit over the time series, but the slope b of the trend component remains roughly the same. In contrast, the value of gamma (0.96) is high, indicating that the estimate of the seasonal component at the current time point is just based upon very recent observations. As for simple exponential smoothing and Holt8217s exponential smoothing, we can plot the original time series as a black line, with the forecasted values as a red line on top of that: We see from the plot that the Holt-Winters exponential method is very successful in predicting the seasonal peaks, which occur roughly in November every year. To make forecasts for future times not included in the original time series, we use the 8220forecast. HoltWinters()8221 function in the 8220forecast8221 package. For example, the original data for the souvenir sales is from January 1987 to December 1993. If we wanted to make forecasts for January 1994 to December 1998 (48 more months), and plot the forecasts, we would type: The forecasts are shown as a blue line, and the orange and yellow shaded areas show 80 and 95 prediction intervals, respectively. We can investigate whether the predictive model can be improved upon by checking whether the in-sample forecast errors show non-zero autocorrelations at lags 1-20, by making a correlogram and carrying out the Ljung-Box test: The correlogram shows that the autocorrelations for the in-sample forecast errors do not exceed the significance bounds for lags 1-20. Furthermore, the p-value for Ljung-Box test is 0.6, indicating that there is little evidence of non-zero autocorrelations at lags 1-20. We can check whether the forecast errors have constant variance over time, and are normally distributed with mean zero, by making a time plot of the forecast errors and a histogram (with overlaid normal curve): From the time plot, it appears plausible that the forecast errors have constant variance over time. From the histogram of forecast errors, it seems plausible that the forecast errors are normally distributed with mean zero. Thus, there is little evidence of autocorrelation at lags 1-20 for the forecast errors, and the forecast errors appear to be normally distributed with mean zero and constant variance over time. This suggests that Holt-Winters exponential smoothing provides an adequate predictive model of the log of sales at the souvenir shop, which probably cannot be improved upon. Furthermore, the assumptions upon which the prediction intervals were based are probably valid. ARIMA Models Exponential smoothing methods are useful for making forecasts, and make no assumptions about the correlations between successive values of the time series. However, if you want to make prediction intervals for forecasts made using exponential smoothing methods, the prediction intervals require that the forecast errors are uncorrelated and are normally distributed with mean zero and constant variance. While exponential smoothing methods do not make any assumptions about correlations between successive values of the time series, in some cases you can make a better predictive model by taking correlations in the data into account. Autoregressive Integrated Moving Average (ARIMA) models include an explicit statistical model for the irregular component of a time series, that allows for non-zero autocorrelations in the irregular component. Differencing a Time Series ARIMA models are defined for stationary time series. Therefore, if you start off with a non-stationary time series, you will first need to 8216difference8217 the time series until you obtain a stationary time series. If you have to difference the time series d times to obtain a stationary series, then you have an ARIMA(p, d,q) model, where d is the order of differencing used. You can difference a time series using the 8220diff()8221 function in R. For example, the time series of the annual diameter of women8217s skirts at the hem, from 1866 to 1911 is not stationary in mean, as the level changes a lot over time: We can difference the time series (which we stored in 8220skirtsseries8221, see above) once, and plot the differenced series, by typing: The resulting time series of first differences (above) does not appear to be stationary in mean. Therefore, we can difference the time series twice, to see if that gives us a stationary time series: Formal tests for stationarity Formal tests for stationarity called 8220unit root tests8221 are available in the fUnitRoots package, available on CRAN, but will not be discussed here. The time series of second differences (above) does appear to be stationary in mean and variance, as the level of the series stays roughly constant over time, and the variance of the series appears roughly constant over time. Thus, it appears that we need to difference the time