Pemodelan Keuangan Kuantitatif amp Kerangka Perdagangan untuk R Jika ada satu area R yang sedikit kurang, itu adalah kemampuan untuk memvisualisasikan data keuangan dengan alat charting keuangan standar. Karena tidak ada paket lain yang menerapkan ini, quantmod menerima telepon tersebut dan mengambil kesempatan untuk memberikan sebuah solusi. Apa yang dimulai dengan satu solusi bagan OHLC tunggal telah berkembang menjadi fasilitas charting yang sangat dapat dikonfigurasi dan dinamis seperti pada versi 0.3-4, dengan lebih banyak kesejukan untuk 0,4-0 dan seterusnya. Untuk saat ini, mari kita lihat apa yang sekarang ada: Diagram Keuangan dalam quantmod: Sebagian besar fungsi pembuatan peta dirancang untuk digunakan secara interaktif. Contoh berikut harus sangat mudah untuk mereplikasi dari command line atau pilihan GUI pribadi Anda. Berjalan dari naskah membutuhkan sedikit perawatan ekstra, tapi sekarang mungkin juga. Mari mulai charting Memperkenalkan chart chartseriesSeries adalah fungsi utama melakukan semua pekerjaan dalam quantmod. Courtesy of as. xts itu bisa menangani objek apa saja yaitu deret waktu seperti, yang berarti objek R kelas xts. Kebun binatang TimeSeries Nya. Ts Ikat Dan lebih Secara default setiap seri yang ada. OHLC dipetakan sebagai rangkaian OHLC. Ada argumen tipe yang memungkinkan pengguna menentukan gaya yang akan diberikan: bagan batang tradisional, bagan lilin, dan diagram batang korek api - lilin tipis. Mendapatkannya :) - serta grafik garis. Pilihan default otomatis memungkinkan perangkat lunak memutuskan, lilin di mana mereka terlihat jelas, korek api jika banyak titik dipetakan, dan garis jika rangkaiannya bukan dari sifat OHLC. Jika Anda tidak ingin selalu menentukan jenisnya untuk menimpa perilaku ini, Anda bebas menggunakan fungsi pembungkus di bagian berikutnya, atau menggunakan setDefault dari paket Default jahat dan berguna (tersedia di CRAN). Fakta bahwa saya menulis itu tidak ada hubungannya dengan dukungan saya :) gt getSymbols (GS) Goldman OHLC dari yahoo 1 GS gt chartSeries (GS) gt melihat gaya korek api otomatis gt dengan baik mengubahnya di bagian berikutnya gt tapi untuk saat ini baik-baik saja. Fungsi dasar charting mencoba untuk tidak menyimpang terlalu jauh dari pola penggunaan standar di R. Meskipun Anda tidak akan dapat menggunakan salah satu alat grafis standar untuk menampilkan grafik. Penulis quantmod oh-so-wise telah mencoba mengantisipasi kebutuhan itu dengan fungsi khusus untuk menebus kekurangan ini. Sebuah langkah cepat kembali, untuk menjelaskan apa yang terjadi di balik layar dalam chartSeries mungkin sesuai. Charting dikelola melalui proses dua langkah. Pertama, data diperiksa dan keputusan dasar tentang cara terbaik untuk menarik seri dihitung. Hasil dari ini adalah objek internal - disebut sebagai chob (ch art ob ject). Objek ini kemudian diteruskan ke fungsi gambar utama (tidak dipanggil langsung) untuk ditarik ke layar. Tujuan pemisahan ini adalah untuk memungkinkan penambahan bagan gaya dinamis yang lebih mengesankan, serta modifikasi, menjadi wajar untuk dicapai. Bila perubahan dilakukan pada grafik saat ini - apakah itu menambahkan indikator teknis, atau mengubah parameter asli, seperti gaya bagan - objek chob yang tersimpan hanya diubah dan kemudian digambar ulang tanpa banyak manipulasi pengguna yang membosankan. Tujuannya adalah untuk membuatnya bekerja tanpa usaha pengguna tambahan - dan untuk kemudian mengakhirinya saja. Memetakan jalan pintas - barChart, lineChart, dan candleChart. Sementara chartSeries adalah fungsi utama yang disebut saat menggambar grafik dalam quantmod - ini sama sekali bukan satu-satunya cara untuk menyelesaikan sesuatu. Ada fungsi wrapper untuk masing-masing jenis grafik utama yang tersedia saat ini dalam quantmod. Fungsi pembungkus ada untuk membuat hidup sedikit lebih mudah. Bagan gaya bar, varietas hlc dan ohlc tersedia secara langsung dengan barChart. Charting candlestick datang secara alami melalui fungsi wrapper candleChart, dan garis-garis melalui nama samar - Anda bisa menebaknya - lineChart. Tidak ada yang istimewa dari fungsi-fungsi ini di luar yang jelas. Sebenarnya mereka adalah satu liner yang hanya memanggil chartSeries dengan args default yang berubah sesuai. Tapi mereka membuat tambahan yang bagus ke kandang. Gt pertama beberapa bar gaya high-low-close, tema monokromatik gt barChart (GS, themewhite. mono, bar. typehlc) gt bagaimana dengan beberapa lilin, kali ini dengan warna gt candleChart (GS, multi. color, themewhite) gt gt dan Sekarang garis, dengan skema warna default gt lineChart (GS, line. typeh, TANULL) Seperti yang Anda lihat, ada sedikit fleksibilitas untuk menampilkan informasi Anda. Yang mungkin juga Anda perhatikan adalah argumen yang berbeda untuk setiap panggilan. Nah sekarang kita lihat apa yang beberapa dari mereka lakukan. Argumen Formal: Warna, subsetting, tanda centang. Tempat terbaik untuk informasi lengkap tentang argumen apa yang diperlukan dalam dokumentasi. Tapi sekarang kita lihat beberapa opsi umum yang mungkin Anda ubah. Mungkin yang terpenting dari sudut pandang kegunaan adalah subset argumen. Ini membutuhkan string berbasis waktu xtsISO8601 dan membatasi plot ke rentang datetime yang ditentukan. Ini tidak membatasi data yang tersedia untuk fungsi analisis teknologis, hanya membatasi konten yang ditarik ke layar. Untuk alasan ini, sangat menguntungkan untuk menggunakan data sebanyak yang Anda miliki, dan kemudian menyediakan fungsi bagan bagan dengan subset yang ingin Anda lihat. Subsetting ini juga dapat diakses melalui panggilan untuk zoomChart. Contoh, atau tiga, akan membantu memperjelas penggunaannya. Gt keseluruhan seri gt chartSeries (GS) gt sekarang - sedikit tapi tergantung pada gt (07 Desember sampai pengamatan terakhir di 08) gt candleChart (GS, subset2007-12 :: 2008) Sintaksnya sedikit berbeda - setelah faktanya. Gt juga mengubah sumbu x yang melabeli candleChart (GS, themewhite, typecandles) gt reChart (major. ticksmonths, subsetfirst 16 weeks) Tiga hal yang perlu diperhatikan pada grafik terakhir. Pertama adalah penggunaan reChart untuk memodifikasi chart asli. Ini