Pierwszym krokiem w opracowaniu modelu Box-Jenkins jest określenie, czy seria jest stacjonarna i czy ma miejsce jakakolwiek znacząca sezonowość, którą trzeba modelować. Narodowość można ocenić z wykresu kolejności runnej Wykres sekwencji runnej powinien wskazywać stałą pozycję i skalę Można to wykryć z obszaru autokorelacji W szczególności, niestacjonarność jest często wskazywana na wykresie autokorelacji z bardzo powolnym rozkładem. Odróżnienie w celu osiągnięcia stacjonarności. Boks i Jenkins zalecają podejście różnicujące w celu osiągnięcia stacjonarności. Jednak dopasowanie krzywej i odejmowanie dopasowanych Wartości z pierwotnych danych mogą być również wykorzystane w kontekście modeli Box-Jenkins. Na etapie identyfikacji modelu naszym celem jest wykrycie sezonowości, jeśli istnieje, oraz określenie kolejności sezonowych okresów autoregresji i sezonowych wiele serii, okres jest znany i wystarczy jeden okres sezonowości. Przykładowo, w przypadku danych miesięcznych zazwyczaj zawieramy a sezonowy okres AR 12 lub sezonowy termin MA 12 Dla modeli Box-Jenkins nie jasno usuwamy sezonowości przed dopasowaniem modelu Zamiast tego uwzględnimy kolejność okresów sezonowych w specyfikacji modelu do oprogramowania szacującego ARIMA Jednak może to być pomocne w celu zastosowania sezonowej różnicy do danych i zregenerowania autocorelacji i częściowych części autokorelacji Może to pomóc w identyfikacji modelu nie-sezonowego składnika modelu W niektórych przypadkach sezonowe różnice mogą usunąć większość lub całość efektu sezonowości. Identyfikacja p i q. Każdy stacjonarność i sezonowość zostały omówione, następnym krokiem jest zidentyfikowanie kolejności, tj. P i q wartości autoregresywnych i średnich ruchów. Kompozycje autocorelacji i częściowej autokorelacji. Pierwszym narzędziem do tego jest wykres autokorelacji oraz częściową sekwencję autokorelacji Przykładowy wykres autokorelacyjny i próbka częściowego autokorelacji porównano z teoretycznym zachowaniem z tych działek, gdy zlecenie jest znane. Przedstawem procesu autoregresji p. Nieznacznie, w przypadku procesu AR1, funkcja autokorelacji próbki powinna wykazywać się wykładniczo malejącym wyglądem Jednakże procesy AR wysokiej generacji są często mieszaniną wykładniczo malejących i zwilżonych sinusoidalnych dla autoregresji wyższego rzędu próbka autokorelacji próbki musi być uzupełniona fragmentem autokorelacji częściowej Częściowa autokorelacja procesu AR p staje się zera w punkcie p 1 i większa, więc zbadamy przykładową część autokorelacji, aby sprawdzić, czy istnieje jest dowodem na odejście od zera Zazwyczaj jest to określone przez umieszczenie przedziału ufności 95 na próbce częściowej części autokorelacji większość programów generujących próbkowe wykresy autokorelacji będzie również wykreślać ten przedział ufności Jeśli program nie generuje pasma zaufania, to około 2 m 2, przy czym N oznacza wielkość próbki. Średnia roczna Proces q. Funkcja autokorelacji procesu MA q staje się zerowa przy opóźnieniu q 1 i większa, więc zbadamy przykładową funkcję autokorelacji, aby zobaczyć, gdzie jest zasadniczo zero Czytamy to przez umieszczenie 95 przedziału ufności dla funkcji autokorelacji próbki na przykładowa struktura autocorelacji Większość programów, które mogą wygenerować wykres autokorelacji może również generować ten przedział ufności. Przykładowa, częściowa funkcja autokorelacji nie jest ogólnie przydatna do określenia kolejności przebiegu średniej ruchomej. Kształt funkcji autokorelacji Poniższa tabela zawiera podsumowanie sposobu wykorzystania przykładowa funkcja autokorelacji dla identyfikacji modelu. Miłości średnie - proste i wykładnicze. Średnie przeciętne - proste i wyrównane. Średnie wygody wygładzają dane o cenach do postaci wskaźnika po wskaźniku Nie przewidują kierunku ceny, ale raczej określają bieżący kierunek z opóźnieniem Przeprowadzka średnie opóźniają, ponieważ są oparte na wcześniejszych cenach Pomimo tego opóźnienia, przesuwając a Wady pomagają gładko działać cenowo i odfiltrować hałas Poza tym tworzą bloki wielu innych wskaźników technicznych i nakładek, takich jak pasma Bollingera MACD i oscylator McClellan Dwa najbardziej popularne typy średnich kroczących to: Simple Moving Average SMA i Exponential Przenoszenie średniej EMA Te średnie ruchome mogą być wykorzystane do określenia kierunków trendu lub określenia potencjalnego poziomu wsparcia i oporu. Oto wykres z zarówno SMA, jak i EMA na tym wykresie. Kliknij wykres, aby wyświetlić wersję na żywo. Przeprowadzić obliczanie średniej ruchomej. Prosta średnia ruchoma jest obliczana poprzez obliczenie średniej ceny zabezpieczenia w określonej liczbie okresów. Większość średnich kroczących opiera się na cenach zamknięcia. 5-dniowa prosta średnia ruchoma to pięciodniowa suma cen zamknięcia podzielona przez pięć. Jak sama nazwa wskazuje , średnia ruchoma jest średnią, która porusza Stare dane są pomijane w miarę pojawiania się nowych danych To powoduje, że przeciętny czas przemieszcza się wzdłuż skali czasowej Poniżej znajduje się przykład 5-da r średnia ruchoma zmienia się przez trzy dni. Pierwszy dzień średniej ruchomej obejmuje po prostu ostatnie pięć dni Drugi dzień średniej ruchomej zmniejsza pierwszy punkt danych 11 i dodaje nowy punkt danych 16 Trzeci dzień średniej ruchomej trwa dalej upuszczenie pierwszego punktu danych 12 i dodanie nowego punktu danych 17 W powyższym przykładzie ceny stopniowo zwiększają się z 11 do 17 w ciągu siedmiu dni Uwaga, że średnia ruchoma również wzrasta od 13 do 15 w trzydniowym okresie obliczeniowym że każda średnia ruchoma jest tuż poniżej ostatniej ceny Na przykład, średnia ruchoma dla pierwszego dnia wynosi 13, a ostatnia cena wynosi 15 Ceny poprzednich czterech dni były niższe i powoduje to, że średnia ruchoma jest niższa. Obliczanie średniej ruchomejExponential średnie kroczące zmniejszają opóźnienie, stosując większą wagę do ostatnich cen Waga zastosowana do najnowszej ceny zależy od liczby okresów w średniej ruchomej Istnieją trzy kroki do obliczania mnożona średnia ruchoma Najpierw obliczyć prostą średnią ruchoma Średnia geometryczna wykładnicza EMA musi zaczynać się gdzieś tak, tak jak w poprzednim okresie jest używana prosta średnia ruchoma EMA w pierwszym obliczeniu Drugie obliczenie mnożnika ważącego Trzecie obliczenie średniej ruchomej wykładniczej poniższa formuła jest na dziesięciodniową EMA. A 10-godzinna wykładnicza średnia ruchoma ma zastosowanie do 18 18 ważenia do ostatniej ceny EMA 10-EMA może również być nazwana EMA 18 18 EMA 20 z 20-krotnym ważeniem do ostatniej ceny 2 20 1 0952 Należy zwrócić uwagę, że ważenie krótszego okresu czasu jest większe niż ważenie przez dłuższy okres czasu W rzeczywistości spadek wagi o połowę za każdym razem, gdy średni okres przeciętny jest podwojony. Jeśli chcesz, aby nasi określonego procentu dla EMA, można użyć tej formuły, aby ją przeliczyć na okresy czasu, a następnie wprowadzić tę wartość jako parametr EMA. Poniżej przedstawiono przykład arkusza kalkulacyjnego 10-dniowej prostej średniej ruchomej i dziesięciodniowej wykładniczej mo średnia dla Intel Proste średnie kroczące są proste i wymagają niewielkiego wyjaśnienia Średnia dziesięć dni po prostu porusza się w miarę pojawiania się nowych cen, a stare ceny spadają Średnia wykładnicza zaczyna się od prostej średniej ruchomej 22 22 w pierwszym obliczeniu Po pierwsze obliczenie, normalna formuła przejmuje Ponieważ EMA rozpoczyna się od prostej średniej ruchomej, jego prawdziwa wartość nie zostanie zrealizowana do 20 lub więcej okresów później Innymi słowy, wartość w arkuszu kalkulacyjnym excel może się różnić od wartości wykresu z powodu krótki okres zwrotu Poniższy arkusz kalkulacyjny kończy się tylko 30 okresami, co oznacza, że wpływ prostej średniej ruchomej miało 20 okresów, aby rozproszyć StockCharts co najmniej 250-krotne okresy zazwyczaj znacznie dalszy dla jego obliczeń, więc skutki prostej średniej ruchomej w pierwszym obliczeniu zostały w pełni rozproszone. Czynnik Lag. Im dłużej średnia ruchoma, tym bardziej opóźnia się 10-dniowa średniej ruchomej średniej będzie hu g ceny dość blisko i skręcają wkrótce po obniżce cen Krótkie średnie ruchy są jak łodzie szybkości - zwinny i szybki do zmiany W przeciwieństwie do 100-dniowej średniej ruchomej zawiera wiele poprzednich danych, które spowalniają Dłuższe średnie ruchome są takie jak zbiorniki oceaniczne - letargiczne i powolne do zmiany Większy i dłuższy ruch cenowy dla 100-dniowej średniej ruchomej zmienia kurs. Kliknij na wykresie na żywo. Wykres powyżej pokazuje SP 500 ETF z 10-dniową EMA ściśle według cen i 100-dniowa SMA mieląca wyższa Nawet po spadku z stycznia do lutego, 100-dniowa SMA odbyła kurs i nie skręciła 50-dniowa SMA mieści się gdzieś pomiędzy średnim ruchem 10 i 100 dni, jeśli chodzi o współczynnik opóźnienia . Średni ruch liniowy vs średnie. Mimo że istnieją wyraźne różnice między prostymi ruchowymi średnimi a średnimi przesuwnymi, niekoniecznie lepsze niż inne średnie ruchowe wykładnicze mają mniej opóźnień, a tym samym są bardziej wrażliwe na powtarzalne centa - i ostatnie zmiany cen Mnożące się średnie kroczące obróci się przed średnimi ruchami Proste średnie ruchome z drugiej strony stanowią prawdziwą średnią cen dla całego okresu czasu W związku z tym proste średnie kroczące mogą być lepiej dostosowane do identyfikacji wsparcia lub poziomy oporu. Średnia preferencja zależy od celów, stylu analitycznego i horyzontu czasowego Wykresy powinny eksperymentować z oboma typami średnich kroczących, jak również różne ramy czasowe w celu znalezienia najlepszego dopasowania Poniższy wykres przedstawia IBM z 50-dniowym SMA na czerwono i 50 EMA na zielono Zarówno osiągnęły szczyt pod koniec stycznia, ale spadek EMA był ostrzejszy niż spadek SMA EMA pojawiła się w połowie lutego, ale SMA nadal spadała do końca marca Zawiadomienie, że SMA zwróciła się miesiąc po EMA. Lengths i Timeframes. Długość średniej ruchomej zależy od celów analitycznych Krótkofalowe średnie 5-20 okresów najlepiej nadaje się do krótkoterminowych trendów i tradi Chartistów interesujących się średnio - terminowymi trendami wybierają dłuższe średnie ruchy, które mogą wynosić 20-60 okresów Długoterminowe inwestorzy wolą ruszać średnio 100 lub więcej okresów. Kilka średnich ruchomej długości jest bardziej popularne niż inne 200-dniowa średnia ruchoma jest chyba najbardziej popularna ze względu na długość, jest to wyraźnie długoterminowa średnia ruchoma Następna średnia średnica ruchów 50-dniowych jest dość popularna w średnim okresie. Wielu chrześcijan używa 50-dniowego i 200-dniowego ruchu średniego razem Krótkotrwała średnia ruchoma w ciągu 10 dni była dość popularna w przeszłości, ponieważ łatwo było obliczyć Jeden po prostu dodał liczby i przesunął punkt dziesiętny. Identyfikacja zlecenia. Te same sygnały mogą być generowane przy użyciu prostych lub wykładniczych średnich kroczących Jak zauważyliśmy powyżej, preferencja zależy od każdej z tych osób Poniższe przykłady wykorzystują zarówno proste, jak i wykładnicze średnie ruchome Określenie średnia ruchoma dotyczy zarówno prostych, jak i wykładniczych ruchów średnich. Kierunek movin g średnia przekazuje ważne informacje o cenach Wzrastająca średnia ruchoma wskazuje, że ceny są na ogół rosnące Spadająca średnia ruchoma wskazuje, że średnio spadają ceny Rosnąca długoterminowa średnia ruchoma odzwierciedla długoterminową tendencję wzrostową Spadek długoterminowej średniej ruchomej odzwierciedla długoterminowy spadek. Na wykresie przedstawiono 3M MMM z 150-dniową średnią ruchową średnią. Ten przykład pokazuje, jak dobrze działają średnie ruchome, gdy trend jest silny. 150-dniowa EMA odrzucona w listopadzie 2007 r. i ponownie w styczniu 2008 r. Zauważ, że zajęło 15 spadek odwrócenia kierunku tej średniej ruchomej Te wskaźniki opóźniające wskazują tendencje do odwrócenia w miarę ich wystąpienia w najlepszym razie lub po wystąpieniu najgorszego MMM w dalszym ciągu niższego w marcu 2009 r., A następnie wzrosły o 40-50 Zauważ, że 150-dniowy EMA nie pojawiła się dopiero po tym napływie Kiedy to nastąpi, MMM kontynuował wyższe w ciągu najbliższych 12 miesięcy Ruch średniotem pracy wspaniale w silnych trendach. Double Crossovers. Two średnie ruchome c wykorzystywane razem do generowania sygnałów krzyżowych W analizie technicznej rynków finansowych John Murphy nazywa to podwójną metodą krzyżową Podwójne przejazdy obejmują jedną stosunkowo krótką średnią ruchową i jedną stosunkowo długą średnią ruchoma Podobnie jak w przypadku wszystkich średnich kroczących, ogólna długość średniej ruchomej definiuje ramy czasowe systemu System wykorzystujący 5-dniową EMA i 35-dniową EMA uznaje się za krótkoterminową System z 50-dniowym SMA i 200-dniowym SMA uznawany byłby za średniookresowy, a może nawet długoterminowy . Przecięcie przejściowe następuje wtedy, gdy krótsza średnia ruchoma przecina powyżej dłuższej średniej ruchomej. Jest to również znany jako złoty krzyż. Krzyżówka niedźwiedzia występuje, gdy krótsza średnia ruchoma przecina poniżej dłuższej średniej ruchomej. Jest to znany jako martwy krzyż. produkują stosunkowo późne sygnały Wszakże system zatrudnia dwa wskaźniki opadające Im dłuższe są przeciętne okresy, tym większe opóźnienie w sygnałach Te sygnały działają świetnie, gdy trwa dobry trend Trzeba jednak zauważyć, że średni ruchowy system crossoveru przyniesie wiele pseudonów bez silnego trendu. Jest też potrójna metoda krzyżowa obejmująca trzy średnie ruchome. Znowu generowany jest sygnał, gdy najkrótsza średnia ruchoma przecina dwa dłuższe średnie ruchome Prosty trzycyfrowy system przecięcia może obejmować średnie ruchy w ciągu 5 dni, 10 dni i 20 dni. Wykres powyżej przedstawia Home Depot HD z 10-dniową zieloną linią przerywaną EMA i 50-dniową czerwoną linią EMA Czarna linia jest codziennym zamknięciem Użycie średniej ruchomych zwrotnic doprowadziłoby do trzech pchaczy przed złapaniem dobrego handlu 10-dniowa EMA złamała się pod 50-dniową EMA pod koniec 1 października, ale to nie trwało długo, jak 10 dni przeniósł się powyŜej w połowie listopada 2 Ten krzyŜ trwał dłuŜej, ale następny niedźwiedzia zwrotny w styczniu 3 pojawiły się pod koniec listopada poziom cen, co spowodowało kolejny pysk Wynik ten nie trwał tak długo, jak 10-dniowy EMA powrócił ponad 50-dniowy kilka dni l Po czwartym sygnale zasygnalizowano silny ruch w miarę wzrostu stanu zapasów w ciągu 20 lat. Są dwa starty tutaj: pierwsze, przejazdy są skłonne do whipsaw Można zastosować filtr cenowy lub czasowy, aby zapobiec szarpaniom Handlowcy mogą wymagać przecięcia na 3 dni przed działaniem lub wymagać, aby 10-dniowa EMA przemieszczała się powyżej poniżej 50-dniowej EMA o określoną wartość przed działaniem Drugie, MACD może być użyty do identyfikacji i ilościowego oznaczania zwrotów MACD 10,50,1 pokaże linię reprezentująca różnicę między dwoma wykładniczymi średnicami ruchomymi MACD staje się dodatni podczas złotego krzyża i ujemny podczas martwego krzyża Oscylator Cena Procentowa PPO może być użyta w ten sam sposób, aby pokazać różnice procentowe Zauważ, że MACD i PPO oparte są na średnich ruchach wykładniczych i średnich nie pokrywają się z prostymi średnimi. Wykres ten przedstawia Oracle ORCL z 50-dniową EMA, 200-dniową EMA i MACD 50 200,1. W okresie 2 1 2 lat T było cztery przecięcia średniej ruchomej po raz pierwszy odniósł efekty w postaci whipsów lub złych firm Trwały trend zaczął się od czwartego rozdrożu, ponieważ ORCL wzrósł do połowy lat dwudziestych Po raz kolejny ruchome przecięcia średnie działają świetnie, gdy trend jest silny, ale powoduje straty przy braku tendencji. . Średnie przeciętne mogą być również wykorzystane do generowania sygnałów z prostymi przejściami cenowymi. Wzrost sygnału generowany jest, gdy ceny przewyższają średnią ruchową. Nieprzerwany sygnał jest generowany, gdy ceny spadają poniżej średniej ruchomej Przeceny cen można łączyć ze sobą w ramach większego trendu dłuższa średnia ruchów wyznacza ton dla większego trendu i krótszej średniej ruchomej wykorzystuje się do generowania sygnałów Poszukiwanie gwałtownych krzyżów cenowych tylko wtedy, gdy ceny są już powyżej dłuższej średniej ruchomej To będzie handel w zgodzie z większym trendem Na przykład , jeśli cena przekracza 200-dniową średnią ruchoma, chartiści skoncentrują się tylko na sygnałach, gdy cena przekracza 50-dniową średnią ruchoma. spadek poniżej 50-dniowej średniej ruchomości poprzedzałby taki sygnał, ale takie krzywdy niekorzystne byłyby ignorowane, ponieważ większa jest tendencja Krzyż niedźwiedzi po prostu sugeruje pullback w większym trendzie wstecznym powyżej 50-dniowej średniej ruchomej sygnalizuje wzrost cen i kontynuację większego trendu. Następny wykres przedstawia Emerson Electric EMR z 50-dniową EMA i 200-dniową EMA. Stado wzrosło powyżej i utrzymywało się powyżej średniej ruchowej 200 dni w sierpniu. 