Kapal Base R dengan banyak fungsi berguna untuk deret waktu, khususnya dalam paket statistik. Ini dilengkapi dengan banyak paket di CRAN, yang dirangkum singkat di bawah ini. Ada juga tumpang tindih antara alat untuk deret waktu dan skenario Econometrics and Finance. Paket dalam tampilan ini dapat disusun secara kasar menjadi topik berikut. Jika Anda berpikir bahwa beberapa paket hilang dari daftar, beri tahu kami. Infrastruktur. Base R berisi infrastruktur substansial untuk merepresentasikan dan menganalisa data deret waktu. Kelas dasarnya adalah quottsquot yang dapat mewakili deret waktu secara teratur (menggunakan perangko waktu numerik). Oleh karena itu, ini sangat sesuai untuk data tahunan, bulanan, kuartalan, dan sebagainya. Statistik bergulir. Moving averages dihitung oleh m dari forecast. Dan rollmean dari kebun binatang. Yang terakhir ini juga menyediakan fungsi umum secara bergantian. Bersama dengan fungsi statistik rolling spesifik lainnya. Roll menyediakan fungsi paralel untuk menghitung statistik bergulir. Grafis. Plot time series diperoleh dengan plot () diterapkan pada objek ts. (Parsial) plot fungsi autokorelasi diimplementasikan di acf () dan pacf (). Versi alternatif disediakan oleh Acf () dan Pacf () dalam ramalan. Bersama dengan tampilan kombinasi menggunakan tsdisplay (). SDD menyediakan diagram ketergantungan serial yang lebih umum, sementara dCovTS menghitung dan merencanakan kovarians jarak dan fungsi korelasi deret waktu. Tampilan musiman diperoleh dengan menggunakan monthplot () dalam statistik dan seasonplot dalam perkiraan. Wats menerapkan grafis rangkaian waktu. Ggseas menyediakan grafis ggplot2 untuk seri dan statistik bergulir musiman. Dygraphs menyediakan antarmuka ke koleksi grafik waktu seri Dygraphs interaktif. Plot ZRA memprediksikan objek dari paket perkiraan menggunakan dygraphs. Plot penggemar dasar distribusi perkiraan disediakan oleh perkiraan dan vars. Plot kipas yang lebih fleksibel dari setiap distribusi berurutan diterapkan di fanplot. Class quottsquot hanya bisa menangani perangko waktu numerik, namun masih banyak lagi kelas yang tersedia untuk menyimpan informasi berjangka waktu dan komputasi dengannya. Untuk tinjauan umum lihat R Help Desk: Kelas Tanggal dan Waktu di R oleh Gabor Grothendieck dan Thomas Petzoldt di R News 4 (1). 29-32 Kelas quotyearmonquot dan quotyearqtrquot dari kebun binatang memungkinkan perhitungan yang lebih mudah dengan pengamatan bulanan dan kuartalan. Class quotDatequot dari paket dasar adalah kelas dasar untuk menangani tanggal dalam data harian. Tanggal disimpan secara internal sebagai jumlah hari sejak 1970-01-01. Paket chron menyediakan kelas untuk tanggal (). Jam () dan datetime (intra-hari) di chron (). Tidak ada dukungan untuk zona waktu dan waktu penghematan siang hari. Secara internal, objek quototquot adalah (pecahan) hari sejak 1970-01-01. Kelas quotPOSIXctquot dan quotPOSIXltquot menerapkan standar POSIX untuk informasi datetime (intra-hari) dan juga mendukung zona waktu dan waktu siang hari. Namun, perhitungan zona waktu memerlukan perawatan dan mungkin tergantung pada sistem. Secara internal, objek quotPOSIXctquot adalah jumlah detik sejak 1970-01-01 00:00:00 GMT. Paket lubridate menyediakan fungsi yang memudahkan perhitungan berbasis POSIX tertentu. Class quottimeDatequot disediakan dalam paket timeDate (sebelumnya: fCalendar). Hal ini ditujukan untuk informasi berjangka waktu keuangan dan kesepakatan dengan zona waktu dan waktu siang hari melalui konsep baru dari pusat informasi quotquancial. Secara internal, ia menyimpan semua informasi di quotPOSIXctquot dan melakukan semua perhitungan dalam GMT saja. Fungsi kalender, mis. Termasuk informasi tentang akhir pekan dan hari libur untuk berbagai bursa saham, juga