mengambil sebagian besar argumen tentang panggilan awal, dan memungkinkan modifikasi cepat ke grafik Anda. Jadilah itu mengubah tema warna atau subsetting - ia datang sangat berguna. Item penting kedua adalah penggunaan sintaks pertama di dalam subset. Hal ini memungkinkan ekspresi yang sedikit lebih alami dari apa yang Anda inginkan, dan tidak mengharuskan Anda mengetahui apapun tentang rangkaian tanggal atau waktu. Item akhir dari catatan pada gambar terakhir adalah argumen tick. marks. Ini adalah bagian dari daftar formal fungsi chartSeries asli, dan ini digunakan untuk memodifikasi penempatan label dalam bagan. Seringkali jarak yang dipilih secara otomatis - yang didorong oleh fungsi xts axTicksByTime melakukan pekerjaan yang cukup baik - Anda mungkin merasa perlu untuk menyesuaikan output lebih jauh. Dalam kasus ini, kami menandai kutu utama dengan permulaan bulan. Technical Analysis dan chartSeries Diperbarui dan siap untuk pergi adalah beberapa alat fantastis dari paket TTR oleh Josh Ulrich. Tersedia di CRAN Sekarang mungkin hanya menambahkan puluhan alat analisis teknis untuk membuat grafik dengan perintah yang lebih sederhana. Indikator saat ini dari paket TTR, dan juga beberapa yang berasal dari paket quantmod adalah: Semua pekerjaan di atas mirip dengan fungsi dasar TTR yang mereka sebut. Perbedaan utamanya adalah bahwa penambahan keluarga panggilan tidak termasuk argumen data, karena ini berasal dari grafik saat ini. Beberapa contoh akan menyoroti bagaimana membangun grafik dengan indikator built-in. Gt getSymbols (GS) Goldman OHLC dari yahoo 1 GS gt Argumen TA untuk chartSeries adalah salah satu cara untuk menentukan indikator gt yang akan diterapkan ke grafik. Gt NULL berarti jangan menggambar apapun Gt gt chartSeries (GS, TANULL) gt Sekarang dengan beberapa indikator yang diterapkan gt gt chartSeries (GS, themewhite, TAaddVo () addBBands () addCCI ()) gt Hasil yang sama dapat dicapai dengan sedikit lebih interaktif: gt gt chartSeries (GS , Themewhite) bagikan bagan gt addVo () tambahkan volume gt addBBands () tambahkan Bollinger Bands gt addCCI () tambahkan Commodity Channel Index Salah satu penambahan terbaru dan paling menarik pada rilis quantmod terbaru mencakup dua alat charting baru yang dirancang untuk membuat penambahan custom Indikator jauh lebih cepat dari sebelumnya mungkin. Yang pertama adalah addTA. Ini adalah ekstensi utama untuk fungsi addTA sebelumnya, karena sekarang memungkinkan data sewenang-wenang digambar pada grafik. Bertindak sebagai dasarnya pembungkus data Anda, satu-satunya persyaratan adalah bahwa data memiliki jumlah observasi yang sama seperti aslinya, atau kelas xts dan tanggalnya berada dalam rentang waktu dan skala data asli. Hal ini dimungkinkan untuk memiliki data baru ini diplot dalam subkategaranya TA (default), atau overlay pada seri utama. Fungsi kedua dan yang berpotensi lebih menarik adalah newTA. Ini adalah fungsi kerangka yang telah lama ditunggu untuk membuat indikator TA khusus agar ditambahkan ke grafik manapun. Dibutuhkan konsep kerangka satu langkah lebih jauh, dan secara dinamis menciptakan kode fungsi yang dibutuhkan untuk sebuah indikator baru, berdasarkan fungsi yang Anda berikan padanya. Intinya sedikit pemrograman self-aware membuat menambahkan indikator baru cukup intuitif dan praktis tidak menimbulkan rasa sakit. Mengingat kemampuannya yang agak canggih, ia berada di puncak percobaan. Beruntung jika semuanya gagal, dan apa yang Anda dapatkan bukanlah yang Anda harapkan, Anda selalu dapat memodifikasi kode yang dibuat agar sesuai dengan kebutuhan Anda. Sekilas menambahkan data indikator khusus dan membuat indikator baru dari awal. Gt getSymbols (YHOO) Yahoo OHLC dari yahoo 1 YHOO gt addTA memungkinkan Anda menambahkan indikator dasar ke grafik Anda - bahkan jika mereka mengambil bagian gt dari quantmod. Gt gt chartSeries (YHOO, TANULL) gt Kemudian tambahkan perubahan harga Open to Close gt menggunakan fungsi opCl quantmod gt gt addTA (OpCl (YHOO), colblue, typeh) gt Menggunakan newTA adalah mungkin untuk membuat fungsi TA generik gt Anda sendiri. --- Mari kita panggil addOpCl gt addOpCl lt - newTA (OpCl, colgreen, typeh) gt gt addOpCl () Lebih untuk datang. Ada banyak lagi yang bisa dikatakan tentang chartSeries dan quantmods alat visualisasi saat ini dan masa depan, namun sekarang saatnya untuk menyebutnya satu hari (atau 30) dan menyimpulkan pendahuluan ini untuk mencatat dalam quantmod. Penambahan masa depan ke situs ini dan dokumentasi akan mencakup rincian lebih lanjut tentang berinteraksi dengan grafik - sekarang dan di rilis mendatang, opsi tata letak baru, dan kemungkinan perayapan menjadi alat dan teknik visualisasi yang sama sekali baru. Tapi untuk saat ini hanya yang saya punya. Perangkat lunak ini ditulis dan dikelola oleh Jeffrey A. Ryan. Lihat lisensi untuk rincian tentang penyalinan dan penggunaan. Copyright 2008.IFC - Pelacak Radars Untuk pengenalan radar atau Nike Acquisition Radars, pergilah ke. Kontrol untuk radar ini ditempatkan di Van Kontrol Radar, yang terletak sangat dekat dengan Van Kontrol Baterai dan ukuran yang sama. John Porter, manajer SF-88, melaporkan bahwa The Battery Control Van berukuran 20,5 kaki, lebar 8 kaki, tinggi 7 kaki, dengan lidah 6,5 kaki. Ini adalah radar pelacak Nike Hercules. Bisa jadi Radar Pelacakan Rudal (MTR), Radar Pelacakan Target (Target Tracking Radar / TTR), atau Radar Kisaran Target (TRR). Detilnya tersembunyi di bawah perisai angin, yang mengurangi tekanan angin dan pemukulan. Kendaraan pengangkut beroda masih ada, saat dilepas, antena ini didukung oleh tiga kaki yang bisa diatur dalam segitiga. Pelacak radar Pelacak Sasaran Nike X-Band (sekitar 10 GHz) ini tanpa layar angin. Gambar dari situs Nike yang baru-baru ini (2016) keluar dari layanan di Folgaria, TN, Italia. Sekarang menjadi bagian dari sebuah museum Nike Hercules yang hebat di situs itu. Juga, TM9-5000-18 NIKE I SYSTEMS - TTR TRANSMITTER AND RECEIVER CIRCUITRY tersedia untuk tampilan lain (yang lebih rinci) dari beberapa materi ini. (Ajang Nike 1 Ajax adalah pendahulu Nike Hercules, namun prinsip yang sama berlaku.) Dari