Base R Schiffe mit viel Funktionalität nützlich für Zeitreihen, insbesondere im Stats-Paket. Dies wird durch viele Pakete auf CRAN ergänzt, die im Folgenden kurz zusammengefasst sind. Es gibt auch eine beträchtliche Überschneidung zwischen den Werkzeugen für Zeitreihen und denen in der Ökonometrie und Finanzen Aufgabe Ansichten. Die Pakete in dieser Ansicht können grob in folgende Themen eingeteilt werden. Wenn Sie denken, dass etwas Paket aus der Liste fehlt, lassen Sie es uns bitte wissen. Infrastruktur. Base R enthält eine wesentliche Infrastruktur zur Darstellung und Analyse von Zeitreihendaten. Die grundlegende Klasse ist quottsquot, die regelmäßig beabstandete Zeitreihen darstellen kann (mit numerischen Zeitstempeln). Daher eignet es sich besonders gut für jährliche, monatliche, vierteljährliche Daten etc. Rolling Statistics. Umzugsdurchschnitte werden von der Prognose berechnet. Und Rollmean aus dem Zoo. Letztere bietet auch eine allgemeine Funktion. Zusammen mit anderen spezifischen Rolling Statistics Funktionen. Roll bietet parallele Funktionen für die Berechnung von Rollstatistiken. Grafiken Zeitreihenplots werden mit Plot (), die auf ts-Objekte angewendet werden, erhalten. (Partielle) Autokorrelationsfunktionen werden in acf () und pacf () implementiert. Alternative Versionen werden von Acf () und Pacf () in der Prognose zur Verfügung gestellt. Zusammen mit einer Kombinationsanzeige mit tsdisplay (). SDD liefert allgemeinere serielle Abhängigkeitsdiagramme, während dCovTS die Distanzkovarianz - und Korrelationsfunktionen der Zeitreihen berechnet und abbildet. Saisonale Displays werden mit monthplot () in Stats und Saisonplot in Prognose erhalten. Wats implementiert Wrap-around-Zeitreihen-Grafik. Ggseas liefert ggplot2-Grafiken für saisonbereinigte Serien - und Rollstatistiken. Dygraphs bietet eine Schnittstelle zu den Dygraphs interaktive Zeitreihen-Charting-Bibliothek. ZRA-Plots prognostizieren Objekte aus dem Prognosepaket mit Dygraphen. Grundlegende Fächerplots von Prognoseverteilungen werden durch Prognose und Vars bereitgestellt. Mehr flexibler Fan-Plots von beliebigen sequentiellen Verteilungen sind in Fanplot implementiert. Klasse quottsquot kann nur mit numerischen Zeitstempeln umgehen, aber viele weitere Klassen sind verfügbar für die Speicherung von zeitlichen Informationen und Computing mit ihm. Für eine Übersicht siehe R Help Desk: Datum und Uhrzeit Klassen in R von Gabor Grothendieck und Thomas Petzoldt in R News 4 (1). 29-32 Klassen quotainarmonquot und quotyearqtrquot aus dem Zoo erlauben eine bequemere Berechnung mit monatlichen und vierteljährlichen Beobachtungen. Klasse quotDatequot aus dem Basispaket ist die Grundklasse für den Umgang mit Daten in täglichen Daten. Die Termine sind intern als die Anzahl der Tage seit 1970-01-01 gespeichert. Das chron-Paket bietet Klassen für Termine (). Stunden () und datetime (intra-day) in chron (). Es gibt keine Unterstützung für Zeitzonen und Sommerzeit. Innerlich sind quotContracot-Objekte (fraktional) Tage seit 1970-01-01. Klassen quotPOSIXctquot und quotPOSIXltquot implementieren den POSIX-Standard für datetime (Intra-Day) Informationen und unterstützen auch Zeitzonen und Sommerzeit. Allerdings erfordern die Zeitzonenberechnungen etwas Sorgfalt und können systemabhängig sein. Intern sind quotPOSIXctquot Objekte die Anzahl der Sekunden seit 1970-01-01 00:00:00 GMT. Paket lubridate bietet Funktionen, die bestimmte POSIX-basierte Berechnungen erleichtern. Klasse quottimeDatequot wird im timeDate-Paket (vorher: fCalendar) zur Verfügung gestellt. Es richtet sich an finanzielle zeitliche Informationen und befasst sich mit Zeitzonen und Sommerzeiten über ein neues Konzept der quotalen Finanzplätze. Intern speichert es alle Informationen in quotPOSIXctquot und tut alle Berechnungen nur in GMT. Kalender-Funktionalität, z. B. Einschließlich Informationen über Wochenenden und Feiertage für verschiedene Börsen, ist auch enthalten. Das tis-Paket bietet die quottiquot-Klasse für zeitliche Informationen. Die