50-dniowa EMA na początku listopada i ponownie na początku lutego Ceny szybko przeszły ponad 50-dniowy EMA, aby zapewnić sygnały o uproszczonej zielonej strzałce w harmonii z większym trendem wzrostowym MACD 1,50,1 jest wyświetlany w oknie wskaźników w celu potwierdzenia krzyżów cenowych powyżej lub poniżej 50-dniowej EMA Jednorodzona EMA jest równa cenie zamykania MACD 1,50,1 jest dodatnia, gdy wartość graniczna przekracza 50-dniową EMA i jest ujemna, gdy wartość graniczna jest niższa niż 50-dniowa EMA. Opór. Średnie mogą działać również jako suppo rt w trendzie wzrostowym i oporu w downtrendu Krótkoterminowe trenowanie może znaleźć wsparcie w pobliżu 20-dniowej prostej średniej ruchomej, która jest również wykorzystywana w zespołach Bollingera Długoterminowa tendencja wzrostowa może znaleźć wsparcie blisko 200-dniowej prostej średniej ruchomej, która jest najbardziej popularną długoterminową średnią ruchoma Jeśli rzeczywiście, 200-dniowa średnia ruchoma może oferować wsparcie lub opór tylko dlatego, że jest tak szeroko stosowany To prawie jak samospełniający się proroctwo. Gratka pokazuje NY Composite z 200-dniowa prosta średnia ruchoma od połowy 2004 r. Do końca 2008 r. 200-dniowe wsparcie udzielane wiele razy podczas wyprzedzenia Kiedy tendencja odwróciła się z podwójnym górnym złamaniem wsparcia, 200-dniowa średnia ruchoma działała jako opór wokół 9500. Nie oczekuj dokładnych poziomów wsparcia i oporu od średnich kroczących, zwłaszcza dłuższych ruchów średnich Rynki są pod wpływem emocji, co czyni je bardziej skłonne do przeoczenia zamiast dokładnych poziomów, średnie ruchome mogą być wykorzystane do identyfikacji podparcia lub oporu es. Korzyści płynące ze stosowania ruchomych średnich należy zważać na wady. Przekazywanie średnich trendów jest następujące lub opóźnione, wskaźniki, które zawsze będą krok za sobą. Niekoniecznie jest to zła rzeczą. Chociaż przecież trend jest Twoim przyjacielem i to najlepiej jest prowadzić handel w kierunku tendencji Przechodząc średnie upewnij się, że przedsiębiorca jest zgodny z obecną tendencją Choć trend jest Twoim przyjacielem, papiery wartościowe spędzają dużo czasu w zakresie handlu, co powoduje, że ruchome średnie nieefektywne Kiedyś trend, średnie kroczące pozostaną w Twoim stanie, ale także dają późne sygnały Don t spodziewają się sprzedaży na górze i kupić na dole przy użyciu średnich kroczących Jak w przypadku większości narzędzi analizy technicznej, średnie ruchy nie powinny być używane samodzielnie, ale w połączeniu z innymi narzędziami uzupełniającymi Chartiści mogą używać średnich kroczących, aby zdefiniować ogólny trend, a następnie użyć RSI, aby zdefiniować poziomy przewyższania lub przeterminowania. Dodawanie średnich ruchów do wykresów Wykresy. Średnie ruchy są równe ailable jako funkcja nakładania się cen na stół roboczy programu SharpCharts Korzystając z menu rozwijanego, użytkownicy mogą wybierać albo prostą średnią ruchomej lub średnią ruchową wykładniczą Pierwszy parametr służy do określania liczby przedziałów czasu. Można dodać opcjonalny parametr aby określić, które pole ceny powinno być stosowane w obliczeniach - O dla otwartych, H dla wysokich, L dla niskich, a C dla przecięcia Przecinka służy do oddzielania parametrów. Można dodać inny parametr opcjonalny, aby przesunąć ruch średnie do lewej przeszłości lub w prawo przyszłość Ujemna liczba -10 przesuwa średnią ruchomej do lewej 10 okresów Liczba dodatnia 10 zmieni średnią ruchomej w prawo 10 okresów. Wiele średnich kroczących można położyć na wykresie poprzez dodanie inna linia nakładki na stół roboczy Użytkownicy StockCharts mogą zmieniać kolory i style, aby rozróżnić wiele średnich kroczących Po wybraniu wskaźnika, otwórz Opcje zaawansowane, klikając zielony trójkąt kąt. Opcje zaawansowane mogą być również użyte do dodania ruchomej przeciętnej nakładki na inne wskaźniki techniczne, takie jak RSI, CCI i Volume. Click tutaj dla wykresu na żywo z kilkoma różnymi średnimi ruchoma. Używanie średnich ruchów za pomocą skanowania w StockCharts. Oto kilka przykładowych skanów, które StockCharts członkowie mogą używać do skanowania w różnych średnich ruchliwych sytuacjach. Bullish Moving Average Cross Skanuje on w poszukiwaniu zapasów o wzrastającej 150-dniowej prostej średniej ruchomej i upartym krzyżu 5-dniowej EMA i 35-dniowej EMA 150-dniowa średnia ruchoma wzrasta, dopóki będzie to sprzedawać powyżej jego poziomu pięć dni temu Utrzymujący krzyż ma miejsce, gdy 5-dniowa EMA przekracza 35-dniową EMA przy przeciętnej wielkości. Bearish Moving Average Cross To skanuje szuka zapasów ze spadkiem 150- dziennie średnia ruchoma i krzywa nieuzasadniona 5-dniowej EMA i 35-dniowej EMA 150-dniowa średnia ruchoma spadnie, dopóki spadnie poniżej jego poziomu pięć dni temu Ujemny krzyż ma miejsce, gdy 5-dniowa ruch EMA poniżej 35-dniowej EMA na abo średniej wielkości. Następna nauka. John Murphy s książki ma rozdział poświęcony średnich kroczących i ich różnych zastosowań Murphy obejmuje plusy i minusy ruchomych średnich Oprócz tego, Murphy pokazuje, jak ruchome średnie pracy z Bollinger Bands i kanału handlu opartych systems. Technical Analiza rynków finansowych John Murphy. Statistics Current - Textbook. Structural Equation Modeling. A Conceptual Overview. Structural Equation Modeling jest bardzo ogólną, bardzo potężną techniką analizy wielowymiarowej, która obejmuje wyspecjalizowane wersje wielu innych metod analizy jako specjalnych przypadków przyjąć, że znasz podstawową logikę rozumowania statystycznego opisaną w Koncepcji Elementarnej Ponadto przyjmiemy, że znasz pojęcia o wariancji, kowariancji i korelacji, jeśli nie, radzimy przeczytać sekcję Podstawowe statystyki na ten punkt Chociaż nie jest to absolutnie konieczne, bardzo pożądane jest, aby mieć jakieś tło w rzeczywistości r analiza przed przystąpieniem do modelowania strukturalnego. Majorowe zastosowania modelowania równań strukturalnych obejmują modelowanie lub modelowanie aspxów, które hipotezuje związek przyczynowo-skutkowy między zmiennymi i testuje modele przyczynowe z liniowym systemem równań Modele przypadków mogą obejmować zmienne manifestujące, zmienne utajone lub analiza czynników podwójnych. analiza czynników korygujących przedłużenie analizy czynników, w których badane są konkretne hipotezy dotyczące struktury obciążenia czynnikiem i interkorelacji. druga analiza czynników zamówienia analiza czynników, w której analizowana jest analiza współczynników korelacji wspólnych czynników w celu zapewnienia modele regresji drugiego rzędu. rozszerzenie analizy regresji liniowej, w której odważniki regresji mogą być ograniczone do siebie równe sobie wzajemnie lub do określonych liczbowych modeli struktury wartości zmienności, które hipotetyzują, że macierz zmienności ma określoną formę Na przykład można przetestować hipotezę, że zbiór varia bles mają takie same odchylenia w tej procedurze. Modele struktury korelacji, które hipotezują, że macierz korelacji ma szczególną postać. Klasycznym przykładem jest hipoteza, że macierz korelacji ma strukturę okręgu Guttman, 1954 Wiggins, Steiger, Gaelick, 1981.Many różnego rodzaju modele należą do każdej z powyższych kategorii, więc modelowanie strukturalne jako przedsiębiorstwo jest bardzo trudne do scharakteryzowania. Większość modeli równań strukturalnych może być wyrażona jako diagramy ścieżek. Nawet początkujący modelowania strukturalnego mogą wykonywać skomplikowane analizy z minimalnym poziomem szkolenia. Podstawowe pojęcie modelowania strukturalnego. Niektóre z podstawowych idei wykładanych w pośrednich kursach statystycznych stosowanych są efektem transformacji addytywnych i multiplikatywnych na liście liczb Uczniowie uczą się, że jeśli pomnożysz każdą liczbę na liście przez niektórych stałych K, pomnożyć średnią liczb przez K Podobnie, pomnożysz odchylenie standardowe przez absol wartość ute K. Na przykład, załóżmy, że masz listę numerów 1,2,3 Te liczby mają średnią 2 i standardowe odchylenie 1 Teraz, przypuśćmy, że masz te trzy liczby i pomnoż je przez 4 Następnie średnia oznacza 8, a odchylenie standardowe wynosiłoby 4, a więc wariacja 16. Punkt jest, jeśli masz zestaw liczb X odnoszących się do innego zbioru liczb Y przez równanie Y4X, wówczas wariancja Y musi być 16 razy większą od X, więc można przetestować hipotezę, że Y i X są związane przez równanie Y4X pośrednio przez porównanie wariancji zmiennych Y i X. Pomysł ten uogólnia na różne sposoby różne zmienne powiązane przez grupa równań liniowych Reguły stają się bardziej złożone, obliczenia trudniejsze, ale podstawowa wiadomość pozostaje taka sama - można sprawdzić, czy zmienne są ze sobą powiązane poprzez zestaw zależności liniowych, analizując wariancje i kowariancje zmiennych. Staciści mają opracowano procedury testowania w hether zestaw wariancji i kowariancji w matrycy kowariancji pasuje do określonej struktury Sposób pracy modelowania strukturalnego jest następujący. Wyjesz, w jaki sposób uważasz, że zmienne są ze sobą powiązane, często przy użyciu diagramu ścieżki. , za pomocą pewnych złożonych reguł wewnętrznych, jakie są ich konsekwencje dla wariancji i kowariancji zmiennych. Sprawdzasz, czy wariancje i kowariancje pasują do tego modelu. Wyniki badań statystycznych, a także szacunki parametrów i standardowe błędy dla współczynniki numeryczne w równaniach liniowych są zgłaszane. Na podstawie tych informacji decydujesz, czy model wydaje się być dobrym pomysłem na twoje dane. Są pewne ważne i bardzo podstawowe punkty logiczne, które należy pamiętać o tym procesie. Po pierwsze, choć matematyczny maszyny wymagane do wykonania modelowania równań strukturalnych są niezwykle skomplikowane, podstawowa logika jest zawarta w powyższych 5 krokach Poniżej przedstawiamy proces Drugi, musimy r że nierozsądne jest oczekiwanie, że model strukturalny idealnie pasuje do wielu przyczyn Model strukturalny z zależnościami liniowymi jest tylko aproksymacją Świat jest mało prawdopodobny Jest liniowy Prawdziwe relacje między zmiennymi są prawdopodobnie nieliniowe Ponadto wiele z nich Założenia statystyczne są też nieco wątpliwe. Prawdziwe pytanie nie jest tak bardzo, czy model idealnie pasuje, czy raczej pasuje wystarczająco dobrze, aby był użytecznym przybliżeniem do rzeczywistości i rozsądnym wyjaśnieniem trendów w naszych danych. musi pamiętać, że po prostu dlatego, że model pasuje do danych dobrze nie oznacza, że model jest koniecznie poprawny Nie można udowodnić, że model jest prawdą twierdzić, że jest to błędne potwierdzenie konsekwencji Na przykład można by powiedzieć Jeśli Joe jest kotem, Joe ma włosy Jednak Joe ma włosy nie oznacza, że Joe jest kotem Podobnie można powiedzieć, że jeśli pewien model przyczyny jest prawdziwy, to pasuje do danych Jednak model dopasowania danych nie ma że model jest właściwy Może istnieć inny model, który dobrze dopasuje się do danych. Modelowanie równań strukturalnych i Diagram ścieżki. Path Diagramy odgrywają zasadniczą rolę w modelowaniu strukturalnym Diagramy ścieżek są podobne do diagramów przepływowych Pokazują zmienne połączone liniami, które są wykorzystywane do wskazywania przepływu przyczynowego. Można myśleć o diagramie ścieżki jako narzędzia do pokazania, które zmienne powodują zmiany innych zmiennych. Jednak nie trzeba tak dokładnie traktować diagramów ścieżek, jak również mogą być one węższe i bardziej szczegółowe. Weźmy pod uwagę klasyczne równanie regresji liniowej. Niektóre równanie może być przedstawione w diagramie ścieżki w następujący sposób. Takie schematy tworzą prosty izomorfizm Wszystkie zmienne w układzie równań są umieszczone na diagramie, w pudełkach lub owalnych Każdy równanie jest przedstawione na diagramie jak następuje: Wszystkie zmienne niezależne zmienne po prawej stronie równania mają strzałki wskazujące na zmienną zależną Wei współczynnik Ghting jest umieszczony nad strzałką Na powyższym diagramie przedstawiono prosty system równań liniowych i jego reprezentację diagramu ścieżki. Niektórej, oprócz reprezentowania równań liniowych ze strzałkami, diagramy zawierają także pewne dodatkowe aspekty Po pierwsze, różnice wariantów niezależnych, które musimy wiedzieć, aby przetestować model relacji strukturalnych, są pokazane na schematach przy użyciu linii krzywych bez łączników strzałowych Odnoszą się do takich linii jak druty Drugi, niektóre zmienne są reprezentowane w owych, inne w prostokątnych polach Manifest zmienne są umieszczane w polach na schemacie ścieżkowym zmienne utajone są umieszczone w owalu lub okręgu Na przykład zmienna E na powyższym diagramie może być traktowana jako reszta regresji liniowej, gdy Y jest przewidywana z X Taka reszta nie jest bezpośrednio obserwowana, ale obliczana z Y i X, traktujemy je jako zmienną ukrytą i umieścimy ją w owalny. Przykład omówiony powyżej jest bardzo prosty. Ogólnie, Interesuje nas testowanie modeli, które są znacznie bardziej skomplikowane niż te. W miarę jak systemy równań, które badamy, stają się coraz bardziej skomplikowane, a więc struktura kowariancji, którą one zawierają. Ostatecznie, złożoność może stać się tak zdumiewająca, że tracimy pewne bardzo podstawowe zasady. pociąg rozumowania, który wspiera testowanie modeli przyczynowych z liniowymi testami równań strukturalnych ma kilka słabych linków Zmienne mogą być nieliniowe Mogą być liniowo powiązane z przyczyn niezwiązanych z tym, co powszechnie uważamy za przyczynowość Starożytne powiedzenie, korelacja nie jest przyczynowością , nawet jeśli korelacja jest złożona i wielowymiarowa Co powoduje, że modelowanie przyczynowe pozwala nam zrobić, to zbadać stopień, w jakim dane nie zgadzają się z jedną racjonalnie opłacalną konsekwencją modelu sprawiedliwości Jeśli układ liniowych równań isomorficznych do schematu ścieżki pasuje do danych, jest to zachęcające, ale ledwie dowód prawdy modelu przyczynowego. Chociaż diagramy ścieżek mogą być używane do reprezentowania przepływu przyczynowego w systemie zmiennych, nie muszą one oznaczać takiego przepływu przyczynowego Takie diagramy mogą być postrzegane jako prosta reprezentacja izomorficzna układu równań liniowych jako taka mogą przekazywać relacje liniowe, gdy nie zakłada się relacji przyczynowych , chociaż można zinterpretować diagram na powyższym rysunku, co oznacza, że X powoduje Y, diagram może być interpretowany jako wizualna reprezentacja zależności regresji liniowej pomiędzy X i Y. Was ten temat jest pomocny. Podstawowa odpowiedź. Zespół. Sprawialność analizy czasu awarii Ogólne informacje. Te techniki rozwijały się przede wszystkim w dziedzinie nauk medycznych i biologicznych, ale są również szeroko stosowane w naukach społecznych i ekonomicznych, a także w analizie niezawodności i analizy błędów. jest badanie skuteczności nowego leczenia ogólnie chorych z końcem choroby Główną zmienną interesującą jest liczba dni, które odpowiadają Przeżycie pacjentów Zasadniczo można użyć standardowych statystyk parametrycznych i nieparametrycznych do opisania przeciętnego przeżywalności i porównania nowych metod leczenia z tradycyjnymi metodami, patrz Podstawowe dane statystyczne i nieparametria i dopasowanie dystrybucyjne Jednak po zakończeniu badania pacjenci, przetrwały przez cały okres badania, zwłaszcza wśród pacjentów, którzy wstąpili do szpitala i projekt badawczy późno w badaniu nie będzie innych pacjentów, z którymi utracilibyśmy kontakt Na pewno nie chciałbyś wykluczyć wszystkich tych pacjentów z study by declaring them to be missing data since most of them are survivors and, therefore, they reflect on the success of the new treatment method Those observations, which contain only partial information are called censored observations eg patient A survived at least 4 months before he moved away and we lost contact the term censoring was first used by Hald, 1949.Censored Observations. In general, censored observations arise whenever the dependent variable of interest represents the time to a terminal event, and the duration of the study is limited in time Censored observations may occur in a number of different areas of research For example, in the social sciences we may study the survival of marriages, high school drop-out rates time to drop-out , turnover in organizations, etc In each case, by the end of the study period, some subjects will still be married, will not have dropped out, or are still working at the same company thus, those subjects represent censored observations. In economics we may study the survival of new businesses or the survival times of products such as automobiles In quality control research, it is common practice to study the survival of parts under stress failure time analysis. Analytic Techniques. Essentially, the methods offered in Survival Analysis address the same research questions as many of the other procedures however, all methods in Survival An alysis will handle censored data The life table, survival distribution and Kaplan-Meier survival function estimation are all descriptive methods for estimating the distribution of survival times from a sample Several techniques are available for comparing the survival in two or more groups Finally, Survival Analysis offers several regression models for estimating the relationship of multiple continuous variables to survival times. Life Table Analysis. The most straightforward way to describe the survival in a sample is to compute the Life Table The life table technique is one of the oldest methods for analyzing survival failure time data e g see Berkson Gage, 1950 Cutler Ederer, 1958 Gehan, 1969 This table can be thought of as an enhanced frequency distribution table The distribution of survival times is divided into a certain number of intervals For each interval we can then compute the number and proportion of cases or objects that entered the respective interval alive, the number and proportion of cases that failed in the respective interval i e number of terminal events, or number of cases that died , and the number of cases that were lost or censored in the respective interval. Based on those numbers and proportions, several additional statistics can be computed. Number of Cases at Risk This is the number of cases that entered the respective interval alive, minus half of the number of cases lost or censored in the respective interval. Proportion Failing This proportion is computed as the ratio of the number of cases failing in the respective interval, divided by the number of cases at risk in the interval. Proportion Surviving This proportion is computed as 1 minus the proportion failing. Cumulative Proportion Surviving Survival Function This is the cumulative proportion of cases surviving up to the respective interval Since the probabilities of survival are assumed to be independent across the intervals, this probability is computed by multiplying out the probabiliti es of survival across all previous intervals The resulting function is also called the survivorship or survival function. Probability Density This is the estimated probability of failure in the respective interval, computed per unit of time, that is. In this formula, F i is the respective probability density in the i th interval, P i is the estimated cumulative proportion surviving at the beginning of the i th interval at the end of interval i-1 , P i 1 is the cumulative proportion surviving at the end of the i th interval, and h i is the width of the respective interval. Hazard Rate The hazard rate the term was first used by Barlow, 1963 is defined as the probability per time unit that a case that has survived to the beginning of the respective interval will fail in that interval Specifically, it is computed as the number of failures per time units in the respective interval, divided by the average number of surviving cases at the mid-point of the interval. Median Survival Time This is th e survival time at which the cumulative survival function is equal to 0 5 Other percentiles 25th and 75th percentile of the cumulative survival function can be computed accordingly Note that the 50th percentile median for the cumulative survival function is usually not the same as the point in time up to which 50 of the sample survived This would only be the case if there were no censored observations prior to this time. Required Sample Sizes In order to arrive at reliable estimates of the three major functions survival, probability density, and hazard and their standard errors at each time interval the minimum recommended sample size is 30.Distribution Fitting. General Introduction In summary, the