Buku Pegangan FM 44-1-2 ADA Referennce, 15 Juni 1984, lihat halaman 21 Rings of Antrian Pelacakan Baja Supersonik Antena Antena Nike perlu Akurasi penunjuk ekstrim Nike Hercules memiliki jangkauan lebih dari 90 mil dan sistem harus bisa mengarahkan rudal ke dengan 10 atau 12 yard dari tempat Target Tracking Radar mengatakan targetnya adalah. Ini berarti Target Tracking Radar (memberi target lokasi) dan Radar Pelacakan Rudal (memberikan lokasi rudal Hercules) harus diselaraskan dengan hati-hati (dibahas di bawah). Juga, efek dari - panas matahari yang mendistorsi pemasangan antena - kekuatan angin yang mendistorsi pemasangan antena harus diminimalkan juga. Untuk membantu mencapai ini, sebuah menara ganda digunakan: - menara dalam mendukung antena - menara luar mendukung layar angin dan nuansa menara dalam Foto-foto di bawah adalah area IFC dari SF-88, utara San Francisco Mari kita mulai dengan Menara dalam, beton - dengan tiga proyeksi di bagian atas untuk tiga kaki antena. Foto milik Greg Brown Menara luar menaungi menara dalam yang mengurangi pemanasan diferensial di sisi matahari menara dalam. Ini juga mendukung tangga akses, layar angin, platform kerja dan rel penjaga. Tiga bantalan yang ditandai 0 adalah bantalan yang didukung oleh menara dalam, dan dukung ketiga bantalan antena. Lingkaran putih parsial adalah sisa-sisa dukungan gelembung layar angin. Gambar ini menunjukkan dua rakitan dasar yang terpisah, persegi panjang luar yang menopang menara baja luar, balok dalam yang menopang menara beton bulat dalam. Foto courtesy of Greg Brown Pelacakan Perangkat Gambar Base Antena yang digunakan untuk BoreSighting Ini adalah kotak kontrol dan kabel yang digunakan untuk melakukan penyesuaian penunjukan antena yang baik selama pengamatan bore dan prosedur penyelarasan lainnya. Ini adalah teleskop yang digunakan dalam prosedur penampakan dan prosedur pelurusan lainnya. Layanan Taman Nasional telah dilengkapi dengan unit ini - Konektor pada antena pelacak Ini bisa disebut sisi konektor Antena Pelacakan Nike. Ini juga mencakup dua kompartemen penyimpanan untuk peralatan yang sering digunakan. Sebagian besar konektor menggunakan konektor arus rendah (sinyal), berikut adalah contohnya. Konektor yang terpapar adalah untuk tiga kabel koaksial yang membawa sinyal Medium Frequency (60 MHz) ke Control Radar Control untuk amplifikasi lebih lanjut. Mereka adalah sinyal Sum, Azimuth Error and Elevation Error. Pelacakan Antena Base Electronics Ini adalah magnetron dan suplai daya tegangan tinggi lainnya, dan amplifier yang digunakan untuk menggerakkan antena azimuth dan elevasi motor. Mereka berada di sisi berlawanan relatif terhadap sisi konektor. Posisi Operator Pelacakan Target Ini adalah Pelacakan Pelacakan Pelacakan, masing-masing dengan cakupan dan kontrol berlabel. Mereka terlihat sangat mirip dengan Nike Ajax dan Nike Hercules. Di Nike Hercules, sebuah posisi tambahan (stand-up), Supervisor Pelacakan, ditambahkan untuk membantu mengkoordinasikan kegiatan dan mengoperasikan kontrol anti-jamming. Gambar Radar Control van di Ft. Sill museum dari Al Harvard Pan-adaptor ruang lingkup untuk anti jamming dan kontrol di bawah kiri, TTR Magnetron kontrol di bawah kanan, Azimuth lingkup kanan dari Greg Brown Gambar ini adalah chassis A-Scope posisi target track elevasi ditarik keluar. (Semua chassis A-Scope identik) Gambar ini adalah konsol Pelacakan Sasaran bertenaga dan kontrol anti-jamming dari sistem Italia, dari Pengontrol Pelacakan Pelacakan Ramiro Carli Ballola Ini adalah Nike Hercules saja - orang yang berdiri di belakang tiga operator pelacakan Dan menggunakan panel kontrol anti-jamming yang dapat dilepas ini. Saya telah menunjukkan ini dengan sangat rinci saat saya terpesona membayangkan apa yang bisa dilakukan untuk menghindari sinyal gangguan. Magnetron TTR - MTR, tipe WE 5780 A magnetron adalah tabung vakum khusus yang mampu menghasilkan pulsa tenaga kuda yang sangat kuat (100 kilowatt) dari pulsa kuat tegangan tinggi (sekitar 30 Kv) saat ini (sekitar 30 ampli). The Western Electric type WE 5780 magnetron digunakan pada radar Nike Target Tracking. Ini adalah magnetron merdu dengan frekuensi terpusat pada 10 GHz (panjang gelombang 3 cm). Jon Elson mengirim foto-foto ini dari Jon: ARGhhhh. Saya telah memiliki magnetron ini dengan saya selama lebih dari 40 tahun, telah duduk di sebuah kotak di garasi saya saat ini selama 25 tahun. Ambillah untuk mengambil gambar, dan saya DROPPED itu DAMN Saya kira saya bisa fudge itu kembali bersama-sama untuk sebuah gambar. Ini adalah peralatan tuning luar - ke kiri adalah motor kecil yang digerakkan, lalu kabel fleksibel dengan mekanisme memutar bagian dalam, hingga sudut 90 derajat, hingga drive cacing. Hal ini memungkinkan TTR untuk mencoba menghindari gangguan dan kemacetan dengan mengubah frekuensi - 10 Penerima secara otomatis melacak perubahan frekuensi magnetron (AFC) Berikut adalah lembar data dari frank. pocnetsheets20155780.pdf Lengan lebar biasanya didefinisikan sebagai lebar antara setengah titik daya Dari berkas utama antena. Wikipedia memberikan sebuah formula untuk lebar balok parabola yang khas seperti: BeamWidthInDegrees 70 WaveLength AntennaDiameter dimana WaveLength dan AntennaDiameter berada pada unit yang sama. Menggunakan faktor - panjang gelombang 3 cm - diameter antena (sekitar 5 kaki) 152 cm memberi lebar balok antena Nike sekitar 1,4 derajat Lebar balok kecil (sempit) pada antena pelacak adalah hal yang baik, memberi: - lebih banyak energi radar Pada target dan keuntungan (kisaran yang lebih baik) - penentuan sudut sasaran yang lebih baik - resistensi yang lebih baik terhadap geseran sumbu belakang Antena pelacak Nike memiliki lebar balok sekitar 1,4 derajat, yaitu sebagian besar daya pemancar terkonsentrasi di sekitar 1,4 derajat dan 1,4 Derajat tinggi Meskipun TRR (Target Ranging Radar) memiliki panjang gelombang yang lebih pendek, namun tidak digunakan untuk penentuan sudut. Radar Pelacakan Target Ajax Pelacakan radar Nike Ajax sebelumnya memiliki jangkauan efektif sekitar 50 mil. Ikhtisar Antena ini (empat gambar berikutnya) ada di Historical Electronics