disertakan. Paket tis menyediakan kelas quottiquot untuk informasi berjangka waktu. Kelas quotmondatequot dari paket mondate memfasilitasi komputasi dengan tanggal dalam hal bulan. Paket tempdisagg mencakup metode untuk pemilahan sementara dan interpolasi dari deret waktu frekuensi rendah ke rangkaian frekuensi yang lebih tinggi. Disagregasi seri waktu juga disediakan oleh tsdisagg2. TimeProjection mengambil komponen waktu yang berguna dari objek tanggal, seperti hari minggu, akhir pekan, hari libur, hari bulan, dll, dan memasukkannya ke dalam bingkai data. Seperti disebutkan di atas, quottsquot adalah kelas dasar untuk rangkaian waktu secara spasial dengan menggunakan perangko waktu numerik. Paket kebun binatang menyediakan infrastruktur untuk deret waktu yang teratur dan tidak teratur dengan menggunakan kelas sewenang-wenang untuk perangko waktu (yaitu mengizinkan semua kelas dari bagian sebelumnya). Ini dirancang agar konsisten dengan quottsquot. Pemaksaan dari dan ke quotzooquot tersedia untuk semua kelas lain yang disebutkan di bagian ini. Paket xts didasarkan pada kebun binatang dan menyediakan penanganan seragam dari berbagai kelas data berbasis waktu yang berbeda. Berbagai paket menerapkan deret waktu tidak teratur berdasarkan prangko waktu quotPOSIXctquot, yang ditujukan khusus untuk aplikasi keuangan. Ini termasuk kutipan dari tseries. Dan quotftsquot dari fts. Kelas quottimeSeriesquot di timeSeries (sebelumnya: fSeries) menerapkan deret waktu dengan perangko waktu quotimeDatequot. Kelas quottisquot di tis menerapkan deret waktu dengan perangko quottiquot. Paket tframe berisi infrastruktur untuk mengatur kerangka waktu dalam format yang berbeda. Pemodelan Peramalan dan Univariat Paket perkiraan menyediakan kelas dan metode untuk perkiraan waktu seri univariat, dan menyediakan banyak fungsi yang menerapkan model peramalan yang berbeda termasuk semua yang ada dalam paket statistik. Pemulusan eksponensial HoltWinters () dalam statistik menyediakan beberapa model dasar dengan pengoptimalan parsial, ets () dari paket perkiraan menyediakan seperangkat model dan fasilitas yang lebih besar dengan pengoptimalan penuh. Robot menyediakan alternatif yang kuat untuk fungsi ets (). Kelancaran menerapkan beberapa generalisasi pemulusan eksponensial. Paket MAPA menggabungkan model pemulusan eksponensial pada berbagai tingkat agregasi temporal untuk meningkatkan akurasi perkiraan. Metode theta diimplementasikan dalam fungsi thetaf dari paket perkiraan. Implementasi alternatif dan perluasan disediakan di forecTheta. Model autoregresif Ar () dalam statistik (dengan pemilihan model) dan FitAR untuk model AR bawaan. Model ARIMA Arima () dalam statistik adalah fungsi dasar untuk model ARIMA, SARIMA, ARIMAX, dan subset ARIMA. Hal ini ditingkatkan dalam paket ramalan melalui fungsi Arima () bersama dengan auto. arima () untuk pemilihan pesanan otomatis. Arma () dalam paket tseries menyediakan algoritma yang berbeda untuk model ARMA ARMA dan subset. FitARMA menerapkan algoritma MLE yang cepat untuk model ARMA. Paket gsarima berisi fungsionalitas untuk simulasi seri waktu Generalized SARIMA. Paket mar1 menangani AR multiplikatif (1) dengan proses musiman. TSTutorial menyediakan tutorial interaktif untuk pemodelan Box-Jenkins. Interval prediksi yang disempurnakan untuk model seri ARIMA dan structural time series disediakan oleh tsPI. Model ARMA periodik. Pir dan partsm untuk model deret waktu autoregresif berkala, dan perARMA untuk pemodelan ARMA periodik dan prosedur lain untuk analisis deret periodik. Model ARFIMA Beberapa fasilitas untuk model ARFIMA fraksional berbeda tersedia dalam paket fracdiff. Paket arfima memiliki fasilitas yang lebih maju dan umum untuk model ARFIMA dan ARIMA, termasuk model regresi