life table gives us a good indication of the distribution of failures over time However, for predictive purposes it is often desirable to understand the shape of the underlying survival function in the population The major distributions that have been proposed for modeling survival or failure times are the exponential and linear exponential distribution, the Weibull distribution of extreme events, and the Gompertz distribution. Estimation The parameter estimation procedure for estimating the parameters of the theoretical survival functions is essentially a least squares linear regression algorithm see Gehan Siddiqui, 1973 A linear regression algorithm can be used because all four theoretical distributions can be made linear by appropriate transformations Such transformations sometimes produce different variances for the residuals at different times, leading to biased estimates. Goodness-of-Fit Given the parameters for the different distribution functions and the respective model, we can compute the likelihood of the data One can also compute the likelihood of the data under the null model, that is, a model that allows for different hazard rates in each interval Without going into details, these two likelihoods can be compared via an incremental Chi-square test statistic If th is Chi-square is statistically significant, then we conclude that the respective theoretical distribution fits the data significantly worse than the null model that is, we reject the respective distribution as a model for our data. Plots You can produce plots of the survival function, hazard, and probability density for the observed data and the respective theoretical distributions These plots provide a quick visual check of the goodness-of-fit of the theoretical distribution The example plot below shows an observed survivorship function and the fitted Weibull distribution. Specifically, the three lines in this plot denote the theoretical distributions that resulted from three different estimation procedures least squares and two methods of weighted least squares. Kaplan-Meier Product-Limit Estimator. Rather than classifying the observed survival times into a life table, we can estimate the survival function directly from the continuous survival or failure times Intuitively, imagine that w e create a life table so that each time interval contains exactly one case Multiplying out the survival probabilities across the intervals i e for each single observation we would get for the survival function. In this equation, S t is the estimated survival function, n is the total number of cases, and denotes the multiplication geometric sum across all cases less than or equal to t j is a constant that is either 1 if the j th case is uncensored complete , and 0 if it is censored This estimate of the survival function is also called the product-limit estimator and was first proposed by Kaplan and Meier 1958 An example plot of this function is shown below. The advantage of the Kaplan-Meier Product-Limit method over the life table method for analyzing survival and failure time data is that the resulting estimates do not depend on the grouping of the data into a certain number of time intervals Actually, the Product-Limit method and the life table method are identical if the intervals of the life table contain at most one observationparing Samples. General Introduction One can compare the survival or failure times in two or more samples In principle, because survival times are not normally distributed, nonparametric tests that are based on the rank ordering of survival times should be applied A wide range of nonparametric tests can be used in order to compare survival times however, the tests cannot handle censored observations. Available Tests The following five different mostly nonparametric tests for censored data are available Gehan s generalized Wilcoxon test, the Cox-Mantel test, the Cox s F test the log-rank test, and Peto and Peto s generalized Wilcoxon test A nonparametric test for the comparison of multiple groups is also available Most of these tests are accompanied by appropriate z - values values of the standard normal distribution these z - values can be used to test for the statistical significance of any differences between groups However, note that most of these tests will only yield reliable results with fairly large samples sizes the small sample behavior is less well understood. Choosing a Two Sample Test There are no widely accepted guidelines concerning which test to use in a particular situation Cox s F test tends to be more powerful than Gehan s generalized Wilcoxon test when. Sample sizes are small i e n per group less than 50.If samples are from an exponential or Weibull. If there are no censored observations see Gehan Thomas, 1969.Lee, Desu, and Gehan 1975 compared Gehan s test to several alternatives and showed that the Cox-Mantel test and the log-rank test are more powerful regardless of censoring when the samples are drawn from a population that follows an exponential or Weibull distribution under those conditions there is little difference between the Cox-Mantel test and the log-rank test Lee 1980 discusses the power of different tests in greater detail. Multiple Sample Test There is a multiple-sample test that is an extensi on or generalization of Gehan s generalized Wilcoxon test, Peto and Peto s generalized Wilcoxon test, and the log-rank test First, a score is assigned to each survival time using Mantel s procedure Mantel, 1967 next a Chi - square value is computed based on the sums for each group of this score If only two groups are specified, then this test is equivalent to Gehan s generalized Wilcoxon test, and the computations will default to that test in this case. Unequal Proportions of Censored Data When comparing two or more groups it is very important to examine the number of censored observations in each group Particularly in medical research, censoring can be the result of, for example, the application of different treatments patients who get better faster or get worse as the result of a treatment may be more likely to drop out of the study, resulting in different numbers of censored observations in each group Such systematic censoring may greatly bias the results of comparisons. Regression Mo dels. General Introduction. A common research question in medical, biological, or engineering failure time research is to determine whether or not certain continuous independent variables are correlated with the survival or failure times There are two major reasons why this research issue cannot be addressed via straightforward multiple regression techniques as available in Multiple Regression First, the dependent variable of interest survival failure time is most likely not normally distributed -- a serious violation of an assumption for ordinary least squares multiple regression Survival times usually follow an exponential or Weibull distribution Second, there is the problem of censoring that is, some observations will be incomplete. Cox s Proportional Hazard Model. The proportional hazard model is the most general of the regression models because it is not based on any assumptions concerning the nature or shape of the underlying survival distribution The model assumes that the underlyin g hazard rate rather than survival time is a function of the independent variables covariates no assumptions are made about the nature or shape of the hazard function Thus, in a sense, Cox s regression model may be considered to be a nonparametric method The model may be written as. where h t denotes the resultant hazard, given the values of the m covariates for the respective case z 1 z 2 z m and the respective survival time t The term h 0 t is called the baseline hazard it is the hazard for the respective individual when all independent variable values are equal to zero We can linearize this model by dividing both sides of the equation by h 0 t and then taking the natural logarithm of both sides. We now have a fairly simple linear model that can be readily estimated. Assumptions While no assumptions are made about the shape of the underlying hazard function, the model equations shown above do imply two assumptions First, they specify a multiplicative relationship between the underlyin g hazard function and the log-linear function of the covariates This assumption is also called the proportionality assumption In practical terms, it is assumed that, given two observations with different values for the independent variables, the ratio of the hazard functions for those two observations does not depend on time The second assumption of course, is that there is a log-linear relationship between the independent variables and the underlying hazard function. Cox s Proportional Hazard Model with Time-Dependent Covariates. An assumption of the proportional hazard model is that the hazard function for an individual i e observation in the analysis depends on the values of the covariates and the value of the baseline hazard Given two individuals with particular values for the covariates, the ratio of the estimated hazards over time will be constant -- hence the name of the method the proportional hazard model The validity of this assumption may often be questionable For example, age is often included in studies of physical health Suppose you studied survival after surgery It is likely, that age is a more important predictor of risk immediately after surgery, than some time after the surgery after initial recovery In accelerated life testing one sometimes uses a stress covariate e g amount of voltage that is slowly increased over time until failure occurs e g until the electrical insulation fails see Lawless, 1982, page 393 In this case, the impact of the covariate is clearly dependent on time The user can specify arithmetic expressions to define covariates as functions of several variables and survival time. Testing the Proportionality Assumption As indicated by the previous examples, there are many applications where it is likely that the proportionality assumption does not hold In that case, one can explicitly define covariates as functions of time For example, the analysis of a data set presented by Pike 1966 consists of survival times for two groups of rats th at had been exposed to a carcinogen see also Lawless, 1982, page 393, for a similar example Suppose that z is a grouping variable with codes 1 and 0 to denote whether or not the respective rat was exposed One could then fit the proportional hazard model. Thus, in this model the conditional hazard at time t is a function of 1 the baseline hazard h 0 2 the covariate z and 3 of z times the logarithm of time Note that the constant 5 4 is used here for scaling purposes only the mean of the logarithm of the survival times in this data set is equal to 5 4 In other words, the conditional hazard at each point in time is a function of the covariate and time thus, the effect of the covariate on survival is dependent on time hence the name time-dependent covariate This model allows one to specifically test the proportionality assumption If parameter b 2 is statistically significant e g if it is at least twice as large as its standard error , then one can conclude that, indeed, the effect of the cov ariate z on survival is dependent on time, and, therefore, that the proportionality assumption does not hold. Exponential Regression. Basically, this model assumes that the survival time distribution is exponential, and contingent on the values of a set of independent variables z i The rate parameter of the exponential distribution can then be expressed as. S z denotes the survival times, a is a constant, and the b i s are the regression parameters. Goodness-of-fit The Chi-square goodness-of-fit value is computed as a function of the log-likelihood for the model with all parameter estimates L 1 , and the log-likelihood of the model in which all covariates are forced to 0 zero L 0 If this Chi-square value is significant, we reject the null hypothesis and assume that the independent variables are significantly related to survival times. Standard exponential order statistic One way to check the exponentiality assumption of this model is to plot the residual survival times against the standard exponential order statistic theta If the exponentiality assumption is met, then all points in this plot will be arranged roughly in a straight line. Normal and Log-Normal Regression. In this model, it is assumed that the survival times or log survival times come from a normal distribution the resulting model is basically identical to the ordinary multiple regression model, and may be stated as. where t denotes the survival times For log-normal regression, t is replaced by its natural logarithm The normal regression model is particularly useful because many data sets can be transformed to yield approximations of the normal distribution Thus, in a sense this is the most general fully parametric model as opposed to Cox s proportional hazard model which is non-parametric , and estimates can be obtained for a variety of different underlying survival distributions. Goodness-of-fit The Chi-square value is computed as a function of the log-likelihood for the model with all independent variables L1 , and the log-likelihood of the model in which all independent variables are forced to 0 zero, L0.Stratified Analyses. The purpose of a stratified analysis is to test the hypothesis whether identical regression models are appropriate for different groups, that is, whether the relationships between the independent variables and survival are identical in different groups To perform a stratified analysis, one must first fit the respective regression model separately within each group The sum of the log-likelihoods from these analyses represents the log-likelihood of the model with different regression coefficients and intercepts where appropriate in different groups The next step is to fit the requested regression model to all data in the usual manner i e ignoring group membership , and compute the log-likelihood for the overall fit The difference between the log-likelihoods can then be tested for statistical significance via the Chi-square statistic. Was this topic helpful. Feedback Submit ted. Text Mining Big Data, Unstructured Data. Text Mining Introductory Overview. The purpose of Text Mining is to process unstructured textual information, extract meaningful numeric indices from the text, and, thus, make the information contained in the text accessible to the various data mining statistical and machine learning algorithms Information can be extracted to derive summaries for the words contained in the documents or to compute summaries for the documents based on the words contained in them Hence, you can analyze words, clusters of words used in documents, etc or you could analyze documents and determine similarities between them or how they are related to other variables of interest in the data mining project In the most general terms, text mining will turn text into numbers meaningful indices , which can then be incorporated in other analyses such as predictive data mining projects, the application of unsupervised learning methods clustering , etc These methods are descri bed and discussed in great detail in the comprehensive overview work by Manning and Schtze 2002 , and for an in-depth treatment of these and related topics as well as the history of this approach to text mining, we highly recommend that source. Typical Applications for Text Mining. Unstructured text is very common, and in fact may represent the majority of information available to a particular research or data mining project. Analyzing open-ended survey responses In survey research e g marketing , it is not uncommon to include various open-ended questions pertaining to the topic under investigation The idea is to permit respondents to express their views or opinions without constraining them to particular dimensions or a particular response format This may yield insights into customers views and opinions that might otherwise not be discovered when relying solely on structured questionnaires designed by experts For example, you may discover a certain set of words or terms that are commonly used by respondents to describe the pro s and con s of a product or service under investigation , suggesting common misconceptions or confusion regarding the items in the study. Automatic processing of messages, emails, etc Another common application for text mining is to aid in the automatic classification of texts For example, it is possible to filter out automatically most undesirable junk email based on certain terms or words that are not likely to appear in legitimate messages, but instead identify undesirable electronic mail In this manner, such messages can automatically be discarded Such automatic systems for classifying electronic messages can also be useful in applications where messages need to be routed automatically to the most appropriate department or agency e g email messages with complaints or petitions to a municipal authority are automatically routed to the appropriate departments at the same time, the emails are screened for inappropriate or obscene messages, which are automatically returned to the sender with a request to remove the offending words or content. Analyzing warranty or insurance claims, diagnostic interviews, etc In some business domains, the majority of information is collected in open-ended, textual form For example, warranty claims or initial medical patient interviews can be summarized in brief narratives, or when you take your automobile to a service station for repairs, typically, the attendant will write some notes about the problems that you report and what you believe needs to be fixed Increasingly, those notes are collected electronically, so those types of narratives are readily available for input into text mining algorithms This information can then be usefully exploited to, for example, identify common clusters of problems and complaints on certain automobiles, etc Likewise, in the medical field, open-ended descriptions by patients of their own symptoms might yield useful clues for the actual medical diagnosis. Investiga ting competitors by crawling their web sites Another type of potentially very useful application is to automatically process the contents of Web pages in a particular domain For example, you could go to a Web page, and begin crawling the links you find there to process all Web pages that are