Museum near Ft. Meade, MD. Tempat yang bagus untuk dikunjungi -)) Ini menggunakan jenis lensa Fresnel yang fokus seperti ini. Boresighting identik dengan antena Pelacak Hercules yang kemudian. Leveling tentu saja merupakan masalah besar. Pelacakan Pelacakan Pelacakan dan Pelacak Rudal HARUS memiliki referensi vertikal yang umum. Dua tingkat di sudut kanan digunakan untuk kenyamanan. Ini adalah satu kaki dukungan tripod dari antena pelacak. Tutup penutup kepala hex baut 1 inci untuk diputar ke level antena. Saya benar-benar terkesan dengan kelancaran rotasi di azimuth. Bantalan itu tidak memiliki permainan yang jelas, tapi mudah dan lancar untuk bergerak. Pada tahun 2012 saya bertanya kepada Kennith Behr tentang ini. Dia mengatakan ini adalah Bearing Kaydon dan memberikan gambar ini. Menunjukkan evolusi bentuk Fresnel antena pelat pelat logam. Bab 3, ANTENA LEMBAR METAL-PLATE oleh Paul Wade. Ini berisi diagram berikut. Diskusi tentang Lensa Delay Metalic, seperti yang digunakan oleh radar pelacakan Nike Ajax, dipresentasikan dalam terbitan Bell System Technical Journal ini. Bell and Western Electric (anak perusahaan) merancang dan membangun peralatan Nike IFC. Sebuah foto Majalah Hidup - di Red Canyon - mungkin pasukan menembaki peralatan Nike Ajax mereka sebelum membawanya ke beberapa tempat di beberapa kota. Kemungkinan lainnya adalah pasukan kembali menembak ulang api tahunan dari peralatan penduduk. Radar Pelacak Hercules Radar Pelacak Hercules Nike memiliki jangkauan maksimum 200.000 yard, sedikit di atas 110 mil. (Ini adalah batas sulit karena tampilan pelacakan dan penskalaan komputer memiliki batas itu.) Untuk perbandingan yang menarik dengan SCR-584 sebelumnya (PDII) klik di sini. Sistem Nike Hercules memiliki dua radar pelacak target yang serupa secara eksternal. Perbedaan internal termasuk penggunaan pita frekuensi yang berbeda. Kedua radar tersebut digunakan sebagai pengganti radar biasa untuk membantu mengatasi kemacetan musuh. Berbagai strategi membuat kehidupan penyintas musuh sangat sulit. (Sistem Nike Ajax memiliki satu radar pelacakan target). Rope Goerigk Rope Goerigk) Tali dan dukungan tali hanya ada selama perawatan untuk mengurangi kecelakaan - dilepaskan selama operasi normal. Selama operasi normal, layar angin bulat mengelilingi antena untuk mengurangi kekuatan angin dan kesalahan pelacakan. Tampilan elektronik (foto kredit Rolf Goerigk) Gaston Dessornes ingin. Tahu perkiraan berat menara mobile MTR TTR (lihat gambar terlampir) Gambar berikut berasal dari pindaian TM9-1430-253-34 oleh GoogleBooks Semua elektronik praktis (dan berat) ditempatkan di basis tripod. Ini termasuk pasokan listrik, amplifier magnetik untuk menggerakkan motor penggerak. Ini adalah bagian belakang antena TTR - yang dibuat seringan praktis - Sambungan listrik, yaitu kontrol, tegangan sudut elevasi, daya, saluran frekuensi menengah. Dibuat dengan dasar melalui cincin slip. Berikut adalah rangkaian cincin slip, yang menghubungkan bagian-bagian yang berputar secara elektrik, yang digunakan baik di azimut dan yang lainnya untuk elevasi - Pemancar Azimuth adalah potensiometer sinusine presisi yang presisi yang membantu mengubah koordinat kutub (sudut) ke koordinat Cartesian (x, y): -)) Meskipun ada kompleksitas yang cukup besar, kita jarang memiliki masalah dengan radar (atau keseluruhan sistem IFC Nike). Itu sangat dirancang dan diproduksi dengan baik :-)) Cukup banyak kegembiraan untuk terus berlari. Sayangnya, Western Electric, perancang dan pabrikan telah dibongkar oleh pemerintah - Dalam situasi khusus, seperti bahaya topan atau kondisi Arktik, sebuah penutup pelindung besar disertakan yang dapat memberikan perlindungan tambahan. Lapisan ini berbentuk seperti kerang kerang yang bisa ditutup dalam kondisi sangat buruk. Lihat gambar Sisi tampilan. Quarter view dan informasi lokasi Alaska Situs Peter and Site Summit. Saya menebak (tolong perbaiki saya) bahwa jika kerang kerang ditutup, radar pelacakan tidak dapat digunakan. Dari Rolf Goerigk. Spesifikasi Radar Penargetan Pelacakan (TTR) meliputi: Pulsa Pendek (SP) 0,25 mikrodetik Long Pulse (LP) 2,5 mikrodetik Radar Receiver Cabinets Ada dua lemari, berdampingan, hampir saling cermin satu sama lain, satu untuk Radar Pelacakan Sasaran (TTR), dan satu untuk Radar Pelacakan Rudal (MTR). Pra-Modernisasi, Tabung Vakum Pintu pertama di kabinet di sebelah kiri Pelacakan Pelacakan Target masuk ke dalam Pelacakan Radar Penerima Radar. Kabinet yang diperlihatkan adalah untuk Hercules, tapi Ajax sangat mirip. Pada suatu waktu saya bisa memberi tahu Anda fungsi dan cara menyesuaikan semuanya di sini - Sasis di pintu adalah Chassis Uji, digunakan untuk menguji fungsi yang benar dari segala sesuatu di kabinet. Pintu tertutup di sebelah kiri sangat mirip dengan radar pelacak Rudal. Masing-masing berisi amplifier 60 MHz Intermediate Frequency (IF) untuk Sum, Elevation Error dan Azimuth Error, Automatic Gain Control untuk amplifier IF, Range Unit, sirkuit yang memungkinkan pelacakan otomatis pada elevasi, azimuth dan range, dan Test Panel yang juga dapat mengendalikan Bor situs tiang elektronik. Post-Modernisasi, Solid State, Transistors Semua informasi dan gambar dari Pak Ramiro Carli Ballola Im mengirimkan beberapa foto yang relevan dengan BC Van yang sebenarnya terletak di Base Tuono, ini adalah konfigurasi terakhir yang dibekukan pada tahun 2005 (tidak ada lagi modifikasi yang diramalkan atau disahkan) Untuk semua negara yang berpartisipasi dalam WSPC, Seperti yang telah saya jelaskan, tiga negara yang tersisa itu adalah ITALIA-YUNANI dan TURKEY, akhir tahun 2005 YUNANI mulai memudarkan sistem ini, diikuti oleh ITALIA pada tahun 2007, setelah itu WSPC memasuki fase likuidasi. Untuk kedua sub-menu MTRTTR dimulai dari daftar teratas item adalah sebagai berikut: 1) Beacon AFC 2) JIKA penguat SUM 3) JIKA penguat AZ 4) JIKA penguat El 5) Penguat video dan AGC 6) Kesalahan Servo Konverter AZ 7) Kesalahan Servo Konverter EL Pada dua pintu, baik MTR atau TTR dari daftar teratas adalah sebagai berikut: 1) JIKA generator uji (seperti pelat tua) 2) Error voltage Monitor (seperti chassy lama) 