quotmondatequot-Klasse aus dem Mondate-Paket erleichtert das Rechnen mit Daten in Monaten. Das Tempdisagg-Paket beinhaltet Methoden zur zeitlichen Disaggregation und Interpolation einer niederfrequenten Zeitreihe zu einer höheren Frequenzreihe. Zeitreihen-Disaggregation wird auch von tsdisagg2 zur Verfügung gestellt. TimeProjection extrahiert nützliche Zeitkomponenten eines Datumsobjekts, wie zB Wochentag, Wochenende, Urlaub, Tag des Monats usw., und legen Sie es in einen Datenrahmen. Wie oben erwähnt, ist quottsquot die Grundklasse für regelmäßig beabstandete Zeitreihen mit numerischen Zeitstempeln. Das Zoo-Paket bietet Infrastruktur für regelmäßig und unregelmäßig beabstandete Zeitreihen mit beliebigen Klassen für die Zeitstempel (d. h. alle Klassen aus dem vorherigen Abschnitt). Es ist so konsequent wie möglich mit quottsquot. Zwang von und zu quotzooquot ist für alle anderen Klassen, die in diesem Abschnitt erwähnt werden. Das Paket xts basiert auf dem Zoo und bietet eine einheitliche Handhabung von Rs verschiedenen zeitbasierten Datenklassen. Verschiedene Pakete implementieren unregelmäßige Zeitreihen auf der Basis von quotPOSIXctquot Zeitstempeln, die speziell für Finanzanwendungen gedacht sind. Dazu gehören quotirtsquot aus tseries. Und quittiert aus fts. Die Klasse quottimeSeriesquot in timeSeries (früher: fSeries) implementiert Zeitreihen mit quottimeDatequot Zeitstempeln. Die Klasse quottisquot in tis implementiert Zeitreihen mit quottiquot Zeitstempeln. Das Paket tframe enthält die Infrastruktur für die Festlegung von Zeitrahmen in verschiedenen Formaten. Prognose und Univariate Modellierung Das Prognosepaket bietet eine Klasse und Methoden für univariate Zeitreihenprognosen und bietet viele Funktionen, die verschiedene Prognosemodelle implementieren, einschließlich aller im Stats-Paket. Exponentielle Glättung . HoltWinters () in Stats bietet einige Basismodelle mit partieller Optimierung, ets () aus dem Prognosepaket bietet einen größeren Satz von Modellen und Einrichtungen mit voller Optimierung. Robets bietet eine robuste Alternative zur ets () - Funktion. Glatt implementiert einige Verallgemeinerungen der exponentiellen Glättung. Das MAPA-Paket kombiniert exponentielle Glättungsmodelle auf verschiedenen Ebenen der zeitlichen Aggregation, um die Prognosegenauigkeit zu verbessern. Die theta-Methode wird in der thetaf-Funktion aus dem Prognosepaket implementiert. Eine alternative und erweiterte Implementierung ist in der Prognose vorgesehen. Autoregressive Modelle. Ar () in stats (mit Modellauswahl) und FitAR für Subset AR Modelle. ARIMA Modelle. Arima () in stats ist die Grundfunktion für ARIMA, SARIMA, ARIMAX und Subset ARIMA Modelle. Es wird im Prognosepaket über die Funktion Arima () zusammen mit auto. arima () zur automatischen Auftragsauswahl erweitert. Arma () im tseries-Paket bietet verschiedene Algorithmen für ARMA - und Subset-ARMA-Modelle. FitARMA implementiert einen schnellen MLE-Algorithmus für ARMA-Modelle. Paket gsarima enthält Funktionalität für generalisierte SARIMA Zeitreihensimulation. Das mar1s Paket verarbeitet multiplikative AR (1) mit saisonalen Prozessen. TSTutorial bietet ein interaktives Tutorial für Box-Jenkins Modellierung. Verbesserte Vorhersageintervalle für ARIMA und strukturelle Zeitreihenmodelle werden von tsPI bereitgestellt. Periodische ARMA-Modelle. Birne und Teile für periodische autoregressive Zeitreihenmodelle und perARMA für periodische ARMA-Modellierung und andere Verfahren zur periodischen Zeitreihenanalyse. ARFIMA Modelle. Einige Einrichtungen für fraktionierte differenzierte ARFIMA-Modelle sind im Fracdiff-Paket enthalten. Das Arfima-Paket verfügt über erweiterte und allgemeine Einrichtungen für ARFIMA - und ARIMA-Modelle, einschließlich dynamischer Regressions - (Übertragungsfunktion) - Modelle. ArmaFit () aus dem fArma-Paket ist eine Schnittstelle für ARIMA - und ARFIMA-Modelle. Fraktionale Gaußsche Geräusche und einfache Modelle für hyperbolische Zerfallszeitreihen werden im FGN-Paket