referenced In this manner, you could automatically derive a list of terms and documents available at that site, and hence quickly determine the most important terms and features that are described It is easy to see how these capabilities could efficiently deliver valuable business intelligence about the activities of competitors. Approaches to Text Mining. To reiterate, text mining can be summarized as a process of numericizing text At the simplest level, all words found in the input documents will be indexed and counted in order to compute a table of documents and words, i e a matrix of frequencies that enumerates the number of times that each word occurs in each document This basic process can be f urther refined to exclude certain common words such as the and a stop word lists and to combine different grammatical forms of the same words such as traveling, traveled, travel, etc stemming However, once a table of unique words terms by documents has been derived, all standard statistical and data mining techniques can be applied to derive dimensions or clusters of words or documents, or to identify important words or terms that best predict another outcome variable of interest. Using well-tested methods and understanding the results of text mining Once a data matrix has been computed from the input documents and words found in those documents, various well-known analytic techniques can be used for further processing those data including methods for clustering, factoring, or predictive data mining see, for example, Manning and Schtze, 2002. Black-box approaches to text mining and extraction of concepts There are text mining applications which offer black-box methods to extract deep me aning from documents with little human effort to first read and understand those documents These text mining applications rely on proprietary algorithms for presumably extracting concepts from text, and may even claim to be able to summarize large numbers of text documents automatically, retaining the core and most important meaning of those documents While there are numerous algorithmic approaches to extracting meaning from documents, this type of technology is very much still in its infancy, and the aspiration to provide meaningful automated summaries of large numbers of documents may forever remain elusive We urge skepticism when using such algorithms because 1 if it is not clear to the user how those algorithms work, it cannot possibly be clear how to interpret the results of those algorithms, and 2 the methods used in those programs are not open to scrutiny, for example by the academic community and peer review and, hence, we simply don t know how well they might perform in differ ent domains As a final thought on this subject, you may consider this concrete example Try the various automated translation services available via the Web that can translate entire paragraphs of text from one language into another Then translate some text, even simple text, from your native language to some other language and back, and review the results Almost every time, the attempt to translate even short sentences to other languages and back while retaining the original meaning of the sentence produces humorous rather than accurate results This illustrates the difficulty of automatically interpreting the meaning of text. Text mining as document search There is another type of application that is often described and referred to as text mining - the automatic search of large numbers of documents based on key words or key phrases This is the domain of, for example, the popular internet search engines that have been developed over the last decade to provide efficient access to Web page s with certain content While this is obviously an important type of application with many uses in any organization that needs to search very large document repositories based on varying criteria, it is very different from what has been described here. Issues and Considerations for Numericizing Text. Large numbers of small documents vs small numbers of large documents Examples of scenarios using large numbers of small or moderate sized documents were given earlier e g analyzing warranty or insurance claims, diagnostic interviews, etc On the other hand, if your intent is to extract concepts from only a few documents that are very large e g two lengthy books , then statistical analyses are generally less powerful because the number of cases documents in this case is very small while the number of variables extracted words is very large. Excluding certain characters, short words, numbers, etc Excluding numbers, certain characters, or sequences of characters, or words that are shorter or longe r than a certain number of letters can be done before the indexing of the input documents starts You may also want to exclude rare words, defined as those that only occur in a small percentage of the processed documents. Include lists, exclude lists stop-words Specific list of words to be indexed can be defined this is useful when you want to search explicitly for particular words, and classify the input documents based on the frequencies with which those words occur Also, stop-words, i e terms that are to be excluded from the indexing can be defined Typically, a default list of English stop words includes the , a , of , since, etc, i e words that are used in the respective language very frequently, but communicate very little unique information about the contents of the document. Synonyms and phrases Synonyms, such as sick or ill , or words that are used in particular phrases where they denote unique meaning can be combined for indexing For example, Microsoft Windows might be such a phr ase, which is a specific reference to the computer operating system, but has nothing to do with the common use of the term Windows as it might, for example, be used in descriptions of home improvement projects. Stemming algorithms An important pre-processing step before indexing of input documents begins is the stemming of words The term stemming refers to the reduction of words to their roots so that, for example, different grammatical forms or declinations of verbs are identified and indexed counted as the same word For example, stemming will ensure that both traveling and traveled will be recognized by the text mining program as the same word. Support for different languages Stemming, synonyms, the letters that are permitted in words, etc are highly language dependent operations Therefore, support for different languages is important. Transforming Word Frequencies. Once the input documents have been indexed and the initial word frequencies by document computed, a number of additional tr ansformations can be performed to summarize and aggregate the information that was extracted. Log-frequencies First, various transformations of the frequency counts can be performed The raw word or term frequencies generally reflect on how salient or important a word is in each document Specifically, words that occur with greater frequency in a document are better descriptors of the contents of that document However, it is not reasonable to assume that the word counts themselves are proportional to their importance as descriptors of the documents For example, if a word occurs 1 time in document A, but 3 times in document B, then it is not necessarily reasonable to conclude that this word is 3 times as important a descriptor of document B as compared to document A Thus, a common transformation of the raw word frequency counts wf is to compute. f wf 1 log wf , for wf 0.This transformation will dampen the raw frequencies and how they will affect the results of subsequent computations. Binary frequencies Likewise, an even simpler transformation can be used that enumerates whether a term is used in a document i e. f wf 1, for wf 0.The resulting documents-by-words matrix will contain only 1s and 0s to indicate the presence or absence of the respective words Again, this transformation will dampen the effect of the raw frequency counts on subsequent computations and analyses. Inverse document frequencies Another issue that you may want to consider more carefully and reflect in the indices used in further analyses are the relative document frequencies df of different words For example, a term such as guess may occur frequently in all documents, while another term such as software may only occur in a few The reason is that we might make guesses in various contexts, regardless of the specific topic, while software is a more semantically focused term that is only likely to occur in documents that deal with computer software A common and very useful transformation that reflects both the specificity of words document frequencies as well as the overall frequencies of their occurrences word frequencies is the so-called inverse document frequency for the i th word and j th document. In this formula see also formula 15 5 in Manning and Schtze, 2002 , N is the total number of documents, and dfi is the document frequency for the i th word the number of documents that include this word Hence, it can be seen that this formula includes both the dampening of the simple word frequencies via the log function described above , and also includes a weighting factor that evaluates to 0 if the word occurs in all documents log N N 1 0 and to the maximum value when a word only occurs in a single document log N 1 log N It can easily be seen how this transformation will create indices that both reflect the relative frequencies of occurrences of words, as well as their semantic specificities over the documents included in the analysis. Latent Semantic Indexing via Singular Value Decomposi tion. As described above, the most basic result of the initial indexing of words found in the input documents is a frequency table with simple counts, i e the number of times that different words occur in each input document Usually, we would transform those raw counts to indices that better reflect the relative importance of words and or their semantic specificity in the context of the set of input documents see the discussion of inverse document frequencies, above. A common analytic tool for interpreting the meaning or semantic space described by the words that were extracted, and hence by the documents that were analyzed, is to create a mapping of the word and documents into a common space, computed from the word frequencies or transformed word frequencies e g inverse document frequencies In general, here is how it works. Suppose you indexed a collection of customer reviews of their new automobiles e g for different makes and models You may find that every time a review includes the wo rd gas-mileage, it also includes the term economy Further, when reports include the word reliability they also include the term defects e g make reference to no defects However, there is no consistent pattern regarding the use of the terms economy and reliability, i e some documents include either one or both In other words, these four words gas-mileage and economy, and reliability and defects, describe two independent dimensions - the first having to do with the overall operating cost of the vehicle, the other with the quality and workmanship The idea of latent semantic indexing is to identify such underlying dimensions of meaning , into which the words and documents can be mapped As a result, we may identify the underlying latent themes described or discussed in the input documents, and also identify the documents that mostly deal with economy, reliability, or both Hence, we want to map the extracted words or terms and input documents into a common latent semantic space. Singular valu e decomposition The use of singular value decomposition in order to extract a common space for the variables and cases observations is used in various statistical techniques, most notably in Correspondence Analysis The technique is also closely related to Principal Components Analysis and Factor Analysis In general, the purpose of this technique is to reduce the overall dimensionality of the input matrix number of input documents by number of extracted words to a lower-dimensional space, where each consecutive dimension represents the largest degree of variability between words and documents possible Ideally, you might identify the two or three most salient dimensions, accounting for most of the variability differences between the words and documents and, hence, identify the latent semantic space that organizes the words and documents in the analysis In some way, once such dimensions can be identified, you have extracted the underlying meaning of what is contained discussed, described in the documents. Incorporating Text Mining Results in Data Mining Projects. After significant e g frequent words have been extracted from a set of input documents, and or after singular value decomposition has been applied to extract salient semantic dimensions, typically the next and most important step is to use the extracted information in a data mining project. Graphics visual data mining methods Depending on the purpose of the analyses, in some instances the extraction of semantic dimensions alone can be a useful outcome if it clarifies the underlying structure of what is contained in the input documents For example, a study of new car owners comments about their vehicles may uncover the salient dimensions in the minds of those drivers when they think about or consider their automobile or how they feel about it For marketing research purposes, that in itself can be a useful and significant result You can use the graphics e g 2D scatterplots or 3D scatterplots to help you visualize a nd identify the semantic space extracted from the input documents. Clustering and factoring You can use cluster analysis methods to identify groups of documents e g vehicle owners who described their new cars , to identify groups of similar input texts This type of analysis also could be extremely useful in the context of market research studies, for example of new car owners You can also use Factor Analysis and Principal Components and Classification Analysis to factor analyze words or documents. Predictive data mining Another possibility is to use the raw or transformed word counts as predictor variables in predictive data mining projects. Was this topic helpful. Thank you We appreciate your feedback. Time Series Analysis. How To Identify Patterns in Time Series Data Time Series Analysis. In the following topics, we will first review techniques used to identify patterns in time series data such as smoothing and curve fitting techniques and autocorrelations , then we will introduce a general class of models that can be used to represent time series data and generate predictions autoregressive and moving average models Finally, we will review some simple but commonly used modeling and forecasting techniques based on linear regression For more information see the topics below. General Introduction. In the following topics, we will review techniques that are useful for analyzing time series data, that is, sequences of measurements that follow non-random orders Unlike the analyses of random samples of observations that are discussed in the context of most other statistics, the analysis of time series is based on the assumption that successive values in the data file represent consecutive measurements taken at equally spaced time intervals. Detailed discussions of the methods described in this section can be found in Anderson 1976 , Box and Jenkins 1976 , Kendall 1984 , Kendall and Ord 1990 , Montgomery, Johnson, and Gardiner 1990 , Pankratz 1983 , Shumway 1988 , Vandaele 1983 , Walker 1991 , and Wei 1989.Two Main Goals. There are two main goals of time series analysis a identifying the nature of the phenomenon represented by the sequence of observations, and b forecasting predicting future values of the time series variable Both of these goals require that the pattern of observed time series data is identified and more or less formally described Once the pattern is established, we can interpret and integrate it with other data i e use it in our theory of the investigated phenomenon, e g seasonal commodity prices Regardless of the depth of our understanding and the validity of our interpretation theory of the phenomenon, we can extrapolate the identified pattern to predict future events. Identifying Patterns in Time Series Data. For more information on simple autocorrelations introduced in this section and other auto correlations, see Anderson 1976 , Box and Jenkins 1976 , Kendall 1984 , Pankratz 1983 , and Vandaele 1983 See also. Systematic Pattern and Random Noi se. As in most other analyses, in time series analysis it is assumed that the data consist of a systematic pattern usually a set of identifiable components and random noise error which usually makes the pattern difficult to identify Most time series analysis techniques involve some form of filtering out noise in order to make the pattern more salient. Two General Aspects of Time Series Patterns. Most time series patterns can be described in terms of two basic classes of components trend and seasonality The former represents a general systematic linear or most often nonlinear component that changes over time and does not repeat or at least does not repeat within the time range captured by our data e g a plateau followed by a period of exponential growth The latter may have a formally similar nature e g a plateau followed by a period of exponential growth , however, it repeats itself in systematic intervals over time Those two general classes of time series components may coexist in real-li fe data For example, sales of a company can rapidly grow over years but they still follow consistent seasonal patterns e g as much as 25 of yearly sales each year are made in December, whereas only 4 in August. This general pattern is well illustrated in a classic Series G data set Box and Jenkins, 1976, p 531 representing monthly international airline passenger totals measured in thousands in twelve consecutive years from 1949 to 1960 see example data file and graph above If you plot the successive observations months of airline passenger totals, a clear, almost linear trend emerges, indicating that the airline industry enjoyed a steady growth over the years approximately 4 times more passengers traveled in 1960 than in 1949 At the same time, the monthly figures will follow an almost identical pattern each year e g more people travel during holidays than during any other time of