3) Error Error Modultaor 4) EL Error Bagian TTR Modulator sisi kanan Filter LPSP dari atas: 1) SUM LP dan filter SP 2) Filter AZ LP dan SP 3) EL LP dan SP menyaring bagian MTR sisi kiri Filter SP dari atas: 1) Sum 2) AZ 3) El Relatif ke rangkaian LIN LOG, masih dari atas: 1) IF Amplifier 2) Filter Pulse Panjang 3) Penguat Lin Log Masalah Multi-jalur Bekerja-sekitar OK - Anda bisa mendapatkan sebagian besar dari buku radar yang serius. Sekarang untuk detail praktis yang mengerikan: - ((Gelombang radar hanya gelombang gelombang frekuensi rendah SANGAT. Ini adalah kabar baik dan buruk. Kabar baiknya adalah bahwa gelombang radar memantulkan permukaan konduktif, seperti pesawat terbang. Kabar buruknya Adalah bahwa gelombang radar memantulkan permukaan yang Anda inginkan, tidak seperti bumi atau air (dengan konstanta dielektrik yang berbeda relatif terhadap udara) antara Anda dan pesawat terbang yang ingin Anda lacak. Berikut adalah dua cerita satu cerita dan cerita lain yang melibatkan Pelacakan Rudal Penglihatan Radar - Pada refleksi rudal, ini adalah visualisasi masalah dan grafik yang menunjukkan kesalahan pelacakan pada pesawat dengan ketinggian yang tidak ditentukan dengan radar pada ketinggian dan frekuensi yang tidak ditentukan dengan lebar balok sekitar 3 kali panjang gelombang Nike 3 sentimeter. Berlaku untuk sekitar 3 halaman lagi tentang efek ini Sampel di atas adalah dari edisi 1980 dari Introduction to Radar Systems oleh Merrill I Skolnik. Sementara mencoba menghubungi penulis untuk izin menerbitkannya, (belum ada kontak) Saya menemukan bahwa dia masih hidup, masih mengajar dan menulis. Menurut radarcon2008.orgbioSkolnik. html. Merrill Skolnik menjabat sebagai pengawas Divisi Radar di Laboratorium Riset Angkatan Laut AS di Washington, D. C. dari tahun 1965 sampai 1996. dan memiliki segala macam penghargaan. Jika Anda benar-benar tertarik dengan radar modern (termasuk Lidar), Anda harus mendapatkan salah satu bukunya yang lebih modern. Paragraf di atas menunjukkan alasan bahwa misi Nike melawan pesawat terbang yang tinggi, dan beberapa motivasi untuk mengembangkan rudal homing seperti HAWK, dan metode yang lebih canggih di PATRIOT. Good News Bagan dapat ditafsirkan untuk menunjukkan bahwa kesalahan antara dua radar jarak dekat, yang terlibat dengan sudut ketinggian elevasi yang rendah, akan memiliki kesalahan serupa - bahwa kesalahan MTR yang melacak rudal ke Titik Intercept akan mendapat kompensasi sebagian. Oleh kesalahan Target Tracking Radar yang melacak target ke titik mencegat yang sama. - Dua kesalahan sebagian besar dibatalkan - Berita Buruk Jika Radar Pelacakan Rudal melacak bayangan atau bayangan - ini akan kehilangan jejak rudal, dan tidak ada cara praktis untuk mendapatkan kembali jalur sebelum rudal merusak diri setelah kehilangan sinyal pelacakan dan pelacakan dari MTR . Good News -)) Efeknya diketahui dan dikoreksi sampai dinilai tidak menimbulkan masalah. Ini biasanya mudah di laporan Rolf Goerigk A. S. dari Jerman Contoh yang bagus adalah prosedur penggunaan MSL harian. Karena sudut penggembalaan antena yang biasanya rendah, efek multipath dipantau di ketinggian. Dengan menggunakan NIKE amplitudo monopulse tidak ada obatnya. MSL dinilai tidak operasional. Bayangkan saja, melihat melalui teleskop yang terpasang melihat beberapa sapi dan bukan rudal yang dilacak secara otomatis. Tetapi titik pandang atau titik arahan bukanlah titik pantulan, namun itu adalah jumlah energi RF langsung dan tidak langsung dari MSL. Titik refleksi berubah seiring musim dan siang hari dan kadang-kadang hilang sama sekali. Itu adalah kasus yang sangat menarik dan saya belajar banyak. Saya terlibat (lagi) dalam cerita rahasia itu. Kasus itu benar-benar panas dan sensitif. Hal ini dibahas di beberapa buku radar. Solusi masa perang Memindahkan truk antara MTR dan MSL, ia bekerja Radar Aiming Alignment termasuk boringighting Radar alignment cukup kompleks. Setiap radar (target tracking missile tracking) harus diratakan secara individual dan boresight. Kemudian kedua radar tersebut harus diselaraskan sehingga potensiometer mereka membelah yang sama saat keduanya menunjuk ke arah yang sama. Kemudian perbedaan posisi radar pelacak rudal dari radar pelacak target harus ditempatkan di komputer untuk koreksi itu. Ini adalah tiang situs bor (15 K byte). Gelombang radar sinyal uji keluar dari tanduk pakan di bagian tengah X di atas, dan 4 sisi benda X di atas adalah target optik untuk teleskop di radar (TTR, TRR, dan MTR). Lebih detail disini Klystron refleks kecil, seperti ini, digunakan untuk menghasilkan pulsa gema melalui tanduk umpan ke antena pelacakan yang boresight. Sensitivitas penerima radar juga bisa diperiksa dengan menipiskan output tabung ini. Tabungnya sekitar 3 inci, 8 sentimeter, dari dasar tabung sampai atas. Pin panjang di bawah alas adalah struktur output koaksial yang memberi panduan gelombang. Ini adalah tiang situs bor (34 K byte) yang diturunkan. Bila di tempat, tiang panjangnya vertikal. Gambar dari Rolfs NIKE Halaman oleh goerigkonlinehome. de. Dalam mode daftar, ini dapat diatur menjadi beberapa langkah utama berikut: Perataan individu (semua radar pelacakan) Instrumen penunjuk tingkat adalah versi ruggedized dari tingkat insinyur Presisi yang terpasang pada bagian antena yang berputar. Penyelarasan instrumen diperiksa dengan meratakan antena kemudian memutar antena 180 derajat dan memastikan bahwa instrumen masih menunjukkan tingkatnya. Jika tidak, Anda menyesuaikan kembali level tersebut sampai menunjukkan kesalahan yang sama ke depan dan ke belakang. Anda kemudian kembali meratakan antena dan mengeceknya lagi. (Penyesuaian ini cukup stabil.) Tingkat antena pada umumnya tidak stabil. Individu bore sighting (semua radar pelacakan) Nyalakan Uji Bore Sight Test Mast Osilator - Saat alat uji mast control box senses pulsa X-Band, - akan menghasilkan pulsa echo pada frekuensi yang sama dari tanduk umpan Mengunci osilator Melacak secara otomatis Teleskop ke posisi mount, posisi 1, amati target Tetapkan teleskop ke posisi semula, atur 2, amati target Sesuaikan teleskop (bukan radar) untuk pemandangan yang