abgewickelt. Transfer-Funktionsmodelle werden von der Arimax-Funktion im TSA-Paket und der arfima-Funktion im arfima-Paket zur Verfügung gestellt. Ausreißer-Erkennung nach dem Chen-Liu-Ansatz wird von tsoutliers zur Verfügung gestellt. Strukturelle Modelle werden in StructTS () in Stats und in stsm und stsm. class implementiert. KFKSDS bietet eine naive Implementierung des Kalman-Filters und Glättchens für univariate State Space Modelle. Bayesische strukturelle Zeitreihenmodelle werden in bsts implementiert. Nicht-Gaußsche Zeitreihen können mit GLARMA-Zustandsraummodellen über Glarma behandelt werden. Und mit Generalized Autoregressive Score-Modelle im GAS-Paket. Bedingte Auto-Regressionsmodelle mit Monte Carlo Likelihood Methoden werden in mclcar implementiert. GARCH Modelle. Garch () von tseries passt zu grundlegenden GARCH-Modellen. Viele Variationen zu GARCH-Modellen werden von Rugarch angeboten. Andere univariate GARCH-Pakete beinhalten fGarch, die ARIMA-Modelle mit einer breiten Klasse von GARCH-Innovationen implementiert. Es gibt viele weitere GARCH-Pakete, die in der Finanzaufgabenansicht beschrieben wurden. Stochastische Volatilitätsmodelle werden von stochvol in einem Bayes'schen Rahmen behandelt. Count-Zeitreihenmodelle werden in den tscount - und acp-Paketen behandelt. ZIM bietet Zero-Inflated Models für Zählzeitreihen an. Tsintermittierende implementiert verschiedene Modelle für die Analyse und Prognose der zeitweiligen Bedarfszeitreihen. Zensierte Zeitreihen können mit Cents und Carx modelliert werden. Portmanteau-Tests werden über Box. test () im Statistik-Paket zur Verfügung gestellt. Zusätzliche Tests werden von Portes und WeightedPortTest gegeben. Die Änderung der Punktdetektion erfolgt im Aufbau (mit linearen Regressionsmodellen), im Trend (mit nichtparametrischen Tests) und in wbsts (mit wilder binärer Segmentierung). Das Changepoint-Paket bietet viele populäre Changepoint-Methoden, und ecp macht nichtparametrische Changepoint-Erkennung für univariate und multivariate Serien. Online-Änderungspunkt-Erkennung für univariate und multivariate Zeitreihen wird von onlineCPD bereitgestellt. InspectChangepoint verwendet spärliche Projektionen, um Änderungspunkte in hochdimensionalen Zeitreihen abzuschätzen. Die Zeitreihen-Imputation wird durch das paketische Paket bereitgestellt. Einige begrenzte Einrichtungen sind mit na. interp () aus dem Prognosepaket verfügbar. Prognosen können mit ForecastCombinations kombiniert werden, die die am häufigsten verwendeten Methoden unterstützen, um Prognosen zu kombinieren. ForecastHybrid bietet Funktionen für Ensemble-Prognosen und kombiniert Ansätze aus dem Prognosepaket. GeomComb bietet Eigenvektor-basierte (geometrische) Prognose Kombination Methoden, sowie andere Ansätze. Oper hat Einrichtungen für Online-Vorhersagen auf der Grundlage von Kombinationen von Prognosen, die vom Benutzer zur Verfügung gestellt werden. Die Prognoseauswertung erfolgt in der Genauigkeit () - Funktion aus der Prognose. Verteilungsprognose Auswertung mit Scoring-Regeln ist verfügbar in ScoringRules Verschiedenes. Ltsa enthält Methoden für die lineare Zeitreihenanalyse, Timsac für die Zeitreihenanalyse und - steuerung sowie taffugs für Zeitreihen-BUGS-Modelle. Die spektrale Dichteabschätzung wird durch das Spektrum () im Statistikpaket, einschließlich des Periodogramms, des geglätteten Periodogramms und der AR-Schätzungen, bereitgestellt. Bayesische Spektralfolgerung wird von bspec zur Verfügung gestellt. Quantspec enthält Methoden zur Berechnung und Darstellung von Laplace-Periodogrammen für univariate Zeitreihen. Das Lomb-Scargle-Periodogramm für ungleichmäßig abgetastete Zeitreihen wird von Lombe berechnet. Spektral verwendet Fourier - und Hilbert-Transformationen für die spektrale Filterung. Psd produziert adaptive, Sinus-Multitaper-Spektraldichte-Schätzungen. Kza liefert Kolmogorov-Zurbenko Adaptive Filter einschließlich Break Detection, Spektralanalyse, Wavelets und KZ Fourier Transforms. Multitaper bietet auch einige Multitaper