the year This example data file also illustrates a very common general type of pattern in time series data, where the amplitude of the seasonal changes increases with the overall trend i e the variance is correlated with the mean over the segments of the series This pattern which is called multiplicative seasonality indicates that the relative amplitude of seasonal changes is constant over time, thus it is related to the trend. Trend Analysis. There are no proven automatic techniques to identify trend components in the time series data however, as long as the trend is monotonous consistently increasing or decreasing that part of data analysis is typically not very difficult If the time series data contain considerable error, then the first step in the process of trend identification is smoothing. Smoothing Smoothing always involves some form of local averaging of data such that the nonsystematic components of individual observations cancel each other out The most common technique is moving average smoothing which replaces each element of the series by either the simple or weighted average of n surrounding elements, where n is the width of the smoothing window see Box Jenkins, 1976 Velleman Hoaglin, 1981 Medians can be used instead of means The main advantage of median as compared to moving average smoothing is that its results are less biased by outliers within the smoothing window Thus, if there are outliers in the data e g due to measurement errors , median smoothing typically produces smoother or at least more reliable curves than moving average based on the same window width The main disadvantage of median smoothing is that in the absence of clear outliers it may produce more jagged curves than moving average and it does not allow for weighting. In the relatively less common cases in time series data , when the measurement error is very large, the distance weighted least squares smoothing or negative exponentially weighted smoothing techniques can be used All those methods will filter out the noise and convert the data into a smooth curve that is relatively unbiased by ou tliers see the respective sections on each of those methods for more details Series with relatively few and systematically distributed points can be smoothed with bicubic splines. Fitting a function Many monotonous time series data can be adequately approximated by a linear function if there is a clear monotonous nonlinear component, the data first need to be transformed to remove the nonlinearity Usually a logarithmic, exponential, or less often polynomial function can be used. Analysis of Seasonality. Seasonal dependency seasonality is another general component of the time series pattern The concept was illustrated in the example of the airline passengers data above It is formally defined as correlational dependency of order k between each i th element of the series and the i-k th element Kendall, 1976 and measured by autocorrelation i e a correlation between the two terms k is usually called the lag If the measurement error is not too large, seasonality can be visually identified in t he series as a pattern that repeats every k elements. Autocorrelation correlogram Seasonal patterns of time series can be examined via correlograms The correlogram autocorrelogram displays graphically and numerically the autocorrelation function ACF , that is, serial correlation coefficients and their standard errors for consecutive lags in a specified range of lags e g 1 through 30 Ranges of two standard errors for each lag are usually marked in correlograms but typically the size of auto correlation is of more interest than its reliability see Elementary Concepts because we are usually interested only in very strong and thus highly significant autocorrelations. Examining correlograms While examining correlograms, you should keep in mind that autocorrelations for consecutive lags are formally dependent Consider the following example If the first element is closely related to the second, and the second to the third, then the first element must also be somewhat related to the third one, e tc This implies that the pattern of serial dependencies can change considerably after removing the first order auto correlation i e after differencing the series with a lag of 1.Partial autocorrelations Another useful method to examine serial dependencies is to examine the partial autocorrelation function PACF - an extension of autocorrelation, where the dependence on the intermediate elements those within the lag is removed In other words the partial autocorrelation is similar to autocorrelation, except that when calculating it, the auto correlations with all the elements within the lag are partialled out Box Jenkins, 1976 see also McDowall, McCleary, Meidinger, Hay, 1980 If a lag of 1 is specified i e there are no intermediate elements within the lag , then the partial autocorrelation is equivalent to auto correlation In a sense, the partial autocorrelation provides a cleaner picture of serial dependencies for individual lags not confounded by other serial dependencies. Removing seria l dependency Serial dependency for a particular lag of k can be removed by differencing the series, that is converting each i th element of the series into its difference from the i-k th element There are two major reasons for such transformations. First, we can identify the hidden nature of seasonal dependencies in the series Remember that, as mentioned in the previous paragraph, autocorrelations for consecutive lags are interdependent Therefore, removing some of the autocorrelations will change other auto correlations, that is, it may eliminate them or it may make some other seasonalities more apparent. The other reason for removing seasonal dependencies is to make the series stationary which is necessary for ARIMA and other techniques. For more information on Time Series methods, see also. General Introduction. The modeling and forecasting procedures discussed in Identifying Patterns in Time Series Data involved knowledge about the mathematical model of the process However, in real-life research and practice, patterns of the data are unclear, individual observations involve considerable error, and we still need not only to uncover the hidden patterns in the data but also generate forecasts The ARIMA methodology developed by Box and Jenkins 1976 allows us to do just that it has gained enormous popularity in many areas and research practice confirms its power and flexibility Hoff, 1983 Pankratz, 1983 Vandaele, 1983 However, because of its power and flexibility, ARIMA is a complex technique it is not easy to use, it requires a great deal of experience, and although it often produces satisfactory results, those results depend on the researcher s level of expertise Bails Peppers, 1982 The following sections will introduce the basic ideas of this methodology For those interested in a brief, applications-oriented non - mathematical , introduction to ARIMA methods, we recommend McDowall, McCleary, Meidinger, and Hay 1980.Two Common Processes. Autoregressive process Most time se ries consist of elements that are serially dependent in the sense that you can estimate a coefficient or a set of coefficients that describe consecutive elements of the series from specific, time-lagged previous elements This can be summarized in the equation x t 1 x t-1 2 x t-2 3 x t-3.is a constant intercept , and 1 2 3 are the autoregressive model parameters. Put into words, each observation is made up of a random error component random shock, and a linear combination of prior observations. Stationarity requirement Note that an autoregressive process will only be stable if the parameters are within a certain range for example, if there is only one autoregressive parameter then is must fall within the interval of -1 1 Otherwise, past effects would accumulate and the values of successive x t s would move towards infinity, that is, the series would not be stationary If there is more than one autoregressive parameter, similar general restrictions on the parameter values can be defined e g see Box Jenkins, 1976 Montgomery, 1990.Moving average process Independent from the autoregressive process, each element in the series can also be affected by the past error or random shock that cannot be accounted for by the autoregressive component, that is. Where is a constant, and 1 2 3 are the moving average model parameters. Put into words, each observation is made up of a random error component random shock, and a linear combination of prior random shocks. Invertibility requirement Without going into too much detail, there is a duality between the moving average process and the autoregressive process e g see Box Jenkins, 1976 Montgomery, Johnson, Gardiner, 1990 , that is, the moving average equation above can be rewritten inverted into an autoregressive form of infinite order However, analogous to the stationarity condition described above, this can only be done if the moving average parameters follow certain conditions, that is, if the model is invertible Otherwise, the series wi ll not be stationary. ARIMA Methodology. Autoregressive moving average model The general model introduced by Box and Jenkins 1976 includes autoregressive as well as moving average parameters, and explicitly includes differencing in the formulation of the model Specifically, the three types of parameters in the model are the autoregressive parameters p , the number of differencing passes d , and moving average parameters q In the notation introduced by Box and Jenkins, models are summarized as ARIMA p, d, q so, for example, a model described as 0, 1, 2 means that it contains 0 zero autoregressive p parameters and 2 moving average q parameters which were computed for the series after it was differenced once. Identification As mentioned earlier, the input series for ARIMA needs to be stationary that is, it should have a constant mean, variance, and autocorrelation through time Therefore, usually the series first needs to be differenced until it is stationary this also often requires log tra nsforming the data to stabilize the variance The number of times the series needs to be differenced to achieve stationarity is reflected in the d parameter see the previous paragraph In order to determine the necessary level of differencing, you should examine the plot of the data and autocorrelogram Significant changes in level strong upward or downward changes usually require first order non seasonal lag 1 differencing strong changes of slope usually require second order non seasonal differencing Seasonal patterns require respective seasonal differencing see below If the estimated autocorrelation coefficients decline slowly at longer lags, first order differencing is usually needed However, you should keep in mind that some time series may require little or no differencing, and that over differenced series produce less stable coefficient estimates. At this stage which is usually called Identification phase, see below we also need to decide how many autoregressive p and moving average q parameters are necessary to yield an effective but still parsimonious model of the process parsimonious means that it has the fewest parameters and greatest number of degrees of freedom among all models that fit the data In practice, the numbers of the p or q parameters very rarely need to be greater than 2 see below for more specific recommendations. Estimation and Forecasting At the next step Estimation , the parameters are estimated using function minimization procedures, see below for more information on minimization procedures see also Nonlinear Estimation , so that the sum of squared residuals is minimized The estimates of the parameters are used in the last stage Forecasting to calculate new values of the series beyond those included in the input data set and confidence intervals for those predicted values The estimation process is performed on transformed differenced data before the forecasts are generated, the series needs to be integrated integration is the inverse of differ encing so that the forecasts are expressed in values compatible with the input data This automatic integration feature is represented by the letter I in the name of the methodology ARIMA Auto-Regressive Integrated Moving Average. The constant in ARIMA models In addition to the standard autoregressive and moving average parameters, ARIMA models may also include a constant, as described above The interpretation of a statistically significant constant depends on the model that is fit Specifically, 1 if there are no autoregressive parameters in the model, then the expected value of the constant is , the mean of the series 2 if there are autoregressive parameters in the series, then the constant represents the intercept If the series is differenced, then the constant represents the mean or intercept of the differenced series For example, if the series is differenced once, and there are no autoregressive parameters in the model, then the constant represents the mean of the differenced series, and therefore the linear trend slope of the un-differenced series. Identification Phase. Number of parameters to be estimated Before the estimation can begin, we need to decide on identify the specific number and type of ARIMA parameters to be estimated The major tools used in the identification phase are plots of the series, correlograms of auto correlation ACF , and partial autocorrelation PACF The decision is not straightforward and in less typical cases requires not only experience but also a good deal of experimentation with alternative models as well as the technical parameters of ARIMA However, a majority of empirical time series patterns can be sufficiently approximated using one of the 5 basic models that can be identified based on the shape of the autocorrelogram ACF and partial auto correlogram PACF The following brief summary is based on practical recommendations of Pankratz 1983 for additional practical advice, see also Hoff 1983 , McCleary and Hay 1980 , McDowall, McClear y, Meidinger, and Hay 1980 , and Vandaele 1983 Also, note that since the number of parameters to be estimated of each kind is almost never greater than 2, it is often practical to try alternative models on the same data. One autoregressive p parameter ACF - exponential decay PACF - spike at lag 1, no correlation for other lags. Two autoregressive p parameters ACF - a sine-wave shape pattern or a set of exponential decays PACF - spikes at lags 1 and 2, no correlation for other lags. One moving average q parameter ACF - spike at lag 1, no correlation for other lags PACF - damps out exponentially. Two moving average q parameters ACF - spikes at lags 1 and 2, no correlation for other lags PACF - a sine-wave shape pattern or a set of exponential decays. One autoregressive p and one moving average q parameter ACF - exponential decay starting at lag 1 PACF - exponential decay starting at lag 1.Seasonal models Multiplicative seasonal ARIMA is a generalization and extension of the method introduced in the previous paragraphs to series in which a pattern repeats seasonally over time In addition to the non-seasonal parameters, seasonal parameters for a specified lag established in the identification phase need to be estimated Analogous to the simple ARIMA parameters, these are seasonal autoregressive ps , seasonal differencing ds , and seasonal moving average parameters qs For example, the model 0,1,2 0,1,1 describes a model that includes no autoregressive parameters, 2 regular moving average parameters and 1 seasonal moving average parameter, and these parameters were computed for the series after it was differenced once with lag 1, and once seasonally differenced The seasonal lag used for the seasonal parameters is usually determined during the identification phase and must be explicitly specified. The general recommendations concerning the selection of parameters to be estimated based on ACF and PACF also apply to seasonal models The main difference is that in seasonal series, AC F and PACF will show sizable coefficients at multiples of the seasonal lag in addition to their overall patterns reflecting the non seasonal components of the series. Parameter Estimation. There are several different methods for estimating the parameters All of them should produce very similar estimates, but may be more or less efficient for any given model In general, during the parameter estimation phase a function minimization algorithm is used the so-called quasi-Newton method refer to the description of the Nonlinear Estimation method to maximize the likelihood probability of the observed series, given the parameter values In practice, this requires the calculation of the conditional sums of squares SS of the residuals, given the respective parameters Different methods have been proposed to compute the SS for the residuals 1 the approximate maximum likelihood method according to McLeod and Sales 1983 , 2 the approximate maximum likelihood method with backcasting, and 3 the exact max imum likelihood method according to Melard 1984parison of methods In general, all methods should yield very similar parameter estimates Also, all methods are about equally efficient in most real-world time series applications However, method 1 above, approximate maximum likelihood, no backcasts is the fastest, and should be used in particular for very long time series e g with more than 30,000 observations Melard s exact maximum likelihood method number 3 above may also become inefficient when used to estimate parameters for seasonal models with long seasonal lags e g with yearly lags of 365 days On the other hand, you should always use the approximate maximum likelihood method first in order to establish initial parameter estimates that are very close to the actual final values thus, usually only a few iterations with the exact maximum likelihood method 3 above are necessary to finalize the parameter estimates. Parameter standard errors For all parameter estimates, you will compute so - called asymptotic standard errors These are computed from the matrix of second-order partial derivatives that is approximated via finite differencing see also the respective discussion in Nonlinear Estimation. Penalty value As mentioned above, the estimation procedure requires that the conditional sums of squares of the ARIMA residuals be minimized If the model is inappropriate, it may happen during the iterative estimation process that the parameter estimates become very large, and, in