dinamis (fungsi transfer). ArmaFit () dari paket fArma adalah antarmuka untuk model ARIMA dan ARFIMA. Frustrasi Gaussian noise dan model sederhana untuk seri waktu peluruhan hiperbolik ditangani dalam paket FGN. Model fungsi transfer disediakan oleh fungsi arimax dalam paket TSA, dan fungsi arfima dalam paket arfima. Deteksi outlier mengikuti pendekatan Chen-Liu disediakan oleh tsoutliers. Model struktural diimplementasikan dalam StructTS () dalam statistik, dan di stsm dan stsm. class. KFKSDS menyediakan implementasi naif dari filter Kalman dan smoothers untuk model ruang negara univariat. Model time series waktu Bayesian diimplementasikan dalam bsts Seri waktu non-Gaussian dapat ditangani dengan model ruang keadaan GLARMA via glarma. Dan menggunakan model Generalized Autoregressive Score dalam paket GAS. Model Auto-Regression bersyarat dengan menggunakan metode Monte Carlo Likelihood diimplementasikan pada mclcar. Model GARCH Garch () dari tseries sesuai dengan model GARCH dasar. Banyak variasi pada model GARCH disediakan oleh rugarch. Paket GARCH univariat lainnya termasuk fGarch yang menerapkan model ARIMA dengan kelas yang luas dari inovasi GARCH. Masih banyak lagi paket GARCH yang dijelaskan dalam tampilan tugas Keuangan. Model volatilitas stokastik ditangani oleh stochvol dalam kerangka Bayesian. Menghitung waktu seri model yang ditangani dalam paket tscount dan acp. ZIM menyediakan Zero-Inflated Models untuk menghitung time series. Alat tsintermittent menerapkan berbagai model untuk menganalisa dan meramalkan deret waktu permintaan intermiten. Seri waktu yang disensor bisa dimodelkan menggunakan sen dan carx. Tes Portmanteau disediakan melalui Box. test () dalam paket stats. Tes tambahan diberikan oleh portes dan WeightedPortTest. Deteksi titik perubahan diberikan dalam strucchange (menggunakan model regresi linier), dalam tren (menggunakan uji nonparametrik), dan di wbsts (menggunakan segmentasi biner liar). Paket changepoint menyediakan banyak metode changepoint yang populer, dan ecp melakukan deteksi changepoint nonparametrik untuk seri univariat dan multivariat. Deteksi titik perubahan online untuk rangkaian waktu univariat dan multivarian disediakan oleh onlineCPD. InspectChangepoint menggunakan proyeksi yang jarang untuk memperkirakan perubahan dalam deret waktu berdimensi tinggi. Imputasi seri waktu disediakan oleh paket imputeTS. Beberapa fasilitas yang lebih terbatas tersedia dengan na. interp () dari paket perkiraan. Prakiraan dapat dikombinasikan dengan menggunakan ForecastCombinations yang mendukung metode yang paling sering digunakan untuk menggabungkan prakiraan. ForecastHybrid menyediakan fungsi untuk ansambel prakiraan, menggabungkan pendekatan dari paket perkiraan. GeomComb menyediakan metode kombinasi prediksi berbasis geometri (berbasis geometri), serta pendekatan lainnya. Opera memiliki fasilitas untuk prediksi online berdasarkan kombinasi prakiraan yang disediakan oleh pengguna. Perkiraan evaluasi diberikan dalam akurasi () fungsi dari perkiraan. Evaluasi perkiraan distribusi dengan menggunakan aturan penilaian tersedia dalam penilaian Rule Miscellaneous. Ltsa berisi metode untuk analisis deret waktu linier, timsac untuk analisis dan pengendalian deret waktu, dan tsbugs untuk model BUGS time series. Estimasi kerapatan spektral disediakan oleh spektrum () dalam paket statistik, termasuk periodogram, estimasi periode perataan dan AR yang diratakan. Bayesian spectral inference disediakan oleh bspec. Quantspec mencakup metode untuk menghitung dan merencanakan periodogram Laplace untuk rangkaian waktu univariat. Periodogram Lomb-Scargle untuk rangkaian waktu sampel yang tidak merata dihitung dengan bentuk rahim. Spektral menggunakan transformasi Fourier dan Hilbert untuk