Spektralanalyse-Tools. Wavelet-Methoden. Das Wavelets-Paket umfasst die Berechnung von Wavelet-Filtern, Wavelet-Transformationen und Multiresolution-Analysen. Wavelet-Methoden für die Zeitreihenanalyse auf Basis von Percival und Walden (2000) sind in wmtsa angegeben. WaveletComp bietet einige Werkzeuge für die Wavelet-basierte Analyse von univariaten und bivariaten Zeitreihen, einschließlich Cross-Wavelets, Phasendifferenz und Signifikanztests. Biwavelet kann verwendet werden, um die Wavelet-Spektren, die Cross-Wavelet-Spektren und die Wavelet-Kohärenz von nicht-stationären Zeitreihen zu zeichnen und zu berechnen. Es enthält auch Funktionen zur Cluster-Zeitreihe auf der Grundlage der (dis) Ähnlichkeiten in ihrem Spektrum. Tests von weißem Rauschen mit Wavelets werden von hwwntest zur Verfügung gestellt. Weitere Wavelet-Methoden finden Sie im Paket Brainwaver. Rwt Waveslim Wavethresh und mvcwt. Eine harmonische Regression mit Fourier-Terme wird in HarmonicRegression implementiert. Das Prognosepaket bietet auch einige einfache harmonische Regressionseinrichtungen über die Fourier-Funktion. Zersetzung und Filterung von Filtern und Glättung. Filter () in stats bietet autoregressive und gleitende durchschnittliche lineare Filterung von mehreren univariate Zeitreihen. Das robfilter-Paket bietet mehrere robuste Zeitreihenfilter, während mFilter verschiedene Zeitreihenfilter enthält, die zum Glätten und Extrahieren von Trend - und zyklischen Komponenten geeignet sind. Glatt () aus dem Stats-Paket berechnet Tukeys mit medianen Glätten, 3RS3R, 3RSS, 3R, etc. Sleekts berechnet die 4253H zweimal Glättung Methode. Zersetzung . Die saisonale Zersetzung wird unten diskutiert. Autoregressiv-basierte Zersetzung wird von ArDec zur Verfügung gestellt. Tsdecomp implementiert ARIMA-basierte Zerlegung von vierteljährlichen und monatlichen Daten. Rmaf verwendet einen raffinierten gleitenden Mittelfilter zur Zersetzung. Singular Spectrum Analysis ist in Rssa und Spektralmethoden implementiert. Empirische Moduszerlegung (EMD) und Hilbert-Spektralanalyse wird von EMD bereitgestellt. Zusätzliche Werkzeuge, einschließlich Ensemble EMD, sind in hht erhältlich. Eine alternative Implementierung von Ensemble EMD und seiner kompletten Variante gibt es in Rlibeemd. Saisonale Zersetzung. Das Stats-Paket bietet klassische Zerlegung in Zerlegung (). Und STL-Zerlegung in stl (). Eine verbesserte STL-Zerlegung ist in stlplus verfügbar. StR bietet saisonale-Trend-Zersetzung auf der Grundlage von Regression. X12 bietet einen Wrapper für die X12-Binärdateien, die zuerst installiert werden müssen. X12GUI bietet eine grafische Benutzeroberfläche für x12. X-13-ARIMA-SEATS-Binärdateien werden im x13binary-Paket zur Verfügung gestellt, mit saisonalen Bereitstellung einer R-Schnittstelle und saisonalen Ansicht, die eine GUI bereitstellt. Analyse der Saisonalität Das Bfast-Paket bietet Methoden zur Erkennung und Charakterisierung von abrupten Veränderungen innerhalb des Trends und der saisonalen Komponenten, die aus einer Zersetzung gewonnen werden. Npst bietet eine Verallgemeinerung von Hewitts Saisonalitätstest. Jahreszeit. Saisonale Analyse der Gesundheitsdaten einschließlich Regressionsmodelle, zeitlich geschichtete Fallübergang, Plottenfunktionen und Restkontrollen. Meere Saisonanalyse und Grafik, insbesondere für Klimatologie. Entschärfen Optimale Entkopplung für geophysikalische Zeitreihen mit AR-Armatur. Stationarity, Unit Roots und Cointegration Stationarity und Einheit Wurzeln. Tseries bietet verschiedene stationäre und Unit-Root-Tests, darunter Augmented Dickey-Fuller, Phillips-Perron und KPSS. Alternative Implementierungen der ADF - und KPSS-Tests sind im urca-Paket, das auch weitere Methoden wie Elliott-Rothenberg-Stock, Schmidt-Phillips und Zivot-Andrews-Tests beinhaltet. Das Paket fUnitRoots bietet auch den MacKinnon-Test, während uroot saisonale Wurzeltests liefert. CADFtest bietet Implementierungen sowohl des Standard ADF als auch eines Covariate-Augmented ADF (CADF) Tests. Lokale