fact, invalid In that case, it will assign a very large value a so-called penalty value to the SS This usually entices the iteration process to move the parameters away from invalid ranges However, in some cases even this strategy fails, and you may see on the screen during the Estimation procedure very large values for the SS in consecutive iterations In that case, carefully evaluate the appropriateness of your model If your model contains many parameters, and perhaps an intervention component see below , you may try again with different parameter start values. Evaluation of the Model. Parameter estimates You will report approximate t values, computed from the parameter standard errors see above If not significant, the respective parameter can in most cases be dropped from the model without affecting substantially the overall fit of the model. Other quality criteria Another straightforward and common measure of the reliability of the model is the accuracy of its forecasts generated based on partial data so that the forecasts can be compared with known original observations. However, a good model should not only provide sufficiently accurate forecasts, it should also be parsimonious and produce statistically independent residuals that contain only noise and no systematic components e g the correlogram of residuals should not reveal any serial dependencies A good test of the model is a to plot the residuals and inspect them for any systematic trends, and b to examine the autocorrelogram of residuals there should be no serial dependency between residuals. Analysis of residuals The major concern here is that the residuals are systematically distributed across the series e g they could be negative in the first part of the series and approach zero in the second part or that they contain some serial dependency which may suggest that the ARIMA model is inadequate The analysis of ARIMA residuals constitutes an important test of the model The estimation procedure assumes that the residual are not auto - correlated and that they are normally distributed. Limitations The ARIMA method is appropriate only for a time series that is stationary i e its mean, variance, and autocorrelation should be approximately constant through time and it is recommended that there are at least 50 observations in the input data It is also assumed that the values of the estimated parameters are constant throughout the series. Interrupted Time Series ARIMA. A common research questions in time series analysis is whether an outside event affected subsequent observations For example, did the implementation of a new economic policy improve economic performance did a new anti-crime law affect subsequent crime rates and so on In general, we would like to evaluate the impact of one or more discrete events on the values in the time series This type of interrupted time series analysis is described in detail in McDowall, McCleary, Meidinger, Hay 1980 McDowall, et al distinguish between three major types of impacts that are possible 1 permanent abrupt, 2 permanent gradual, and 3 abrupt temporary See also. Exponential Smoothing. General Introduction. Exponential smoothing has become very popular as a forecasting method for a wide variety of time series data Historically, the method was independently developed by Brown and Holt Brown worked for the US Navy during World War II, where his assignment was to design a tracking system for fire-control information to compute the location of submarines Later, he applied this technique to the forecasting of demand for spare parts an inventory control problem He described those ideas in his 1959 book on inventory control Holt s research was sponsored by the Office of Naval Research independently, he developed exponential smoothing models for constant processes, processes with linear trends, and for seasonal data. Gardner 1985 proposed a unified classification of exponential smoothing methods Excellent introductions can also be found in Makridakis, Wheelwright, and McGee 1983 , Makridakis and Wheelwright 1989 , Montgomery, Johnson, Gardiner 1990.Simple Exponential Smoothing. A simple and pragmatic model for a time series would be to consider each observation as consisting of a constant b and an error component epsilon , that is X t b t The constant b is relatively stable in each segment of the series, but may change slowly over time If appropriate, then one way to isolate the true value of b and thus the systematic or predictable part of the serie s, is to compute a kind of moving average, where the current and immediately preceding younger observations are assigned greater weight than the respective older observations Simple exponential smoothing accomplishes exactly such weighting, where exponentially smaller weights are assigned to older observations The specific formula for simple exponential smoothing is. When applied recursively to each successive observation in the series, each new smoothed value forecast is computed as the weighted average of the current observation and the previous smoothed observation the previous smoothed observation was computed in turn from the previous observed value and the smoothed value before the previous observation, and so on Thus, in effect, each smoothed value is the weighted average of the previous observations, where the weights decrease exponentially depending on the value of parameter alpha If is equal to 1 one then the previous observations are ignored entirely if is equal to 0 zero , t hen the current observation is ignored entirely, and the smoothed value consists entirely of the previous smoothed value which in turn is computed from the smoothed observation before it, and so on thus all smoothed values will be equal to the initial smoothed value S 0 Values of in-between will produce intermediate results. Even though significant work has been done to study the theoretical properties of simple and complex exponential smoothing e g see Gardner, 1985 Muth, 1960 see also McKenzie, 1984, 1985 , the method has gained popularity mostly because of its usefulness as a forecasting tool For example, empirical research by Makridakis et al 1982, Makridakis, 1983 , has shown simple exponential smoothing to be the best choice for one-period-ahead forecasting, from among 24 other time series methods and using a variety of accuracy measures see also Gross and Craig, 1974, for additional empirical evidence Thus, regardless of the theoretical model for the process underlying the observ ed time series, simple exponential smoothing will often produce quite accurate forecasts. Choosing the Best Value for Parameter alpha. Gardner 1985 discusses various theoretical and empirical arguments for selecting an appropriate smoothing parameter Obviously, looking at the formula presented above, should fall into the interval between 0 zero and 1 although, see Brenner et al 1968, for an ARIMA perspective, implying 0 2 Gardner 1985 reports that among practitioners, an smaller than 30 is usually recommended However, in the study by Makridakis et al 1982 , values above 30 frequently yielded the best forecasts After reviewing the literature on this topic, Gardner 1985 concludes that it is best to estimate an optimum from the data see below , rather than to guess and set an artificially low value. Estimating the best value from the data In practice, the smoothing parameter is often chosen by a grid search of the parameter space that is, different solutions for are tried starting, for examp le, with 0 1 to 0 9, with increments of 0 1 Then is chosen so as to produce the smallest sums of squares or mean squares for the residuals i e observed values minus one-step-ahead forecasts this mean squared error is also referred to as ex post mean squared error, ex post MSE for short. Indices of Lack of Fit Error. The most straightforward way of evaluating the accuracy of the forecasts based on a particular value is to simply plot the observed values and the one-step-ahead forecasts This plot can also include the residuals scaled against the right Y - axis , so that regions of better or worst fit can also easily be identified. This visual check of the accuracy of forecasts is often the most powerful method for determining whether or not the current exponential smoothing model fits the data In addition, besides the ex post MSE criterion see previous paragraph , there are other statistical measures of error that can be used to determine the optimum parameter see Makridakis, Wheelwright, an d McGee, 1983.Mean error The mean error ME value is simply computed as the average error value average of observed minus one-step-ahead forecast Obviously, a drawback of this measure is that positive and negative error values can cancel each other out, so this measure is not a very good indicator of overall fit. Mean absolute error The mean absolute error MAE value is computed as the average absolute error value If this value is 0 zero , the fit forecast is perfect As compared to the mean squared error value, this measure of fit will de-emphasize outliers, that is, unique or rare large error values will affect the MAE less than the MSE value. Sum of squared error SSE , Mean squared error These values are computed as the sum or average of the squared error values This is the most commonly used lack-of-fit indicator in statistical fitting procedures. Percentage error PE All the above measures rely on the actual error value It may seem reasonable to rather express the lack of fit in terms of the relative deviation of the one-step-ahead forecasts from the observed values, that is, relative to the magnitude of the observed values For example, when trying to predict monthly sales that may fluctuate widely e g seasonally from month to month, we may be satisfied if our prediction hits the target with about 10 accuracy In other words, the absolute errors may be not so much of interest as are the relative errors in the forecasts To assess the relative error, various indices have been proposed see Makridakis, Wheelwright, and McGee, 1983 The first one, the percentage error value, is computed as. where X t is the observed value at time t and F t is the forecasts smoothed values. Mean percentage error MPE This value is computed as the average of the PE values. Mean absolute percentage error MAPE As is the case with the mean error value ME, see above , a mean percentage error near 0 zero can be produced by large positive and negative percentage errors that cancel each other out Thus, a better measure of relative overall fit is the mean absolute percentage error Also, this measure is usually more meaningful than the mean squared error For example, knowing that the average forecast is off by 5 is a useful result in and of itself, whereas a mean squared error of 30 8 is not immediately interpretable. Automatic search for best parameter A quasi-Newton function minimization procedure the same as in ARIMA is used to minimize either the mean squared error, mean absolute error, or mean absolute percentage error In most cases, this procedure is more efficient than the grid search particularly when more than one parameter must be determined , and the optimum parameter can quickly be identified. The first smoothed value S 0 A final issue that we have neglected up to this point is the problem of the initial value, or how to start the smoothing process If you look back at the formula above, it is evident that you need an S 0 value in order to compute the smoothed value forecast fo r the first observation in the series Depending on the choice of the parameter i e when is close to zero , the initial value for the smoothing process can affect the quality of the forecasts for many observations As with most other aspects of exponential smoothing it is recommended to choose the initial value that produces the best forecasts On the other hand, in practice, when there are many leading observations prior to a crucial actual forecast, the initial value will not affect that forecast by much, since its effect will have long faded from the smoothed series due to the exponentially decreasing weights, the older an observation the less it will influence the forecast. Seasonal and Non-Seasonal Models With or Without Trend. The discussion above in the context of simple exponential smoothing introduced the basic procedure for identifying a smoothing parameter, and for evaluating the goodness-of-fit of a model In addition to simple exponential smoothing, more complex models have been developed to accommodate time series with seasonal and trend components The general idea here is that forecasts are not only computed from consecutive previous observations as in simple exponential smoothing , but an independent smoothed trend and seasonal component can be added Gardner 1985 discusses the different models in terms of seasonality none, additive, or multiplicative and trend none, linear, exponential, or damped. Additive and multiplicative seasonality Many time series data follow recurring seasonal patterns For example, annual sales of toys will probably peak in the months of November and December, and perhaps during the summer with a much smaller peak when children are on their summer break This pattern will likely repeat every year, however, the relative amount of increase in sales during December may slowly change from year to year Thus, it may be useful to smooth the seasonal component independently with an extra parameter, usually denoted as delta. Seasonal components can be additive in nature or multiplicative For example, during the month of December the sales for a particular toy may increase by 1 million dollars every year Thus, we could add to our forecasts for every December the amount of 1 million dollars over the respective annual average to account for this seasonal fluctuation In this case, the seasonality is additive. Alternatively, during the month of December the sales for a particular toy may increase by 40 , that is, increase by a factor of 1 4 Thus, when the sales for the toy are generally weak, than the absolute dollar increase in sales during December will be relatively weak but the percentage will be constant if the sales of the toy are strong, than the absolute dollar increase in sales will be proportionately greater Again, in this case the sales increase by a certain factor and the seasonal component is thus multiplicative in nature i e the multiplicative seasonal component in this case would be 1 4.In plots of the series, the distinguishing characteristic between these two types of seasonal components is that in the additive case, the series shows steady seasonal fluctuations, regardless of the overall level of the series in the multiplicative case, the size of the seasonal fluctuations vary, depending on the overall level of the series. The seasonal smoothing parameter In general the one-step-ahead forecasts are computed as for no trend models, for linear and exponential trend models a trend component is added to the model see below. In this formula, S t stands for the simple exponentially smoothed value of the series at time t and I t-p stands for the smoothed seasonal factor at time t minus p the length of the season Thus, compared to simple exponential smoothing, the forecast is enhanced by adding or multiplying the simple smoothed value by the predicted seasonal component This seasonal component is derived analogous to the S t value from simple exponential smoothing as. Put into words, the predicted seaso nal component at time t is computed as the respective seasonal component in the last seasonal cycle plus a portion of the error e t the observed minus the forecast value at time t Considering the formulas above, it is clear that parameter can assume values between 0 and 1 If it is zero, then the seasonal component for a particular point in time is predicted to be identical to the predicted seasonal component for the respective time during the previous seasonal cycle, which in turn is predicted to be identical to that from the previous cycle, and so on Thus, if is zero, a constant unchanging seasonal component is used to generate the one-step-ahead forecasts If the parameter is equal to 1, then the seasonal component is modified maximally at every step by the respective forecast error times 1- which we will ignore for the purpose of this brief introduction In most cases, when seasonality is present in the time series, the optimum parameter will fall somewhere between 0 zero and 1 one. L inear, exponential, and damped trend To remain with the toy example above, the sales for a toy can show a linear upward trend e g each year, sales increase by 1 million dollars , exponential growth e g each year, sales increase by a factor of 1 3 , or a damped trend during the first year sales increase by 1 million dollars during the second year the increase is only 80 over the previous year, i e 800,000 during the next year it is again 80 less than the previous year, i e 800,000 8 640,000 etc Each type of trend leaves a clear signature that can usually be identified in the series shown below in the brief discussion of the different models are icons that illustrate the general patterns In general, the trend factor may change slowly over time, and, again, it may make sense to smooth the trend component with a separate parameter denoted gamma for linear and exponential trend models, and phi for damped trend models. The trend smoothing parameters linear and exponential trend and damped tre nd Analogous to the seasonal component, when a trend component is included in the exponential smoothing process, an independent trend component is computed for each time, and modified as a function of the forecast error and the respective parameter If the parameter is 0 zero , than the trend component is constant across all values of the time series and for all forecasts If the parameter is 1, then the trend component is modified maximally from observation to observation by the respective forecast error Parameter values that fall in-between represent mixtures of those two extremes Parameter is a trend modification parameter, and affects how strongly changes in the trend will affect estimates of the trend for subsequent forecasts, that is, how quickly the trend will be damped or increased. Classical Seasonal Decomposition Census Method 1.General Introduction. Suppose you recorded the monthly passenger load on international flights for a period of 12 years see Box Jenkins, 1976 If you plot those data, it is apparent that 1 there appears to be a linear upwards trend in the passenger loads