series of the diameter of skirts twice in order to achieve a stationary series. If you need to difference your original time series data d times in order to obtain a stationary time series, this means that you can use an ARIMA(p, d,q) model for your time series, where d is the order of differencing used. For example, for the time series of the diameter of women8217s skirts, we had to difference the time series twice, and so the order of differencing (d) is 2. This means that you can use an ARIMA(p,2,q) model for your time series. The next step is to figure out the values of p and q for the ARIMA model. Another example is the time series of the age of death of the successive kings of England (see above): From the time plot (above), we can see that the time series is not stationary in mean. To calculate the time series of first differences, and plot it, we type: The time series of first differences appears to be stationary in mean and variance, and so an ARIMA(p,1,q) model is probably appropriate for the time series of the age of death of the kings of England. By taking the time series of first differences, we have removed the trend component of the time series of the ages at death of the kings, and are left with an irregular component. We can now examine whether there are correlations between successive terms of this irregular component if so, this could help us to make a predictive model for the ages at death of the kings. Selecting a Candidate ARIMA Model If your time series is stationary, or if you have transformed it to a stationary time series by differencing d times, the next step is to select the appropriate ARIMA model, which means finding the values of most appropriate values of p and q for an ARIMA(p, d,q) model. To do this, you usually need to examine the correlogram and partial correlogram of the stationary time series. To plot a correlogram and partial correlogram, we can use the 8220acf()8221 and 8220pacf()8221 functions in R, respectively. To get the actual values of the autocorrelations and partial autocorrelations, we set 8220plotFALSE8221 in the 8220acf()8221 and 8220pacf()8221 functions. Example of the Ages at Death of the Kings of England For example, to plot the correlogram for lags 1-20 of the once differenced time series of the ages at death of the kings of England, and to get the values of the autocorrelations, we type: We see from the correlogram that the autocorrelation at lag 1 (-0.360) exceeds the significance bounds, but all other autocorrelations between lags 1-20 do not exceed the significance bounds. To plot the partial correlogram for lags 1-20 for the once differenced time series of the ages at death of the English kings, and get the values of the partial autocorrelations, we use the 8220pacf()8221 function, by typing: The partial correlogram shows that the partial autocorrelations at lags 1, 2 and 3 exceed the significance bounds, are negative, and are slowly decreasing in magnitude with increasing lag (lag 1: -0.360, lag 2: -0.335, lag 3:-0.321). The partial autocorrelations tail off to zero after lag 3. Since the correlogram is zero after lag 1, and the partial correlogram tails off to zero after lag 3, this means that the following ARMA (autoregressive moving average) models are possible for the time series of first differences: an ARMA(3,0) model, that is, an autoregressive model of order p3, since the partial autocorrelogram is zero after lag 3, and the autocorrelogram tails off to zero (although perhaps too abruptly for this model to be appropriate) an ARMA(0,1) model, that is, a moving average model of order q1, since the autocorrelogram is zero after lag 1 and the partial autocorrelogram tails off to zero an ARMA(p, q) model, that is, a mixed model with p and q greater than 0, since the autocorrelogram and partial correlogram tail off to zero (although the correlogram probably tails off to zero too abruptly for this model to be appropriate) We use the principle of parsimony to decide which model is best: that is, we assum e that the model with the fewest parameters is best. The ARMA(3,0) model has 3 parameters, the ARMA(0,1) model has 1 parameter, and the ARMA(p, q) model has at least 2 parameters. Therefore, the ARMA(0,1) model is taken as the best model. An ARMA(0,1) model is a moving average model of order 1, or MA(1) model. This model can be written as: Xt - mu Zt - (theta Zt-1), where Xt is the stationary time series we are studying (the first differenced series of ages at death of English kings), mu is the mean of time series Xt, Zt is white noise with mean zero and constant variance, and theta is a parameter that can be estimated. A MA (moving average) model is usually used to model a time series that shows short-term dependencies between successive observations. Intuitively, it