Stationarität. Locits stellt einen Test der lokalen Stationarität zur Verfügung und berechnet die lokalisierte Autokovarianz. Zeitreihen-Kalkulationstarifbestimmung wird durch Costat zur Verfügung gestellt. LSTS hat Funktionen für lokal stationäre Zeitreihenanalyse. Lokal stationäre Wavelet-Modelle für nichtstationäre Zeitreihen werden im Wavethresh implementiert (einschließlich Schätzung, Plotten und Simulationsfunktionalität für zeitvariable Spektren). Kointegration Die Engle-Granger-Zwei-Schritt-Methode mit dem Phillips-Ouliaris-Kointegrationstest ist in Tseries und Urca implementiert. Letzteres enthält zusätzlich Funktionalität für die Johansen-Trace - und Lambda-Max-Tests. TsDyn bietet Johansens-Test und AICBIC gleichzeitige Rang-Lag-Auswahl. CommonTrend bietet Werkzeuge, um gemeinsame Trends aus einem Kointegrationssystem zu extrahieren und zu plotten. Parameterschätzung und Schlussfolgerung in einer kointegrierenden Regression werden in cointReg implementiert. Nichtlineare Zeitreihenanalyse Nichtlineare Autoregression. Verschiedene Formen der nichtlinearen Autoregression sind in tsDyn einschließlich additive AR, neuronale Netze, SETAR und LSTAR Modelle, Schwelle VAR und VECM verfügbar. Neuronale Netzwerk-Autoregression ist auch in GMDH zur Verfügung gestellt. BentcableAR implementiert Bent-Cable Autoregression. BAYSTAR bietet eine Bayes-Analyse von Schwellen-autoregressiven Modellen. TseriesChaos stellt eine R-Implementierung der Algorithmen aus dem TISEAN-Projekt zur Verfügung. Autoregression Markov Schaltmodelle sind in MSwM zur Verfügung gestellt. Während abhängige Gemische von latenten Markov-Modellen in depmix und depmixS4 für kategorische und kontinuierliche Zeitreihen gegeben sind. Tests. Verschiedene Tests für Nichtlinearität sind in fNonlinear vorgesehen. TseriesEntropietests für nichtlineare serielle Abhängigkeit auf der Basis von Entropie-Metriken. Zusätzliche Funktionen für nichtlineare Zeitreihen sind in nlts und nonlinearTseries verfügbar. Fraktale Zeitreihenmodellierung und - analyse wird durch Fraktal bereitgestellt. Fraktalrock erzeugt fraktale Zeitreihen mit nicht-normalen Renditenverteilungen. Dynamische Regressionsmodelle Dynamische Linearmodelle. Eine komfortable Schnittstelle zur Anpassung von dynamischen Regressionsmodellen über OLS ist in dynlm ein erweiterter Ansatz, der auch mit anderen Regressionsfunktionen arbeitet und mehr Zeitreihenklassen in dyn implementiert sind. Fortgeschrittene dynamische Systemgleichungen können mit dse eingebaut werden. Gaußsche lineare Zustandsraummodelle können mit dlm (über Maximum Likelihood, Kalman Filteringsmoothing und Bayesian Methoden) oder mit bsts verwendet werden, die MCMC verwenden. Funktionen für verteilte Verzögerung nichtlineare Modellierung sind in dlnm zur Verfügung gestellt. Mit dem tpr-Paket können zeitvariable Parametermodelle eingebaut werden. OrderedLasso passt zu einem spärlichen linearen Modell mit einer Auftragsbeschränkung auf die Koeffizienten, um verzögerte Regressoren zu behandeln, bei denen die Koeffizienten abfallen, wenn die Verzögerung zunimmt. Die dynamische Modellierung von verschiedenen Arten ist in dynr einschließlich diskreter und kontinuierlicher Zeit, lineare und nichtlineare Modelle und verschiedene Arten von latenten Variablen verfügbar. Multivariate Zeitreihenmodelle Vector autoregressive (VAR) Modelle werden über ar () im Basisstatistikpaket inklusive Auftragsauswahl über die AIC zur Verfügung gestellt. Diese Modelle sind beschränkt auf stationär. MTS ist ein Allzweck-Toolkit zur Analyse multivariater Zeitreihen wie VAR, VARMA, saisonale VARMA, VAR Modelle mit exogenen Variablen, multivariate Regression mit Zeitreihenfehlern und vieles mehr. Möglicherweise sind nicht-stationäre VAR-Modelle im mAr-Gehäuse eingebaut, was auch VAR-Modelle im Hauptkomponentenraum ermöglicht. Sparsevar erlaubt die Schätzung von sparsamen VAR - und VECM-Modellen, ecm bietet Funktionen für den Aufbau von VECM-Modellen, während BigVAR VAR - und VARX-Modelle mit strukturierten