over the years, and 2 there is a recurring pattern or seasonality within each year i e most travel occurs during the summer months, and a minor peak occurs during the December holidays The purpose of the seasonal decomposition method is to isolate those components, that is, to de-compose the series into the trend effect, seasonal effects, and remaining variability The classic technique designed to accomplish this decomposition is known as the Census I method This technique is described and discussed in detail in Makridakis, Wheelwright, and McGee 1983 , and Makridakis and Wheelwright 1989.General model The general idea of seasonal decomposition is straightforward In general, a time series like the one described above can be thought of as consisting of four different components 1 A seasonal component denoted as S t where t stands for the particular point in time 2 a trend component T t , 3 a cyclical component C t , and 4 a random, error, or irregular component I t The difference between a cyclical and a seasonal component is that the latter occurs at regular seasonal intervals, while cyclical factors have usually a longer duration that varies from cycle to cycle In the Census I method, the trend and cyclical components are customarily combined into a trend-cycle component TC t The specific functional relationship between these components can assume different forms However, two straightforward possibilities are that they combine in an additive or a multiplicative fashion. Here X t stands for the observed value of the time series at time t Given some a priori knowledge about the cyclical factors affecting the series e g business cycles , the estimates for the different components can be used to compute forecasts for future observations However, the Exponential smoothing method, which can also incorporate seasonality and trend components, is the preferred technique for f orecasting purposes. Additive and multiplicative seasonality Let s consider the difference between an additive and multiplicative seasonal component in an example The annual sales of toys will probably peak in the months of November and December, and perhaps during the summer with a much smaller peak when children are on their summer break This seasonal pattern will likely repeat every year Seasonal components can be additive or multiplicative in nature For example, during the month of December the sales for a particular toy may increase by 3 million dollars every year Thus, we could add to our forecasts for every December the amount of 3 million to account for this seasonal fluctuation In this case, the seasonality is additive Alternatively, during the month of December the sales for a particular toy may increase by 40 , that is, increase by a factor of 1 4 Thus, when the sales for the toy are generally weak, then the absolute dollar increase in sales during December will be relatively weak but the percentage will be constant if the sales of the toy are strong, then the absolute dollar increase in sales will be proportionately greater Again, in this case the sales increase by a certain factor and the seasonal component is thus multiplicative in nature i e the multiplicative seasonal component in this case would be 1 4 In plots of series, the distinguishing characteristic between these two types of seasonal components is that in the additive case, the series shows steady seasonal fluctuations, regardless of the overall level of the series in the multiplicative case, the size of the seasonal fluctuations vary, depending on the overall level of the series. Additive and multiplicative trend-cycle We can extend the previous example to illustrate the additive and multiplicative trend-cycle components In terms of our toy example, a fashion trend may produce a steady increase in sales e g a trend towards more educational toys in general as with the seasonal component, this trend may be additive sales increase by 3 million dollars per year or multiplicative sales increase by 30 , or by a factor of 1 3, annually in nature In addition, cyclical components may impact sales to reiterate, a cyclical component is different from a seasonal component in that it usually is of longer duration, and that it occurs at irregular intervals For example, a particular toy may be particularly hot during a summer season e g a particular doll which is tied to the release of a major children s movie, and is promoted with extensive advertising Again such a cyclical component can effect sales in an additive manner or multiplicative manner. The Seasonal Decomposition Census I standard formulas are shown in Makridakis, Wheelwright, and McGee 1983 , and Makridakis and Wheelwright 1989.Moving average First a moving average is computed for the series, with the moving average window width equal to the length of one season If the length of the season is even, then the user can choose t o use either equal weights for the moving average or unequal weights can be used, where the first and last observation in the moving average window are averaged. Ratios or differences In the moving average series, all seasonal within-season variability will be eliminated thus, the differences in additive models or ratios in multiplicative models of the observed and smoothed series will isolate the seasonal component plus irregular component Specifically, the moving average is subtracted from the observed series for additive models or the observed series is divided by the moving average values for multiplicative models. Seasonal components The seasonal component is then computed as the average for additive models or medial average for multiplicative models for each point in the season. The medial average of a set of values is the mean after the smallest and largest values are excluded The resulting values represent the average seasonal component of the series. Seasonally adjusted series The original series can be adjusted by subtracting from it additive models or dividing it by multiplicative models the seasonal component. The resulting series is the seasonally adjusted series i e the seasonal component will be removed. Trend-cycle component Remember that the cyclical component is different from the seasonal component in that it is usually longer than one season, and different cycles can be of different lengths The combined trend and cyclical component can be approximated by applying to the seasonally adjusted series a 5 point centered weighed moving average smoothing transformation with the weights of 1, 2, 3, 2, 1.Random or irregular component Finally, the random or irregular error component can be isolated by subtracting from the seasonally adjusted series additive m odels or dividing the adjusted series by multiplicative models the trend-cycle component. X-11 Census Method II Seasonal Adjustment. The general ideas of seasonal decomposition and adjustment are discussed in the context of the Census I seasonal adjustment method Seasonal Decomposition Census I The Census method II 2 is an extension and refinement of the simple adjustment method Over the years, different versions of the Census method II evolved at the Census Bureau the method that has become most popular and is used most widely in government and business is the so-called X-11 variant of the Census method II see Hiskin, Young, Musgrave, 1967 Subsequently, the term X-11 has become synonymous with this refined version of the Census method II In addition to the documentation that can be obtained from the Census Bureau, a detailed summary of this method is also provided in Makridakis, Wheelwright, and McGee 1983 and Makridakis and Wheelwright 1989.For more information on this method, see the following topics. For more information on other Time Series methods, see Time Series Analysis - Index and the following topics. Seasonal Adjustment Basic Ideas and Terms. Suppose you recorded the monthly passenger load on international flights for a period of 12 years see Box Jenkins, 1976 If you plot those data, it is apparent that 1 there appears to be an upwards linear trend in the passenger loads over the years, and 2 there is a recurring pattern or seasonality within each year i e most travel occurs during the summer months, and a minor peak occurs during the December holidays The purpose of seasonal decomposition and adjustment is to isolate those components, that is, to de-compose the series into the trend effect, seasonal effects, and remaining variability The classic technique designed to accomplish this decomposition was developed in the 1920 s and is also known as the Census I method see the Census I overview section This technique is also described and discussed in detail in M akridakis, Wheelwright, and McGee 1983 , and Makridakis and Wheelwright 1989.General model The general idea of seasonal decomposition is straightforward In general, a time series like the one described above can be thought of as consisting of four different components 1 A seasonal component denoted as S t where t stands for the particular point in time 2 a trend component T t , 3 a cyclical component C t , and 4 a random, error, or irregular component I t The difference between a cyclical and a seasonal component is that the latter occurs at regular seasonal intervals, while cyclical factors usually have a longer duration that varies from cycle to cycle The trend and cyclical components are customarily combined into a trend-cycle component TC t The specific functional relationship between these components can assume different forms However, two straightforward possibilities are that they combine in an additive or a multiplicative fashion. X t represents the observed value of the time se ries at time t. Given some a priori knowledge about the cyclical factors affecting the series e g business cycles , the estimates for the different components can be used to compute forecasts for future observations However, the Exponential smoothing method, which can also incorporate seasonality and trend components, is the preferred technique for forecasting purposes. Additive and multiplicative seasonality Consider the difference between an additive and multiplicative seasonal component in an example The annual sales of toys will probably peak in the months of November and December, and perhaps during the summer with a much smaller peak when children are on their summer break This seasonal pattern will likely repeat every year Seasonal components can be additive or multiplicative in nature For example, during the month of December the sales for a particular toy may increase by 3 million dollars every year Thus, you could add to your forecasts for every December the amount of 3 million to account for this seasonal fluctuation In this case, the seasonality is additive. Alternatively, during the month of December the sales for a particular toy may increase by 40 , that is, increase by a factor of 1 4 Thus, when the sales for the toy are generally weak, then the absolute dollar increase in sales during December will be relatively weak but the percentage will be constant if the sales of the toy are strong, then the absolute dollar increase in sales will be proportionately greater Again, in this case the sales increase by a certain factor and the seasonal component is thus multiplicative in nature i e the multiplicative seasonal component in this case would be 1 4 In plots of series, the distinguishing characteristic between these two types of seasonal components is that in the additive case, the series shows steady seasonal fluctuations, regardless of the overall level of the series in the multiplicative case, the size of the seasonal fluctuations vary, depending on the overall level of the series. Additive and multiplicative trend-cycle The previous example can be extended to illustrate the additive and multiplicative trend-cycle components In terms of the toy example, a fashion trend may produce a steady increase in sales e g a trend towards more educational toys in general as with the seasonal component, this trend may be additive sales increase by 3 million dollars per year or multiplicative sales increase by 30 , or by a factor of 1 3, annually in nature In addition, cyclical components may impact sales To reiterate, a cyclical component is different from a seasonal component in that it usually is of longer duration, and that it occurs at irregular intervals For example, a particular toy may be particularly hot during a summer season e g a particular doll which is tied to the release of a major children s movie, and is promoted with extensive advertising Again such a cyclical component can effect sales in an additive manner or multiplicative man ner. The Census II Method. The basic method for seasonal decomposition and adjustment outlined in the Basic Ideas and Terms topic can be refined in several ways In fact, unlike many other time-series modeling techniques e g ARIMA which are grounded in some theoretical model of an underlying process, the X-11 variant of the Census II method simply contains many ad hoc features and refinements, that over the years have proven to provide excellent estimates for many real-world applications see Burman, 1979, Kendal Ord, 1990, Makridakis Wheelwright, 1989 Wallis, 1974 Some of the major refinements are listed below. Trading-day adjustment Different months have different numbers of days, and different numbers of trading-days i e Mondays, Tuesdays, etc When analyzing, for example, monthly revenue figures for an amusement park, the fluctuation in the different numbers of Saturdays and Sundays peak days in the different months will surely contribute significantly to the variability in monthly reven ues The X-11 variant of the Census II method allows the user to test whether such trading-day variability exists in the series, and, if so, to adjust the series accordingly. Extreme values Most real-world time series contain outliers, that is, extreme fluctuations due to rare events For example, a strike may affect production in a particular month of one year Such extreme outliers may bias the estimates of the seasonal and trend components The X-11 procedure includes provisions to deal with extreme values through the use of statistical control principles, that is, values that are above or below a certain range expressed in terms of multiples of sigma the standard deviation can be modified or dropped before final estimates for the seasonality are computed. Multiple refinements The refinement for outliers, extreme values, and different numbers of trading-days can be applied more than once, in order to obtain successively improved estimates of the components The X-11 method applies a series of successive refinements of the estimates to arrive at the final trend-cycle, seasonal, and irregular components, and the seasonally adjusted series. Tests and summary statistics In addition to estimating the major components of the series, various summary statistics can be computed For example, analysis of variance tables can be prepared to test the significance of seasonal variability and trading-day variability see above in the series the X-11 procedure will also compute the percentage change from month to month in the random and trend-cycle components As the duration or span in terms of months or quarters for quarterly X-11 increases, the change in the trend-cycle component will likely also increase, while the change in the random component should remain about the same The width of the average span at which the changes in the random component are about equal to the changes in the trend-cycle component is called the month quarter for cyclical dominance or MCD QCD for short For exam ple, if the MCD is equal to 2, then you can infer that over a 2-month span the trend-cycle will dominate the fluctuations of the irregular random component These and various other results are discussed in greater detail below. Result Tables Computed by the X-11 Method. The computations performed by the X-11 procedure are best discussed in the context of the results tables that are reported The adjustment process is divided into seven major steps, which are customarily labeled with consecutive letters A through G. Prior adjustment monthly seasonal adjustment only Before any seasonal adjustment is performed on the monthly time series, various prior user - defined adjustments can be incorporated The user can specify a second series that contains prior adjustment factors the values in that series will either be subtracted additive model from the original series, or the original series will be divided by these values multiplicative model For multiplicative models, user-specified trading-day adj ustment weights can also be specified These weights will be used to adjust the monthly observations depending on the number of respective trading-days represented by the observation. Preliminary estimation of trading-day variation monthly X-11 and weights Next, preliminary trading-day adjustment factors monthly X-11 only and weights for reducing the effect of extreme observations are computed. Final estimation of trading-day variation and irregular weights monthly X - 11 The adjustments and weights computed in B above are then used to derive improved trend-cycle and seasonal estimates These improved estimates are used to compute the final trading-day factors monthly X-11 only and weights. Final estimation of seasonal factors, trend-cycle, irregular, and seasonally adjusted series The final trading-day factors and weights computed in C above are used to compute the final estimates of the components. Modified original, seasonally adjusted, and irregular series The original and final seasonall y adjusted series, and the irregular component are modified for extremes The resulting modified series allow the user to examine the stability of the seasonal adjustment. Month quarter for cyclical dominance MCD, QCD , moving average, and summary measures In this part of the computations, various summary measures see below are computed to allow the user to examine the relative importance of the different components, the average fluctuation from month-to-month quarter-to-quarter , the average number of consecutive changes in the same direction average number of runs , etc. Charts Finally, you will compute various charts graphs to summarize the results For example, the final seasonally adjusted series will be plotted, in chronological order, or by month see below. Specific Description of all Result Tables Computed by the X-11 Method. In each part A through G of the analysis see Results Tables Computed by