benar Matikan Uji Bore Sight Test Mast Osimina rudal rudal dan teleskop pelacakan target saling bertukar silang Sesuaikan keberpihakan potensiometer Pastikan posisi radar pelacakan rudal diimbangi di komputer Penyesuaian leveling adalah yang paling menyulitkan di banyak situs baru. Ini akan melayang sedikit (memerlukan releveling beberapa kali sehari) sampai bantalan beton itu terjatuh ke tanah. Kemudian penyesuaian tidak akan diperlukan lebih dari satu kali dalam satu hari. Hal ini menghasilkan akurasi penglihatan yang nyata antara Radar Pelacakan Rudal dan Pelacak Target radar sekitar 1,5 inci dalam seribu meter (dengan asumsi tiang stasiun bor berada sekitar 250 meter dari radar ). Pada jarak 110 mil (200.000 kaki) yang tingginya sekitar 300 inci atau setinggi 25 kaki. Masih banyak lagi sumber kesalahan di sistem - tentu saja - namun sistemnya sangat akurat. Dalam ukuran sudut, kesalahan boresight sekitar 0,0025 derajat, atau sekitar 10 detik busur. Ukuran sudut bintang di teleskop kuat di bumi sekitar 1 detik busuk karena masalah atmosfer. Untuk perbandingan yang menarik dengan SCR-584 sebelumnya (PDII), klik di sini. Frank E. Rappange menunjukkan bahwa ada cek yang menunjukkan bahwa semua penyesuaian itu BENAR-BENAR BEKERJA. . Tes utama (yang harus dilakukan setiap 6 jam, saat di 30 SOA) adalah Uji Pelacakan Simultan. Dalam tes ini MTR diatur ke mode track kulit dan kedua TTR (dan TRR) dan MTR terkunci pada target yang sama. BCO bisa membaca perbedaan voltase untuk posisi radar masing-masing di BC Van. Readings were made for both TTR and TRR in the difference pulse modes. It was the decision of the BCO to accept the system or not. Radar Range Determination Radar waves (and light and other electromagnetic waves) travel through air almost as fast as in a vacuum. Fortunately for engineers and users, air pressure, humidity, and other atmospheric variables do not affect the speed of travel very much. To make matters even easier for the Nike problem, any variations that do occur are largely canceled out at the end of the flight, as both the radar beams are traveling through very similar air conditions. Errors due to refractive effects due to differences in air pressure along the beams cancel out. So a common crystal oscillator was used to calibrate the range systems of both the missile and target radars. This adjustment was fortunately very stable, rarely needed tweaking unless a component was changed. There is a circuit called a phantastron that has a remarkably linear pulse delay time from a voltage input. The Range Operator (or range tracking servo system) operates a linear potentiometer which provides the range voltage for: Elevation potentiometers, see Height Determination below Azimuth potentiometers, see Radar Azimuth Determination below Range gate for displays and for servo gating This diagram came as a shock when I was looking through technical manuals in 2015. I had never seen it in trainning nor on site. It must have been discussed on one of the days I was on KP. (The Army had a bad habit of making their slave wage students take their turns doing KP (Kitchen Police, washing pots and pans, mopping the mess hall. ) during technical trainning. The Air Force is much more enlightened, hiring civilians to do kitchen chores instead of techie students.) Good thing the range units didnt fail on our site, would have taken me some time to fix things, or call in ordnance. (There were two range units, one for the Target Tracking, The other for the missile tracking.) Maybe Lopresti or Sizlak (the other two IFC techies) didnt have KP on their sequence. Radar Height Determination Digital - post-modernization - from Ramiro Carli Ballola Note: during the post 1975 modernizations, including replacing many analog components with digital components, the elevation trig potentiometers were replaced by digital angle resolvers. Here is an explanation of a digital angle resolver. The output was sent to a little digital computer in the RC van where the height was computed from the digitalslantrange times the sin of this angle and the groundrange determined from the cosine of this angle. Gathering the details, and educating me (Ed Thelen) is an on-going effort (November 2015) by Ramiro Ballola :-)) Please be patient, these were major philosophical, data flow, and processing changes. going back to the RAEMOD, with the change of the potentiometers in the antennas, in the exploded view photo included at nr 26 you should see fisically the optical resolver inside the TTR azimuth encoder assembly, they were the same on the elevation and equally the same for TTRMTRTRR Inside the functional schematic photo you should see a little part of the RSPU Angle encoder section and you should read the input from synchro and resolver from the antenna and the first data conversion, to be sent to the Coordinate Converter Section (via PCS), than to the TDP (Track data processor) and to the Digital computer in the BC Van Ed Thelen here - The above two diagrams provide fascinating hints of the digitized (azimuth and elevation) angle data sent by the antenna circuitry to the RC van to help provide X, Y, and Z information of the radar target. Ramiro is continuing to collect photos, diagrams, and information. from Ramiro Nov 19, 2015 Ed, if you remember after boresight check before the Orientation, one check was mandatory to be performed and it was the KDP (Known Datum Point) to define the TTR (considered as System center) azimuth position respect to the North Geographic, this value recorded inside the TTR RSPU, together with the Orientation Elevation position and the Range zero check value SP and LP, was the reference you recall to intialize the system. Of course in the MTR RSPU the azimuth reference value was the Orientation value. Ed Thelen here - None of the above, except boresighting - done daily -. and determination of north - done once on Ajax sites -, is unfamiliar to this Ajax techie. I (and this description) have a long way to go - we had no RSPU to initialize - Radar Azimuth (horizontal direction) Determination Digital - post-modernization Note: during the post 1975 modernizations, including replacing many analog components with digital components, the azimuth trig potentiometers were replaced by a digital angle resolver. Here is an explanation of a digital angle resolver. The output was sent to a little digital computer in the RC van where the groundrange and angle were used to compute the E-W and N-S ground values. height was computed from the digitalslantrange times the sin of this angle. Tracking Radar Physical Support One of the many keys to precision tracking between the target and missile tracking radars is the fact that (small) identical errors of tracking by both the target and missile tracking radars cancel out. Example, if both the target and missile tracking radars say that their respective tracks are both 100 yards higher than absolute height, the actual miss distance (if every thing else was perfect) would be 0 yards. Very Interesting and Useful This way, errors due to radar wave (like light wave) refraction in the atmosphere cancel out if both radars are tracking the same point in space (in this discussion we ignore the slightly different paths due to the slightly different physical location of the two radars. The Nike Ajax system assumed that wind buffeting of the two tracking radars would be sufficiently similar so that accurate enough tracking could be accomplished. Since the Nike Hercules had an effective range more than 3 times the Ajax, and a real range more than 4 times the Ajax, errors due to wind buffeting and similar errors could be 3 or 4 times larger, and possibly render Hercules ineffective (too inacurate) at longer ranges. Bubbble surrounds each tracking antenna To counter the wind buffeting, the tracking radars were enclosed in an air inflated fabric bubble. This greatly reduced the wind forces on the tracking antennas. Even if the wind gust shifted the bubble a few inches, the air f orces on the antennas would be greatly reduced during the shift of the bubble. The bubble also protected the antenna from much of the differential heating due to the sun heating (expanding) one side of the mount and antenna relative to the other side (shady side) of the mount and antenna. Although both tracking antennas would likely be illuminated by the sun the same way, vertical alignment was usually made by one person at slightly different times (an error source) and one was never confident that everything was identical anyway. Wind Force and Sun Heating on Tower Mount Ideally, the radars could be located on high ground, well above surrounding trees, buildings. However, in flater areas, towers had to be used to get the tracking radars high enough. The wind also supplies forces and torques on radar towers. The forces and especially the torques shift the top of the tower in space, and shift its angle with the vertical. The shift in space (inches) is much much smaller than other errors, but the shift in angle from vertical could result in much more severe errors. To provide improved resistance to angle errors due to torque in the tower, the tower was actually a double tower. The outer tower was buffeted by the wind, and also the differential expansion due to the sun light heating it. The platform at the top of the outer tower also supported the bubble that protected the antenna from the wind. The inner tower supported the antenna. The inner tower was largely isolated from the wind and the sun which resulted in much more stablity. Image of tower showing: - outer tower platform - bubble base - foot pad from inner tower Simultaneous Tracking Test (the proof) Did all of the above boresighting adjustments and alignments REALLY yield a system that could get a missile within kill distance of the target There is a way to check Have BOTH the target tracking radars and the missile tracking radar track the same target (aircraft). If both radars say the aircraft is in the same place . the tracking system is correctly aligned. Period . No guess work, no theory, no it oughta, the tracking system IS correctly aligned . Assuming the computer works, the missile takes commands, etc. that NIKE system is capable of guiding the missile to the target However, you remember that the Missile Tracking Radar (MTR) tracks a beacon in the missile, not the skin of the missile. So, (FOR THIS TEST) the MTR is set to: a mode to track the MTR radar reflection from a target, not the missile beacon remove the delay of the beacon (a fixed delay between receiving the MTR radar signal and the firing of the beacon) from the MTR range system The above two changes permit MTR to track the aircraft just the same as the TTR system. An aircraft flies about, and the computer voltages representing N-S, E-W, UP-DOWN for the MTR and the TTR are compared. They ideally should be identical. Placing a sensitive volt meter between say the target radar N-S and the missile radar N-S should ideally yield zero at all times while tracking the same aircraft. In practice they rarely are completely identical due to at least the following error sources. different parts of the aircraft reflect (glint) differently at different angles different pointing servo gains and damping actual level errors actual boresight and alignment errors errors in the range, elevation, azimuth potentiometers errors in components in the computer a wide variety of mechanical errors such as binding, looseness. In spite of the above long list of possible error sources, people at NIKE sites had to and did prove - frequently - that the tracking system errors were very few yards at ranges in excess of 50 miles. Unbelievable but true Simulated Tracking (and jamming) using the T-1 System Tracking aircraft with a NIKE system is trivial if the aircraft is not using jamming . With no jamming, you can easily teach your junior high school kid to be a good NIKE radar operator in an afternoon. A group of afternoon trained junior high kids could do all the NIKE aircraft radar tracking operations necessary to shoot down a non-jamming aircraft. The airforces of the world spend a great deal of time and money to try to defeat radar - and many interesting