Lasso-Strafen schätzt. Automatisierte VAR-Modelle und Netzwerke sind in autovarCore verfügbar. Mehr aufwändige Modelle werden im Paket Vars angeboten. TsDyn EstVARXls () in dse. Und ein Bayes'scher Ansatz ist in MSBVAR verfügbar. Eine weitere Implementierung mit bootstrackten Vorhersageintervallen ist in VAR. etp angegeben. MlVAR bietet mehrstufige Vektor-Autoregression. VARsignR bietet Routinen zur Identifizierung von strukturellen Schocks in VAR-Modellen mit Zeichenbeschränkungen. Gdpc implementiert verallgemeinerte dynamische Hauptkomponenten. Pcdpca erweitert dynamische Hauptkomponenten auf periodisch korrelierte multivariate Zeitreihen. VARIMA Modelle und Zustandsraummodelle werden im dse Paket zur Verfügung gestellt. EvalEst erleichtert Monte Carlo-Experimente, um die damit verbundenen Schätzmethoden zu bewerten. Vektorfehlerkorrekturmodelle sind über die urca verfügbar. Vars und tsDyn-Pakete, einschließlich Versionen mit strukturellen Einschränkungen und Schwellenwert. Zeitreihenanalyse. Zeitreihenfaktoranalyse wird in tsfa zur Verfügung gestellt. ForeCA implementiert eine prognostizierte Komponentenanalyse, indem sie nach den besten linearen Transformationen sucht, die eine multivariate Zeitreihe wie möglich prognostiziert machen. PCA4TS findet eine lineare Transformation einer multivariaten Zeitreihe, die niedrigdimensionale Subsererien liefert, die nicht korreliert sind. Multivariate Zustandsraummodelle werden im FKF (Fast Kalman Filter) Paket implementiert. Dies bietet relativ flexible Zustandsraummodelle über die Funktion fkf (): Zustandsraumparameter können zeitveränderlich sein und in beiden Gleichungen sind Abschnitte enthalten. Eine alternative Implementierung wird durch das KFAS-Paket zur Verfügung gestellt, das einen schnellen multivariaten Kalman-Filter, eine glattere, Simulation glatter und prognostizieren bietet. Eine weitere Implementierung erfolgt im dlm-Paket, das auch Werkzeuge zur Umwandlung anderer multivariater Modelle in die staatliche Raumform enthält. Dlmodeler stellt eine einheitliche Schnittstelle für dlm zur Verfügung. KFAS und FKF. MARSS passt auf eingeschränkte und unbeschränkte multivariate autoregressive State-Space-Modelle mit einem EM-Algorithmus. Alle diese Pakete nehmen an, dass die Beobachtungs - und Zustandsfehlerbegriffe unkorreliert sind. Teilweise beobachtete Markov-Prozesse sind eine Verallgemeinerung der üblichen linearen multivariaten Zustandsraummodelle, die nicht-Gaußsche und nichtlineare Modelle erlauben. Diese werden im Pomp-Paket implementiert. Multivariate stochastische Volatilitätsmodelle (mit latenten Faktoren) werden von factorstochvol bereitgestellt. Analyse von großen Gruppen von Zeitreihen Die Zeitreihen-Clustering ist in TSclust implementiert. Dwclust BNPTSclust und pdc. TSdist liefert Distanzmessungen für Zeitreihendaten. Jmotif implementiert Werkzeuge basierend auf Zeitreihen symbolische Diskretisierung für das Finden von Motiven in Zeitreihen und erleichtert die interpretierbare Zeitreihenklassifizierung. Rucrdtw bietet R-Bindungen für Funktionen aus der UCR Suite, um ultraschnelle Subsequenz-Suche nach einem bestes Match unter Dynamic Time Warping und Euklidischer Distanz zu ermöglichen. Methoden zur Plotten - und Prognose von Sammlungen von hierarchischen und gruppierten Zeitreihen werden von hts bereitgestellt. Diebstahl verwendet hierarchische Methoden, um Prognosen zeitlich aggregierter Zeitreihen in Einklang zu bringen. Ein alternativer Ansatz zur Abstimmung von Prognosen hierarchischer Zeitreihen wird von gtop bereitgestellt. Dieb Ununterbrochene Zeitmodelle Ununterbrochene Zeit autoregressive Modellierung ist in cts. Sim. DiffProc simuliert und modelliert stochastische Differentialgleichungen. Simulation und Schlußfolgerung für stochastische Differentialgleichungen liefert sde und yuima. Bootstrapping. Das Boot-Paket bietet die Funktion tsboot () für das Zeitreihen-Bootstrapping, einschließlich Block-Bootstrap mit mehreren Varianten. Tsbootstrap () von tseries bietet schnell stationär