spektral filtering. Psd menghasilkan perkiraan kepadatan spektral adaptif, sinus-multitaper. Kza menyediakan Kolmogorov-Zurbenko Adaptive Filters termasuk break detection, spektral analysis, wavelet dan KZ Fourier Transforms. Multitaper juga menyediakan beberapa alat analisis spektral multitaper. Metode wavelet Paket wavelet mencakup filter wavelet komputasi, transformasi wavelet dan analisis multiresolusi. Metode wavelet untuk analisis deret waktu berdasarkan Percival dan Walden (2000) diberikan dalam wmtsa. WaveletComp menyediakan beberapa alat untuk analisis wavelet berbasis seri waktu univariat dan bivariat termasuk uji wavelet, perbedaan fasa dan uji signifikan. Biwavelet dapat digunakan untuk merencanakan dan menghitung spektrum wavelet, spektrum cross-wavelet, dan koherensi wavelet dari rangkaian waktu non-stasioner. Ini juga mencakup fungsi untuk mengelompokkan deret waktu berdasarkan kesamaan (dis) dalam spektrumnya. Pengujian white noise menggunakan wavelet disediakan oleh hwwntest. Metode wavelet lebih lanjut dapat ditemukan pada paket brainwaver. Rwt. Gelombanglim Wavethresh dan mvcwt. Regresi harmonis dengan menggunakan istilah Fourier diimplementasikan dalam HarmonicRegression. Paket perkiraan juga menyediakan beberapa fasilitas regresi harmonik sederhana melalui fungsi Fourier. Dekomposisi dan Penyaringan Filter dan smoothing. Filter () dalam statistik menyediakan pemeringkatan linear autoregresif dan moving average dari beberapa seri waktu univariat. Paket robfilter menyediakan beberapa filter seri waktu yang kuat, sementara mFilter menyertakan filter rangkaian waktu lain-lain yang berguna untuk menghaluskan dan mengekstrak komponen tren dan siklis. Halus () dari paket statistik menghitung Tukeys yang menjalankan smoothie rata-rata, 3RS3R, 3RSS, 3R, dll. Sleekts menghitung metode smoothing 4253H dua kali. Penguraian . Dekomposisi musiman dibahas di bawah ini. Dekomposisi berbasis Autoregressive disediakan oleh ArDec. Pengkodean menggunakan dekomposisi ARIMA berbasis data kuartalan dan bulanan. Rmaf menggunakan filter rata-rata bergerak yang disaring untuk dekomposisi. Analisis Spektrum Singular diimplementasikan pada Rssa dan spektral. Metode. Analisis Empiris Mode Decomposition (EMD) dan Hilbert spectral disediakan oleh EMD. Alat tambahan, termasuk ensemble EMD, tersedia di hht. Implementasi alternatif dari ensemble EMD dan variannya yang lengkap tersedia di Rlibeemd. Dekomposisi musiman Paket stats memberikan dekomposisi klasik pada dekomposisi (). Dan dekomposisi STL pada stl (). Dekomposisi STL yang disempurnakan tersedia di stlplus. StR memberikan dekomposisi Trend Musiman berdasarkan Regresi. X12 menyediakan pembungkus binari X12 yang harus dipasang terlebih dahulu. X12GUI menyediakan antarmuka pengguna grafis untuk x12. Binari X-13-ARIMA-SEATS disediakan dalam paket x13binary, dengan musiman menyediakan antarmuka R dan seasonalview yang menyediakan GUI. Analisis musiman. Paket bfast menyediakan metode untuk mendeteksi dan menandai perubahan mendadak dalam tren dan komponen musiman yang diperoleh dari dekomposisi. Npst memberikan generalisasi uji musiman Hewitts. musim. Analisis musiman data kesehatan termasuk model regresi, time-stratified case-crossover, fungsi plotting dan residual check. Laut. Analisis dan grafis musiman, terutama untuk klimatologi. Deseasonalize. Optimal deseasonalization untuk deret waktu geofisika menggunakan pemasangan AR. Stationarity, Unit Roots, dan Cointegration Stationarity dan unit roots. Tseries menyediakan berbagai uji stasioneritas dan unit akar termasuk Augmented Dickey-Fuller, Phillips-Perron, dan KPSS. Implementasi alternatif dari tes ADF dan KPSS ada dalam paket urca, yang juga mencakup metode lebih lanjut seperti uji