jamming methods have been developed and are used. How do you train NIKE people to track aircraft that are using Electronic CounterMeasures - ECM (jamming) How do you maintain and enhance this difficult skill Using friendly aircraft for this training and skill maintenance has many disadvantages, including: The friendly air force is unlikely to wish to fly aircraft for hours per day around the various sites to assist in trainning and honing the friendly anti-aircraft forces. Occasional tests may be OK, but almost every day The friendly air force may not wish to turn on their latest and greatest ECM (jamming) equipment for analysis by non-friendly agents. I presume these, and other, reasons led to the development of the ANMPQ-T1 Electronic Warfare Simulator (developed by ITT Baltimore, MD ) which was housed in one very large trailer. The operators in the T-1 trailer could exercise the radar operators in both the Battery Control (BC) van (acquisition operators and battery commander) and the Radar Control (RC) van (Target Tracking operators (azimuth, elevation, range) and Missile Tracking operator. with many types and quantities of interesting ECM (jamming) problems. Jammingspoofing slides from the archives of Association of Old Crows aoc. adobeconnectjammingtechniques5-1-14recordinglauncherfalsefcsContenttruepbModenormal (The introductory part, relative to the pulsed, non-coherent techniques used in Nike) With the Improved Nike Hercules, the jammerspoofer had to fool two different frequency radars in range. Note the attempt to both: a) obscurehide the target b) fool the range operatorsystem to track a fake target Also note: this does not include mechanical jamming, such as chaff, corner reflectors, decoys. This is where ECM was for Nike Ajax and modern techniques are much more interesting. For more details on the T-1 unit, see Lesson 8. Target Simulation - 1.2 megabytes There is a T1 manual on-line at T1 ANMPQ-T1 (another site)(.zip -.pdf files)(10 files totaling 6 Mbytes) For a more general discussion of jamming, go here. LOPARHIPAR target video and ECM was created using sync and preknock signals from the radar. Antenna rotation was slaved to the radar by a device called a flying spot scanner and video was triggered using a system called an antenna pattern generator which simulated not only the main antenna lobe but side lobes as well. By changing the position of the main lobe, the target could be moved in azimuth at will and the ECM would also be positioned along the main lobe. TTR and MTR video was generated much the same as the IF test pulse except that the TTR was given sum, azimuth and elevation signals and long and short pulse. Azimuth and elevation signals were controlled through servos which were slaved to the antenna and range simulation was done by delaying the video from sync. The system could generate 6 independent targets with 4 types of Electronic jamming on 4 different carriers in addition to Acq and track chaff and angle deception. Army Navy Mobile Radar Signal Simulator. I worked on them for about 10 years, a great training device. Simulated up to 6 targets, ECM, and ground clutter. The Chaff cabinet was a bitch to maintain. The T1 also had a reusable missile. The T1 was heat sensitive and the IF strips had to be retuned as the trailer got warmer. The Simulations were injected at the RC BC Vans not at the radars. First, let me say something about analyzing the ECMECCM situation for Nike or any other system. That is somewhat akin to painting a moving train. Technology advances tend to make the advantage shift between ECM and ECCM. So, Im not surprised that some point in time the SSKP was as low as 85. However, I have watched as Air Force planes tried to break Hercules System lock after the TRR was added to the system and they could not do that. I have even read letters from Air Force organizations requesting that Nike not track their planes using their ECCM, because it tended to undermine the confidence of their pilots in their ECM equipment. I dont know the time period that the simulations were done. Ability to do meaningful simulations developed as technology did, so the sophistication (accuracy) of the simulation could be called into question. Second, all simulations involve some approximations and assumptions. Thus, simulations have to go through a validation process to determine the accuracy of the simulation. I have never heard of any meaningful simulations involving the Nike Hercules systems. That doesnt mean there werent any, just that if the 20 or so years working with and around Nike, I would have expected to see something about it. I hope this helps. If you have comments or suggestions, Send e-mail to Ed Thelen Updated November 18, 2015Using R for Time Series Analysis Time Series Analysis This booklet itells you how to use the R statistical software to carry out some simple analyses that are common in analysing time series data. This booklet assumes that the reader has some basic knowledge of time series analysis, and the principal focus of the booklet is not to explain time series analysis, but rather to explain how to carry out these analyses using R. If you are new to time series analysis, and want to learn more about any of the concepts presented here, I would highly recommend the Open University book 8220Time series8221 (product code M24902), available from from the Open University Shop . In this booklet, I will be using time series data sets that have been kindly made available by Rob Hyndman in his Time Series Data Library at robjhyndmanTSDL . If you like this booklet, you may also like to check out my booklet on using R for biomedical statistics, a-little-book-of-r-for-biomedical-statistics. readthedocs. org. and my booklet on using R for multivariate analysis, little-book-of-r-for-multivariate-analysis. readthedocs. org . Reading Time Series Data The first thing that you will want to do to analyse your time series data will be to read it into R, and to plot the time series. 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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