und block bootstrapping. Maximaler Entropie-Bootstrap für Zeitreihen ist in meboot verfügbar. Timesboot berechnet das Bootstrap CI für das Sample ACF und das Periodogramm. BootPR berechnet bias-korrigierte Prognose - und Boostrap-Vorhersageintervalle für autoregressive Zeitreihen. Daten von Makridakis, Wheelwright und Hyndman (1998) Prognose: Methoden und Anwendungen werden im Fma-Paket zur Verfügung gestellt. Daten von Hyndman, Köhler, Ord und Snyder (2008) Prognose mit exponentieller Glättung sind im expsmooth Paket. Daten von Hyndman und Athanasopoulos (2013) Prognose: Grundsätze und Praxis sind im Paket fpp. Daten aus dem M-Wettbewerb und M3-Wettbewerb werden im Mcomp-Paket zur Verfügung gestellt. Daten aus dem M4-Wettbewerb sind in M4comp angegeben. Während Tcomp Daten aus dem 2010 IJF Tourism Forecasting Wettbewerb zur Verfügung stellt. Pdfetch bietet Einrichtungen zum Herunterladen von wirtschaftlichen und finanziellen Zeitreihen aus öffentlichen Quellen. Daten aus dem Quandl-Online-Portal zu finanziellen, wirtschaftlichen und sozialen Datensätzen können interaktiv mit dem Quandl-Paket abgefragt werden. Daten aus dem Datamarket Online-Portal können mit dem Paket rdatamarket abgerufen werden. BETS bietet Zugang zu den wichtigsten Wirtschaftszeitreihen in Brasilien. Daten von Cryer und Chan (2010) sind im TSA Paket. Daten von Shumway und Stoffer (2011) sind im astsa Paket. Daten von Tsay (2005) Die Analyse der finanziellen Zeitreihen ist im FinTS-Paket, zusammen mit einigen Funktionen und Skript-Dateien erforderlich, um einige der Beispiele zu arbeiten. Tswge begleitet den Text Angewandte Zeitreihenanalyse mit R. 2. Auflage von Woodward, Grey und Elliott. TSdbi bietet eine gemeinsame Schnittstelle zu Zeitreihen-Datenbanken. Ruhm bietet eine Schnittstelle für FAME Zeitreihen-Datenbanken AER und Ecdat beide enthalten viele Datensätze (einschließlich Zeitreihen-Daten) aus vielen ökonometrischen Textbüchern dtw. Dynamische Zeitverzerrungsalgorithmen zum Rechnen und Plotten von paarweisen Ausrichtungen zwischen Zeitreihen. EnsembleBMA Bayesian Modell Mittelwertbildung, um probabilistische Prognosen aus Ensembleprognosen und Wetterbeobachtungen zu schaffen. Frühwarnungen Frühwarnungen signalisieren Toolbox zur Erkennung von kritischen Übergängen in Zeitreihenereignissen. Verwandelt maschinell extrahierte Ereignisdaten in reguläre aggregierte multivariate Zeitreihen. Rückmeldungen. Analyse der fragmentierten Zeitrichtlinie zur Untersuchung der Rückmeldung in Zeitreihen LPStimeSeries zielt darauf ab, quittiertes Muster Ähnlichkeit für Zeitreihen zu finden. MAR1 bietet Werkzeuge für die Vorbereitung ökologischer Gemeinschafts-Zeitreihendaten für die multivariate AR-Modellierung. Nets Routinen zur Schätzung von spärlichen Langzeit-Teilkorrelationsnetzen für Zeitreihendaten. PaleoTS Modellierung der Evolution in paläontologischen Zeitreihen. Pastecs Regulierung, Zersetzung und Analyse von Raum-Zeit-Reihen. Pw Parametrische Zeitverzerrung. RGENERATE bietet Werkzeuge zur Erzeugung von Vektor-Zeitreihen. RMAWGEN ist von S3- und S4-Funktionen für die räumliche, mehrstufige stochastische Erzeugung von Tageszeitreihen von Temperatur und Niederschlag unter Verwendung von VAR-Modellen. Das Paket kann in der Klimatologie und der statistischen Hydrologie verwendet werden. RSEIS Seismische Zeitreihenanalyse-Werkzeuge. Rts Raster-Zeitreihenanalyse (z. B. Zeitreihen von Satellitenbildern). Sae2 Zeitreihenmodelle für kleine Flächenschätzungen. SpTimer Räumlich-zeitliche Bayessche Modellierung. Überwachung. Zeitliche und räumlich-zeitliche Modellierung und Überwachung von Epidemie-Phänomenen. TED Turbulenz Zeitreihe Event Detection und Klassifizierung. Gezeiten. Funktionen zur Berechnung von Merkmalen von quasi periodischen Zeitreihen, z. B. Beobachtete Müllwasserspiegel. Tiger. Es werden zeitlich aufgelöste Gruppen von typischen Unterschieden (Fehler) zwischen zwei Zeitreihen ermittelt und visualisiert. TSMining Minivieren von univariaten und multivariaten Motiven in