Elliott-Rothenberg-Stock, Schmidt-Phillips dan Zivot-Andrews. Paket fUnitRoots juga menyediakan tes MacKinnon, sementara uroot menyediakan tes akar unit musiman. CADFtest menyediakan implementasi ADF standar dan uji ADF (CADF) yang dikoordinasikan kovariat. Lokalitas stasiun. Tujuannya memberikan pengujian terhadap stasioneritas lokal dan menghitung autocovariance lokal. Penentuan costationarity time series diberikan oleh costat. LSTS memiliki fungsi untuk analisis rangkaian waktu stasioner lokal. Model wavelet stasioner lokal untuk rangkaian waktu nonstasioner diimplementasikan dengan wavethresh (termasuk estimasi, pertarungan, dan fungsi simulasi untuk spektrum dengan variasi waktu). Kointegrasi Metode two-step Engle-Granger dengan uji kointegrasi Phillips-Ouliaris diimplementasikan pada tseries dan urca. Yang terakhir ini juga berisi fungsionalitas untuk tes Johansen dan tes lambda-max. TsDyn menyediakan tes Johansens dan seleksi rank-lag AICBIC simultan. CommonTrend menyediakan alat untuk mengekstrak dan merencanakan kecenderungan umum dari sistem kointegrasi. Estimasi parameter dan inferensi dalam regresi kointegrasi diimplementasikan pada cointReg. Analisis Seri Waktu Nonlinear Autoregression nonlinier. Berbagai bentuk autoregression nonlinier tersedia di tsDyn termasuk AR aditif, jaring saraf, model SETAR dan LSTAR, ambang VAR dan VECM. Autoregression jaringan saraf juga tersedia di GMDH. BentcableAR menerapkan autoregression Bent-Cable. BAYSTAR menyediakan analisis Bayesian terhadap model autoregresif ambang. TseriesChaos menyediakan implementasi R dari algoritma dari proyek TISEAN. Autoregression Markov switching model disediakan di MSwM. Sementara campuran tergantung model Markov laten diberikan di depmix dan depmixS4 untuk rangkaian waktu kategoris dan kontinu. Pengujian. Berbagai tes untuk nonlinier disediakan di fNonlinear. Uji tieriEntropi untuk ketergantungan serial nonlinier berdasarkan metrik entropi. Fungsi tambahan untuk rangkaian waktu nonlinier tersedia di nlts dan nonlinearTseries. Pemodelan dan analisis deret waktu fraktal disediakan secara fraktal. Fractalrock menghasilkan deret waktu fraktal dengan distribusi kembali non-normal. Model Regresi Dinamis Model linier dinamis. Antarmuka yang mudah digunakan untuk menyesuaikan model regresi dinamis melalui OLS tersedia dalam dynlm pendekatan yang disempurnakan yang juga bekerja dengan fungsi regresi lainnya dan kelas time series lebih banyak diterapkan di dyn. Persamaan sistem dinamis yang lebih maju dapat dipasang menggunakan dse. Model ruang keadaan linier Gaussian dapat dipasang dengan menggunakan dlm (via maximum likelihood, metode filteringsmoothing dan Bayesian Kalman), atau menggunakan bsts yang menggunakan MCMC. Fungsi untuk pemodelan non-linear lag terdistribusi disediakan dalam dlnm. Model parameter waktu bervariasi dapat dipasang menggunakan paket tpr. OrderedLasso cocok dengan model linier yang jarang dengan batasan pesanan pada koefisien untuk menangani regresor yang tertinggal dimana koefisien membusuk seiring dengan lag meningkat. Dinamis pemodelan berbagai jenis tersedia dalam dynr termasuk diskrit dan waktu kontinyu, model linier dan nonlinier, dan berbagai jenis variabel laten. Model Seri Waktu Multivarian Model Vector autoregressive (VAR) disediakan melalui ar () dalam paket statistik dasar termasuk pemilihan pesanan melalui AIC. Model ini dibatasi untuk menjadi tidak bergerak. MTS adalah toolkit semua tujuan untuk menganalisis rangkaian waktu multivarian termasuk VAR, VARMA, VARMA musiman, model VAR dengan variabel eksogen, regresi multivariat dengan kesalahan deret waktu, dan banyak lagi. Mungkin model VAR non-stasioner dipasang dalam paket mAr, yang juga memungkinkan model VAR di ruang komponen utama. Sparsevar memungkinkan