makes good sense that a MA model can be used to describe the irregular component in the time series of ages at death of English kings, as we might expect the age at death of a particular English king to have some effect on the ages at death of the next king or two, but not much effect on the ages at death of kings that reign much longer after that. Shortcut: the auto. arima() function The auto. arima() function can be used to find the appropriate ARIMA model, eg. type 8220library(forecast)8221, then 8220auto. arima(kings)8221. The output says an appropriate model is ARIMA(0,1,1). Since an ARMA(0,1) model (with p0, q1) is taken to be the best candidate model for the time series of first differences of the ages at death of English kings, then the original time series of the ages of death can be modelled using an ARIMA(0,1,1) model (with p0, d1, q1, where d is the order of differencing required). Example of the Volcanic Dust Veil in the Northern Hemisphere Let8217s take another example of selecting an appropriate ARIMA model. The file file robjhyndmantsdldataannualdvi. dat contains data on the volcanic dust veil index in the northern hemisphere, from 1500-1969 (original data from Hipel and Mcleod, 1994). This is a measure of the impact of volcanic eruptions8217 release of dust and aerosols into the environment. We can read it into R and make a time plot by typing: From the time plot, it appears that the random fluctuations in the time series are roughly constant in size over time, so an additive model is probably appropriate for describing this time series. Furthermore, the time series appears to be stationary in mean and variance, as its level and variance appear to be roughly constant over time. Therefore, we do not need to difference this series in order to fit an ARIMA model, but can fit an ARIMA model to the original series (the order of differencing required, d, is zero here). We can now plot a correlogram and partial correlogram for lags 1-20 to investigate what ARIMA model to use: We see from the correlogram that the autocorrelations for lags 1, 2 and 3 exceed the significance bounds, and that the autocorrelations tail off to zero after lag 3. The autocorrelations for lags 1, 2, 3 are positive, and decrease in magnitude with increasing lag (lag 1: 0.666, lag 2: 0.374, lag 3: 0.162). The autocorrelation for lags 19 and 20 exceed the significance bounds too, but it is likely that this is due to chance, since they just exceed the significance bounds (especially for lag 19), the autocorrelations for lags 4-18 do not exceed the signifiance bounds, and we would expect 1 in 20 lags to exceed the 95 significance bounds by chance alone. From the partial autocorrelogram, we see that the partial autocorrelation at lag 1 is positive and exceeds the significance bounds (0.666), while the partial autocorrelation at lag 2 is negative and also exceeds the significance bounds (-0.126). The partial autocorrelations tail off to zero after lag 2. Since the correlogram tails off to zero after lag 3, and the partial correlogram is zero after lag 2, the following ARMA models are possible for the time series: an ARMA(2,0) model, since the partial autocorrelogram is zero after lag 2, and the correlogram tails off to zero after lag 3, and the partial correlogram is zero after lag 2 an ARMA(0,3) model, since the autocorrelogram is zero after lag 3, and the partial correlogram tails off to zero (although perhaps too abruptly for this model to be appropriate) an ARMA(p, q) mixed model, since the correlogram and partial correlogram tail off to zero (although the partial correlogram perhaps tails off too abruptly for this model to be appropriate) Shortcut: the auto. arima() function Again, we can use auto. arima() to find an appropriate model, by typing 8220auto. arima(volcanodust)8221, which gives us ARIMA(1,0,2), which has 3 parameters. However, different criteria can be used to select a model (see auto. arima() help page). If we use the 8220bic8221 criterion, which penalises the number of parameters, we get ARIMA(2,0,0), which is ARMA(2,0): 8220auto. arima(volcanodust, ic8221bic8221)8221. The ARMA(2,0) model has 2 parameters, the ARMA(0,3) model has 3 parameters, and the ARMA(p, q) model has at least 2 parameters. Therefore, using the principle of parsimony, the ARMA(2,0) model and ARMA(p, q) model are equally good candidate models. An ARMA(2,0) model is an autoregressive model of order 2, or AR(2) model. This model can be written as: Xt - mu (Beta1 (Xt-1 - mu)) (Beta2 (Xt-2 - mu)) Zt, where Xt is the stationary time series we are studying (the time series of volcanic dust veil index), mu is the mean of time series Xt, Beta1 and Beta2 are parameters to be estimated, and Zt is white noise with mean zero and constant variance. An AR (autoregressive) model is usually used to model a time series which shows longer term dependencies between successive observations. Intuitively, it makes sense that an AR model could be used to describe the time series of volcanic dust veil index, as we would expect volcanic dust and aerosol levels in one year to affect those in much later years, since the dust and aerosols are unlikely to disappear quickly. If an ARMA(2,0) model (with p2, q0) is used to model the time series of volcanic dust veil index, it would mean that an ARIMA(2,0,0) model can be used (with p2, d0, q0, where d is the order of differencing required). Similarly, if an ARMA(p, q) mixed model is used, where p and q are both greater than zero, than an ARIMA(p,0,q) model can be used. Forecasting Using an ARIMA Model Once you have selected the best candidate ARIMA(p, d,q) model for your time series data, you can estimate the parameters of that ARIMA model, and use that as a predictive model for making forecasts for future values of your time series. You can estimate the parameters of an ARIMA(p, d,q) model using the 8220arima()8221 function in R. Example of the Ages at Death of the Kings of England For example, we discussed above that an ARIMA(0,1,1) model seems a plausible model for the ages at deaths of the kings of England. You can specify the values of p, d and q in the ARIMA model by using the 8220order8221 argument of the 8220arima()8221 function in R. To fit an ARIMA(p, d,q) model to this time series (which we stored in the variable 8220kingstimeseries8221, see above), we type: As mentioned above, if we are fitting an ARIMA(0,1,1) model to our time series, it means we are fitting an an ARMA(0,1) model to the time series of first differences. An ARMA(0,1) model can be written Xt - mu Zt - (theta Zt-1), where theta is a parameter to be estimated. From the output of the 8220arima()8221 R function (above), the estimated value of theta (given as 8216ma18217 in the R output) is -0.7218 in the case of the ARIMA(0,1,1) model fitted to the time series of ages at death of kings. Specifying the confidence level for prediction intervals You can specify the confidence level for prediction intervals in forecast. Arima() by using the 8220level8221 argument. For example, to get a 99.5 prediction interval, we would type 8220forecast. Arima(kingstimeseriesarima, h5, levelc(99.5))8221. We can then use the ARIMA model to make forecasts for future values of the time series, using the 8220forecast. Arima()8221 function in the 8220forecast8221 R package. For example, to forecast the ages at death of the next five English kings, we type: The original time series for the English kings includes the ages at death of 42 English kings. The forecast. Arima() function gives us a forecast of the age of death of the next five English kings (kings 43-47), as well as 80 and 95 prediction intervals for those predictions. The age of death of the 42nd English king was 56 years (the last observed value in our time series), and the ARIMA model gives the forecasted age at death of the next five kings as 67.8 years. We can plot the observed ages of death for the first 42 kings, as well as the ages that would be predicted for these 42 kings and for the next 5 kings using our ARIMA(0,1,1) model, by typing: As in the case of exponential smoothing models, it is a good idea to investigate whether the forecast errors of an ARIMA model are normally distributed with mean zero and constant variance, and whether the are correlations between successive forecast errors. For example, we can make a correlogram of the forecast errors for our ARIMA(0,1,1) model for the ages at death of kings, and perform the Ljung-Box test for lags 1-20, by typing: Since the correlogram shows that none of the sample autocorrelations for lags 1-20 exceed the significance bounds, and the p-value for the Ljung-Box test is 0.9, we can conclude that there is very little evidence for non-zero autocorrelations in the forecast errors at lags 1-20. To investigate whether the forecast errors are normally distributed with mean zero and constant variance, we can make a time plot and histogram (with overlaid normal curve) of the forecast errors: The time plot of the in-sample forecast errors shows that the variance of the forecast errors seems to be roughly constant over time (though perhaps there is slightly higher variance for the second half of the time series). The histogram of the time series shows that the forecast errors are roughly normally distributed and the mean seems to be close to zero. Therefore, it is plausible that the forecast errors are normally distributed with mean zero and constant variance. Since successive forecast errors do not seem to be correlated, and the forecast errors seem to be normally distributed with mean zero