Zeitreihen-Daten. TsModel Zeitreihenmodellierung für Luftverschmutzung und Gesundheit. CRAN-Pakete: Related Links: gt mav (c (4,5,4,6), 3) Zeitreihe: Start 1 Ende 4 Häufigkeit 1 1 NA 4.333333 5.000000 NA Hier habe ich versucht, einen rollenden Durchschnitt zu machen, der das berücksichtigt hat Die letzten 3 Zahlen, so dass ich erwartete, nur zwei Zahlen zurück zu bekommen 8211 4.333333 und 5 8211 und wenn es NA-Werte geben würde, dachte ich, sie sind am Anfang der Sequenz. In der Tat stellt sich heraus, das ist, was die 8216sides8217 Parameter steuert: Seiten für Faltungsfilter nur. Wenn die Seiten 1 die Filterkoeffizienten nur dann für vergangene Werte sind, wenn die Seiten 2 um die Verzögerung 0 zentriert sind. In diesem Fall sollte die Länge des Filters ungerade sein, aber wenn es sogar ist, ist der Filter immer rechtzeitig vorwärts rückwärts. So in unserem 8216mav8217 Funktion der rollende Durchschnitt sieht beide Seiten des aktuellen Wertes anstatt nur bei vergangenen Werten. Wir können das anpassen, um das Verhalten zu bekommen, das wir wollen: gt Bibliothek (Zoo) gt rollmean (c (4,5,4,6), 3) 1 4.333333 5.000000 Ich habe auch erkannt, dass ich alle Funktionen in einem Paket mit dem 8216ls8217 auflisten kann Funktion so I8217ll scannen zoo8217s Liste der Funktionen beim nächsten Mal muss ich etwas Zeitreihe verwandten 8211 there8217ll wahrscheinlich schon eine Funktion für sie gt ls (quotpackage: zooquot) 1 Quoten. Datequot Quoten. Date. numericquot Quoten. Date. tsquot 4 quotas. Date. yearmonquot quotas. Date. yearqtrquot quotas. yearmonquot 7 quotas. yearmon. defaultquot quotas. yearqtrquot quotas. yearqtr. defaultquot 10 quotas. 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 quotMATCHquot 31 quotMATCH. defaultquot quotMATCH. timesquot Quotmedian. zooquot 34 quote. 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. plot. 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. segments. 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 Be Sociable, Serie ShareUsing R für Zeitreihenanalyse Zeit Analyse Diese Broschüre zeigt Ihnen, wie Sie die R-Statistik-Software verwenden, um einige einfache Analysen durchzuführen, die bei der Analyse von Zeitreihendaten üblich sind. Diese Broschüre geht davon aus, dass der Leser einige Grundkenntnisse der Zeitreihenanalyse hat und der Schwerpunkt der Broschüre ist nicht, die Zeitreihenanalyse zu erläutern, sondern vielmehr zu erklären, wie diese Analysen mit R durchgeführt werden können. Wenn Sie neu in der Zeitreihe sind Analyse und möchten mehr über irgendwelche der hier vorgestellten Konzepte erfahren, empfehle ich das Open University Buch 8220Time series8221 (Produktcode M24902), erhältlich ab dem Open University Shop. In dieser Broschüre verwende ich Zeitreihen-Datensätze, die von Rob Hyndman in seiner Time Series Data Library bei robjhyndmanTSDL freundlich zur Verfügung gestellt wurden. Wenn Sie diese Broschüre mögen, können Sie auch gern meine Broschüre über die Verwendung von R für biomedizinische Statistiken, a-luch-of-for-biomedical-statistics. readthedocs. org. Und meine Broschüre über die Verwendung von R für multivariate Analysen, kleine-Mon-für-Multivariate-analysis. readthedocs. org. Lesen von Zeitreihen-Daten Das erste, was Sie tun möchten, um Ihre Zeitreihendaten zu analysieren, wird es sein, es in R zu lesen und die Zeitreihen zu zeichnen. Sie können die Daten in R mit der Funktion scan () lesen, die davon ausgeht, dass sich Ihre Daten für aufeinanderfolgende Zeitpunkte in einer einfachen Textdatei mit einer Spalte befinden. Zum Beispiel enthält die Datei robjhyndmantsdldatamisckings. dat Daten über das Alter des Todes der aufeinanderfolgenden Könige von England, beginnend mit William der Eroberer (ursprüngliche Quelle: Hipel und Mcleod, 1994). Der Datensatz sieht so aus: Nur die ersten Zeilen der Datei wurden angezeigt. 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 assume 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
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