perkiraan model VAR dan VECM yang jarang, ECM menyediakan fungsi untuk membangun model VECM, sementara BigVAR memperkirakan model VAR dan VARX dengan hukuman laso terstruktur. Model VAR otomatis dan jaringan tersedia di autovarCore. Model yang lebih rumit disediakan dalam paket vars. TsDyn EstVARXls () di dse. Dan pendekatan Bayesian tersedia di MSBVAR. Implementasi lain dengan interval prediksi bootstrap diberikan di VAR. etp. MlVAR menyediakan multi level vektor autoregression. VARsignR menyediakan rutinitas untuk mengidentifikasi guncangan struktural pada model VAR yang menggunakan batasan tanda. Gdpc menerapkan komponen utama dinamik yang dinamis. Pcdpca memperluas komponen utama yang dinamis ke rangkaian waktu multivariate yang berkorelasi secara periodik. Model VARIMA dan model ruang negara disediakan dalam paket dse. EvalEst memfasilitasi eksperimen Monte Carlo untuk mengevaluasi metode estimasi yang terkait. Model koreksi kesalahan vektor tersedia melalui urca. Vars dan paket tsDyn, termasuk versi dengan batasan struktural dan thresholding. Analisis komponen seri waktu. Analisis faktor deret waktu diberikan di tsfa. ForeCA menerapkan analisis komponen yang dapat diobati dengan mencari transformasi linier terbaik yang membuat rangkaian waktu multivarian seperti yang diperkirakan. PCA4TS menemukan transformasi linier dari rangkaian waktu multivariat yang memberikan subseries berdimensi rendah yang tidak berkorelasi satu sama lain. Model ruang negara multivariat diimplementasikan dalam paket FKF (Fast Kalman Filter). Ini menyediakan model ruang negara yang relatif fleksibel melalui fungsi fkf (): parameter ruang-negara diperbolehkan untuk menyesuaikan waktu dan penyadapan disertakan dalam kedua persamaan. Implementasi alternatif disediakan oleh paket KFAS yang menyediakan filter Kalman multivarian cepat, lebih halus, simulasi yang lebih halus dan peramalan. Namun implementasi lain diberikan dalam paket dlm yang juga berisi alat untuk mengubah model multivariat lainnya ke dalam bentuk ruang negara. Dlmodeler menyediakan antarmuka terpadu untuk dlm. KFAS dan FKF. MARSS sesuai dengan model ruang negara autoregresif multivariat yang dibatasi dan tidak dibatasi menggunakan algoritma EM. Semua paket ini menganggap istilah kesalahan pengamatan dan keadaan tidak berkorelasi. Proses Markov yang diamati sebagian adalah generalisasi model ruang keadaan multivariat linier yang biasa, yang memungkinkan model non-Gaussian dan nonlinier. Ini diimplementasikan dalam paket pomp. Model volatilitas stokastik multivariat (menggunakan faktor laten) disediakan oleh faktorstochvol. Analisis kelompok besar deret waktu Seriing clustering diimplementasikan di TSclust. Dtwclust BNPTSclust dan pdc. TSdist menyediakan ukuran jarak untuk data deret waktu. Jmotif menerapkan alat berdasarkan diskritisasi simbolis time series untuk menemukan motif dalam deret waktu dan memfasilitasi klasifikasi deret waktu yang dapat ditafsirkan. Rucrdtw menyediakan binding R untuk fungsi dari UCR Suite untuk memungkinkan pencarian awal ultrafast untuk pertandingan terbaik di bawah Dynamic Time Warping dan Euclidean Distance. Metode untuk merencanakan dan meramalkan koleksi rangkaian waktu hirarkis dan dikelompokkan disediakan oleh hts. Pencuri menggunakan metode hirarkis untuk mendamaikan perkiraan rangkaian waktu gabungan temporal. Pendekatan alternatif untuk mendamaikan perkiraan deret waktu hirarkis disediakan oleh gtop. Pencuri Model waktu kontinu Pemodelan autoregresif kontinu disediakan dalam cts. Sim. DiffProc mensimulasikan dan memodelkan persamaan diferensial stokastik. Simulasi dan kesimpulan untuk persamaan diferensial stokastik disediakan oleh sde dan yuima. Bootstrap Paket boot menyediakan fungsi tsboot () untuk bootstrap time