and constant variance, the ARIMA(0,1,1) does seem to provide an adequate predictive model for the ages at death of English kings. Example of the Volcanic Dust Veil in the Northern Hemisphere We discussed above that an appropriate ARIMA model for the time series of volcanic dust veil index may be an ARIMA(2,0,0) model. To fit an ARIMA(2,0,0) model to this time series, we can type: As mentioned above, an ARIMA(2,0,0) model can be written as: written as: Xt - mu (Beta1 (Xt-1 - mu)) (Beta2 (Xt-2 - mu)) Zt, where Beta1 and Beta2 are parameters to be estimated. The output of the arima() function tells us that Beta1 and Beta2 are estimated as 0.7533 and -0.1268 here (given as ar1 and ar2 in the output of arima()). Now we have fitted the ARIMA(2,0,0) model, we can use the 8220forecast. ARIMA()8221 model to predict future values of the volcanic dust veil index. The original data includes the years 1500-1969. To make predictions for the years 1970-2000 (31 more years), we type: We can plot the original time series, and the forecasted values, by typing: One worrying thing is that the model has predicted negative values for the volcanic dust veil index, but this variable can only have positive values The reason is that the arima() and forecast. Arima() functions don8217t know that the variable can only take positive values. Clearly, this is not a very desirable feature of our current predictive model. Again, we should investigate whether the forecast errors seem to be correlated, and whether they are normally distributed with mean zero and constant variance. To check for correlations between successive forecast errors, we can make a correlogram and use the Ljung-Box test: The correlogram shows that the sample autocorrelation at lag 20 exceeds the significance bounds. However, this is probably due to chance, since we would expect one out of 20 sample autocorrelations to exceed the 95 significance bounds. Furthermore, the p-value for the Ljung-Box test is 0.2, indicating that there is little evidence for non-zero autocorrelations in the forecast errors for lags 1-20. To check whether the forecast errors are normally distributed with mean zero and constant variance, we make a time plot of the forecast errors, and a histogram: The time plot of forecast errors shows that the forecast errors seem to have roughly constant variance over time. However, the time series of forecast errors seems to have a negative mean, rather than a zero mean. We can confirm this by calculating the mean forecast error, which turns out to be about -0.22: The histogram of forecast errors (above) shows that although the mean value of the forecast errors is negative, the distribution of forecast errors is skewed to the right compared to a normal curve. Therefore, it seems that we cannot comfortably conclude that the forecast errors are normally distributed with mean zero and constant variance Thus, it is likely that our ARIMA(2,0,0) model for the time series of volcanic dust veil index is not the best model that we could make, and could almost definitely be improved upon Links and Further Reading Here are some links for further reading. For a more in-depth introduction to R, a good online tutorial is available on the 8220Kickstarting R8221 website, cran. r-project. orgdoccontribLemon-kickstart . There is another nice (slightly more in-depth) tutorial to R available on the 8220Introduction to R8221 website, cran. r-project. orgdocmanualsR-intro. html . You can find a list of R packages for analysing time series data on the CRAN Time Series Task View webpage . To learn about time series analysis, I would highly recommend the book 8220Time series8221 (product code M24902) by the Open University, available from the Open University Shop . There are two books available in the 8220Use R8221 series on using R for time series analyses, the first is Introductory Time Series with R by Cowpertwait and Metcalfe, and the second is Analysis of Integrated and Cointegrated Time Series with R by Pfaff. Acknowledgements I am grateful to Professor Rob Hyndman. for kindly allowing me to use the time series data sets from his Time Series Data Library (TSDL) in the examples in this booklet. Many of the examples in this booklet are inspired by examples in the excellent Open University book, 8220Time series8221 (product code M24902), available from the Open University Shop . Thank you to Ravi Aranke for bringing auto. arima() to my attention, and Maurice Omane-Adjepong for bringing unit root tests to my attention, and Christian Seubert for noticing a small bug in plotForecastErrors(). Thank you for other comments to Antoine Binard and Bill Johnston. I will be grateful if you will send me (Avril Coghlan) corrections or suggestions for improvements to my email address alc 64 sanger 46 ac 46 uk