series, termasuk bootstrap blok dengan beberapa varian. Tsbootstrap () dari tseries menyediakan bootstrap stasioner dan blokir yang cepat. Bootstrap entropi maksimum untuk deret waktu tersedia di meboot. Timesboot menghitung bootstrap CI untuk sampel ACF dan periodogram. BootPR menghitung prediksi bias dan interval prediksi boostrap untuk rangkaian waktu autoregresif. Data dari Makridakis, Wheelwright dan Hyndman (1998) Peramalan: metode dan aplikasi disediakan dalam paket fma. Data dari Hyndman, Koehler, Ord dan Snyder (2008) Peramalan dengan eksponensial smoothing ada dalam paket expsmooth. Data dari Hyndman dan Athanasopoulos (2013) Peramalan: prinsip dan praktik ada dalam paket fpp. Data kompetisi M-competition dan M3 disediakan dalam paket Mcomp. Data dari kompetisi M4 diberikan di M4comp. Sementara Tcomp menyediakan data dari Lomba Peramalan Pariwisata IJF 2010. Pdfetch menyediakan fasilitas untuk mendownload rangkaian waktu ekonomi dan keuangan dari sumber publik. Data dari portal online Quandl ke kumpulan data keuangan, ekonomi dan sosial dapat dipertanyakan secara interaktif dengan menggunakan paket Quandl. Data dari portal online Datamarket dapat diambil menggunakan paket rdatamarket. BETS menyediakan akses ke deret waktu ekonomi terpenting di Brasil. Data dari Cryer dan Chan (2010) ada dalam paket TSA. Data dari Shumway dan Stoffer (2011) ada dalam paket astaga. Data dari Tsay (2005) Analisis deret waktu keuangan ada dalam paket FinTS, beserta beberapa fungsi dan file skrip yang diperlukan untuk mengerjakan beberapa contoh. Tswge menyertai teks Applied Time Series Analysis with R. Edisi ke 2 oleh Woodward, Gray, dan Elliott. TSdbi menyediakan antarmuka umum untuk database time series. Ketenaran menyediakan antarmuka untuk database rangkaian waktu FAME AER dan Ecdat keduanya berisi banyak kumpulan data (termasuk data deret waktu) dari banyak buku teks ekonometri dtw. Algoritma warping waktu dinamis untuk komputasi dan merencanakan kesejajaran berpasangan antara deret waktu. EnsembleBMA Bayesian Model Averaging untuk membuat ramalan probabilistik dari prakiraan anambel dan pengamatan cuaca. Earlywarnings. Tanda peringatan awal kotak peralatan untuk mendeteksi transisi kritis dalam rangkaian waktu. Ubah data peristiwa yang diekstraksi dengan mesin menjadi rangkaian waktu multivariasi agregat reguler. Umpan Balik. Analisis keterputusan waktu terfragmentasi untuk menyelidiki umpan balik dalam deret waktu. LPStimeSeries bertujuan untuk mengetahui pola persamaan yang bisa diambil untuk deret waktu. MAR1 menyediakan alat untuk menyiapkan data rangkaian waktu komunitas ekologi untuk pemodelan AR multivarian. Jaring. Rutinitas untuk estimasi jaringan korelasi parsial jangka pendek yang jarang untuk data deret waktu. PaleoTS Pemodelan evolusi dalam rangkaian waktu paleontologis. Pastecs Regulasi, dekomposisi dan analisis rangkaian ruang-waktu. Ptw. Parametrik waktu warping. RGENERATE menyediakan alat untuk menghasilkan deret waktu vektor. RMAWGEN adalah seperangkat fungsi S3 dan S4 untuk generasi stochastic multi-situs spasial dari rangkaian suhu dan presipitasi harian yang menggunakan model VAR. Paket tersebut bisa digunakan dalam klimatologi dan hidrologi statistik. RSEIS. Alat analisis deret waktu seismik. Rts Analisis deret waktu raster (misalnya deret waktu gambar satelit). Sae2 Model deret waktu untuk estimasi area kecil. SpTimer Pemodelan Bayesian spatio-temporal. pengawasan. Pemodelan temporal dan spatio-temporal dan pemantauan fenomena epidemi. TED. Turbulensi time series Event Detection dan klasifikasi. Pasang surut Fungsi untuk menghitung karakteristik deret waktu kuasi periodik, mis. Mengamati tingkat air muara. harimau. Kelompok yang tersusun secara temporal dari perbedaan khas (kesalahan) antara dua seri waktu ditentukan dan divisualisasikan. TSMining Eksperimen Bidik Univariat dan Multivariat dalam Data Seri Waktu. Tsodel Pemodelan deret waktu untuk polusi udara dan kesehatan. Paket CRAN: Tautan terkait: gt mav (c (4,5,4,6), 3) Seri Waktu: Mulai 1 Akhir 4 Frekuensi 1 1 NA 4.333333 5.000000 NA Di sini saya mencoba melakukan rata-rata rolling yang memperhitungkan 3 angka terakhir jadi saya berharap bisa mendapatkan dua nomor lagi 8211 4.333333 dan 5 8211 dan jika akan ada nilai NA saya pikir mereka akan berada di awal urutan. Sebenarnya ternyata inilah kontrol parameter 8216sides8217: hanya sisi untuk filter konvolusi. Jika sisi 1 koefisien filter untuk nilai masa lalu hanya jika sisi 2 berpusat pada lag 0. Dalam kasus ini, panjang saringan harus aneh, namun jika memang demikian, lebih banyak filter yang dimodelkan dalam waktu daripada mundur. Jadi, dalam fungsi 8216mav8217, rata-rata rata-rata bergulir terlihat kedua sisi nilai sekarang daripada hanya pada nilai masa lalu. Kita bisa men-tweaknya untuk mendapatkan tingkah laku yang kita inginkan: gt library (zoo) gt rollmean (c (4,5,4,6), 3) 1 4.333333 5.000000 Saya juga menyadari bahwa saya dapat mencantumkan semua fungsi dalam paket dengan 8216ls8217 Fungsi jadi aku akan memindai daftar fungsi zoo8217s lain kali aku harus melakukan sesuatu yang berhubungan dengan deret waktu 8211 sana mungkin sudah menjadi fungsi untuk itu gt ls (quotpackage: zooquot) 1 kuota. Datequot quota. Date. numericquot quota. Date. tsquot 4 Kuota. Date. yearmonquot quotas. Date. yearqtrquot quotas. yearmonquot 7 kuota. yearmon. defaultquot quotas. yearqtrquot quotas. yearqtr. defaultquot 10 kuota. zooquot kuota. zoo. defaultquot quotas. zooregquot 13 kuota. zooreg. defaultquot kuotautoplot. zooquot quotcbind. Zooquot 16 quotcoredataquot quotcoredata. defaultquot quotcoredatalt-quot 19 quotfacetfreequot quotformat. yearqtrquot quotfortify. zooquot 22 quotfrequencylt-quot quotpeselse. 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 . 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. se Gments. zooquot 73 quotpanel. text. itsquot quotpanel. text. tisquot quotpanel. text. tsquot 76 quotakel. 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 Jadilah Sociable, ShareUsing R untuk Analisis Time Series Analisis Seri Waktu Buklet ini menjelaskan bagaimana Anda menggunakan perangkat lunak statistik R untuk melakukan beberapa langkah sederhana. Nalyses yang umum dalam menganalisa data deret waktu. Buklet ini mengasumsikan bahwa pembaca memiliki beberapa pengetahuan dasar tentang analisis deret waktu, dan fokus utama dari buklet tersebut bukanlah untuk menjelaskan analisis deret waktu, melainkan untuk menjelaskan bagaimana melakukan analisis ini menggunakan R. Jika Anda baru mengenal deret waktu Analisis, dan ingin belajar lebih banyak tentang konsep apa pun yang disajikan di sini, saya akan sangat merekomendasikan buku Open University 8220Time series8221 (kode produk M24902), tersedia dari Open University Shop. Dalam buklet ini, saya akan menggunakan kumpulan data rangkaian waktu yang telah disediakan oleh Rob Hyndman dalam Time Data Library-nya di robjhyndmanTSDL. Jika Anda menyukai buklet ini, Anda mungkin juga ingin memeriksa buklet saya untuk menggunakan R untuk statistik biomedis, a-little-book-of-r-for-biomedical-statistics. readthedocs. org. Dan buklet saya tentang penggunaan R untuk analisis multivariat, sedikit - buku - untuk - untuk memulai - analisis. readthedocs. org. Reading Time Series Data Hal pertama yang ingin Anda lakukan untuk menganalisis data deret waktu Anda adalah membacanya menjadi R, dan untuk merencanakan deret waktu. 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
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