the X-11 Method , different result tables are computed Customarily, these tables are num bered, and also identified by a letter to indicate the respective part of the analysis For example, table B 11 shows the initial seasonally adjusted series C 11 is the refined seasonally adjusted series, and D 11 is the final seasonally adjusted series Shown below is a list of all available tables Those tables identified by an asterisk are not available applicable when analyzing quarterly series Also, for quarterly adjustment, some of the computations outlined below are slightly different for example instead of a 12-term monthly moving average, a 4-term quarterly moving average is applied to compute the seasonal factors the initial trend-cycle estimate is computed via a centered 4-term moving average, the final trend-cycle estimate in each part is computed by a 5-term Henderson average. Following the convention of the Bureau of the Census version of the X-11 method, three levels of printout detail are offered Standard 17 to 27 tables , Long 27 to 39 tables , and Full 44 to 59 tables In the description of each table below, the letters S, L, and F are used next to each title to indicate, which tables will be displayed and or printed at the respective setting of the output option For the charts, two levels of detail are available Standard and All. See the table name below, to obtain more information about that table. A 2 Prior Monthly Adjustment S Factors. Tables B 14 through B 16, B18, and B19 Adjustment for trading-day variation These tables are only available when analyzing monthly series Different months contain different numbers of days of the week i e Mondays, Tuesdays, etc In some series, the variation in the different numbers of trading-days may contribute significantly to monthly fluctuations e g the monthly revenues of an amusement park will be greatly influenced by the number of Saturdays Sundays in each month The user can specify initial weights for each trading-day see A 4 , and or these weights can be estimated from the data the user can also choose to apply those weights conditionally, i e only if they explain a significant proportion of variance. B 14 Extreme Irregular Values Excluded from Trading-day Regression L. B 15 Preliminary Trading-day Regression L. B 16 Trading-day Adjustment Factors Derived from Regression Coefficients F. B 17 Preliminary Weights for Irregular Component L. B 18 Trading-day Factors Derived from Combined Daily Weights F. B 19 Original Series Adjusted for Trading-day and Prior Variation F. C 1 Original Series Modified by Preliminary Weights and Adjusted for Trading-day and Prior Variation L. Tables C 14 through C 16, C 18, and C 19 Adjustment for trading-day variation These tables are only available when analyzing monthly series, and when adjustment for trading-day variation is requested In that case, the trading-day adjustment factors are computed from the refined adjusted series, analogous to the adjustment performed in part B B 14 through B 16, B 18 and B 19. C 14 Extreme Irregular Values Excluded from Trading-day Regression S. C 15 Final Trading-day Regression S. C 16 Final Trading-day Adjustment Factor s Derived from Regression X11 output Coefficients S. C 17 Final Weights for Irregular Component S. C 18 Final Trading-day Factors Derived From Combined Daily Weights S. C 19 Original Series Adjusted for Trading-day and Prior Variation S. D 1 Original Series Modified by Final Weights and Adjusted for Trading-day and Prior Variation L. Distributed Lags Analysis. For more information on other Time Series methods, see Time Series Analysis - Index and the following topics. General Purpose. Distributed lags analysis is a specialized technique for examining the relationships between variables that involve some delay For example, suppose that you are a manufacturer of computer software, and you want to determine the relationship between the number of inquiries that are received, and the number of orders that are placed by your customers You could record those numbers monthly for a one-year period, and then correlate the two variables However, obviously inquiries will precede actual orders, and you c an expect that the number of orders will follow the number of inquiries with some delay Put another way, there will be a time lagged correlation between the number of inquiries and the number of orders that are received. Time-lagged correlations are particularly common in econometrics For example, the benefits of investments in new machinery usually only become evident after some time Higher income will change people s choice of rental apartments, however, this relationship will be lagged because it will take some time for people to terminate their current leases, find new apartments, and move In general, the relationship between capital appropriations and capital expenditures will be lagged, because it will require some time before investment decisions are actually acted upon. In all of these cases, we have an independent or explanatory variable that affects the dependent variables with some lag The distributed lags method allows you to investigate those lags. Detailed discussions of dis tributed lags correlation can be found in most econometrics textbooks, for example, in Judge, Griffith, Hill, Luetkepohl, and Lee 1985 , Maddala 1977 , and Fomby, Hill, and Johnson 1984 In the following paragraphs we will present a brief description of these methods We will assume that you are familiar with the concept of correlation see Basic Statistics , and the basic ideas of multiple regression see Multiple Regression. General Model. Suppose we have a dependent variable y and an independent or explanatory variable x which are both measured repeatedly over time In some textbooks, the dependent variable is also referred to as the endogenous variable, and the independent or explanatory variable the exogenous variable The simplest way to describe the relationship between the two would be in a simple linear relationship. In this equation, the value of the dependent variable at time t is expressed as a linear function of x measured at times t t-1 t-2 , etc Thus, the dependent variable is a linear function of x and x is lagged by 1, 2, etc time periods The beta weights i can be considered slope parameters in this equation You may recognize this equation as a special case of the general linear regression equation see the Multiple Regression overview If the weights for the lagged time periods are statistically significant, we can conclude that the y variable is predicted or explained with the respective lag. Almon Distributed Lag. A common problem that often arises when computing the weights for the multiple linear regression model shown above is that the values of adjacent in time values in the x variable are highly correlated In extreme cases, their independent contributions to the prediction of y may become so redundant that the correlation matrix of measures can no longer be inverted, and thus, the beta weights cannot be computed In less extreme cases, the computation of the beta weights and their standard errors can become very imprecise, due to round-off error In the co ntext of Multiple Regression this general computational problem is discussed as the multicollinearity or matrix ill-conditioning issue. Almon 1965 proposed a procedure that will reduce the multicollinearity in this case Specifically, suppose we express each weight in the linear regression equation in the following manner. Almon could show that in many cases it is easier i e it avoids the multicollinearity problem to estimate the alpha values than the beta weights directly Note that with this method, the precision of the beta weight estimates is dependent on the degree or order of the polynomial approximation. Misspecifications A general problem with this technique is that, of course, the lag length and correct polynomial degree are not known a priori The effects of misspecifications of these parameters are potentially serious in terms of biased estimation This issue is discussed in greater detail in Frost 1975 , Schmidt and Waud 1973 , Schmidt and Sickles 1975 , and Trivedi and Pagan 1979.Single Spectrum Fourier Analysis. Spectrum analysis is concerned with the exploration of cyclical patterns of data The purpose of the analysis is to decompose a complex time series with cyclical components into a few underlying sinusoidal sine and cosine functions of particular wavelengths The term spectrum provides an appropriate metaphor for the nature of this analysis Suppose you study a beam of white sun light, which at first looks like a random white noise accumulation of light of different wavelengths However, when put through a prism, we can separate the different wave lengths or cyclical components that make up white sun light In fact, via this technique we can now identify and distinguish between different sources of light Thus, by identifying the important underlying cyclical components, we have learned something about the phenomenon of interest In essence, performing spectrum analysis on a time series is like putting the series through a prism in order to identify the wave l engths and importance of underlying cyclical components As a result of a successful analysis, you might uncover just a few recurring cycles of different lengths in the time series of interest, which at first looked more or less like random noise. A much cited example for spectrum analysis is the cyclical nature of sun spot activity e g see Bloomfield, 1976, or Shumway, 1988 It turns out that sun spot activity varies over 11 year cycles Other examples of celestial phenomena, weather patterns, fluctuations in commodity prices, economic activity, etc are also often used in the literature to demonstrate this technique To contrast this technique with ARIMA or Exponential Smoothing the purpose of spectrum analysis is to identify the seasonal fluctuations of different lengths, while in the former types of analysis, the length of the seasonal component is usually known or guessed a priori and then included in some theoretical model of moving averages or autocorrelations. The classic text on spec trum analysis is Bloomfield 1976 however, other detailed discussions can be found in Jenkins and Watts 1968 , Brillinger 1975 , Brigham 1974 , Elliott and Rao 1982 , Priestley 1981 , Shumway 1988 , or Wei 1989.For more information, see Time Series Analysis - Index and the following topics. Cross-Spectrum Analysis. For more information, see Time Series Analysis - Index and the following topics. General Introduction. Cross-spectrum analysis is an extension of Single Spectrum Fourier Analysis to the simultaneous analysis of two series In the following paragraphs, we will assume that you have already read the introduction to single spectrum analysis Detailed discussions of this technique can be found in Bloomfield 1976 , Jenkins and Watts 1968 , Brillinger 1975 , Brigham 1974 , Elliott and Rao 1982 , Priestley 1981 , Shumway 1988 , or Wei 1989.Strong periodicity in the series at the respective frequency A much cited example for spectrum analysis is the cyclical nature of sun spot activity e g see Bloomfield, 1976, or Shumway, 1988 It turns out that sun spot activity varies over 11 year cycles Other examples of celestial phenomena, weather patterns, fluctuations in commodity prices, economic activity, etc are also often used in the literature to demonstrate this technique. The purpose of cross-spectrum analysis is to uncover the correlations between two series at different frequencies For example, sun spot activity may be related to weather phenomena here on earth If so, then if we were to record those phenomena e g yearly average temperature and submit the resulting series to a cross-spectrum analysis together with the sun spot data, we may find that the weather indeed correlates with the sunspot activity at the 11 year cycle That is, we may find a periodicity in the weather data that is in-sync with the sun spot cycles We can easily think of other areas of research where such knowledge could be very useful for example, various economic indicators may show similar correlate d cyclical behavior various physiological measures likely will also display coordinated i e correlated cyclical behavior, and so on. Basic Notation and Principles. A simple example Consider the following two series with 16 cases. Results for Each Variable. The complete summary contains all spectrum statistics computed for each variable, as described in the Single Spectrum Fourier Analysis overview section Looking at the results shown above, it is clear that both variables show strong periodicities at the frequencies 0625 and 1875.Cross-Periodogram, Cross-Density, Quadrature-Density, Cross-Amplitude. Analogous to the results for the single variables, the complete summary will also display periodogram values for the cross periodogram However, the cross-spectrum consists of complex numbers that can be divided into a real and an imaginary part These can be smoothed to obtain the cross-density and quadrature density quad density for short estimates, respectively The reasons for smoothing, and th e different common weight functions for smoothing are discussed in the Single Spectrum Fourier Analysis The square root of the sum of the squared cross-density and quad-density values is called the cross - amplitude The cross-amplitude can be interpreted as a measure of covariance between the respective frequency components in the two series Thus we can conclude from the results shown in the table above that the 0625 and 1875 frequency components in the two series covary. Squared Coherency, Gain, and Phase Shift. There are additional statistics that can be displayed in the complete summary. Squared coherency You can standardize the cross-amplitude values by squaring them and dividing by the product of the spectrum density estimates for each series The result is called the squared coherency which can be interpreted similar to the squared correlation coefficient see Correlations - Overview , that is, the coherency value is the squared correlation between the cyclical components in the two se ries at the respective frequency However, the coherency values should not be interpreted by themselves for example, when the spectral density estimates in both series are very small, large coherency values may result the divisor in the computation of the coherency values will be very small , even though there are no strong cyclical components in either series at the respective frequencies. Gain The gain value is computed by dividing the cross-amplitude value by the spectrum density estimates for one of the two series in the analysis Consequently, two gain values are computed, which can be interpreted as the standard least squares regression coefficients for the respective frequencies. Phase shift Finally, the phase shift estimates are computed as tan -1 of the ratio of the quad density estimates over the cross-density estimate The phase shift estimates usually denoted by the Greek letter are measures of the extent to which each frequency component of one series leads the other. How the Ex ample Data were Created. Now, let s return to the example data set presented above The large spectral density estimates for both series, and the cross-amplitude values at frequencies 0 0625 and 1875 suggest two strong synchronized periodicities in both series at those frequencies In fact, the two series were created as. v1 cos 2 0625 v0-1 75 sin 2 2 v0-1.v2 cos 2 0625 v0 2 75 sin 2 2 v0 2. where v0 is the case number Indeed, the analysis presented in this overview reproduced the periodicity inserted into the data very well. Spectrum Analysis - Basic Notation and Principles. For more information, see Time Series Analysis - Index and the following topics. Frequency and Period. The wave length of a sine or cosine function is typically expressed in terms of the number of cycles per unit time Frequency , often denoted by the Greek letter nu some textbooks also use f For example, the number of letters handled in a post office may show 12 cycles per year On the first of every month a large amount of mail is sent many bills come due on the first of the month , then the amount of mail decreases in the middle of the month, then it increases again towards the end of the month Therefore, every month the fluctuation in the amount of mail handled by the post office will go through a full cycle Thus, if the unit of analysis is one year, then n would be equal to 12, as there would be 12 cycles per year Of course, there will likely be other cycles with different frequencies For example, there might be annual cycles 1 , and perhaps weekly cycles 52 weeks per year. The period T of a sine or cosine function is defined as the length of time required for one full cycle Thus, it is the reciprocal of the frequency, or T 1 To return to the mail example in the previous paragraph, the monthly cycle, expressed in yearly terms, would be equal to 1 12 0 0833 Put into words, there is a period in the series of length 0 0833 years. The General Structural Model. As mentioned before, the purpose of spectrum analysis is to decompose the original series into underlying sine and cosine functions of different frequencies, in order to determine those that appear particularly strong or important One way to do so would be to cast the issue as a linear Multiple Regression problem, where the dependent variable is the observed time series, and the independent variables are the sine functions of all possible discrete frequencies Such a linear multiple regression model can be written as. Following the common notation from classical harmonic analysis, in this equation lambda is the frequency expressed in terms of radians per unit time, that is 2 k where is the constant pi 3 14 and k k q What is important here is to recognize that the computational problem of fitting sine and cosine functions of different lengths to the data can be considered in terms of multiple linear regression Note that the cosine parameters a k and sine parameters b k are regression coefficients that tell us the degree to which the respective functions are correlated with the data Overall there are q different sine and cosine functions intuitively as also discussed in Multiple Regression , it should be clear that we cannot have more sine and cosine functions than there are data points in the series Without going into detail, if there are N data points in the series, then there will be N 2 1 cosine functions and N 2-1 sine functions In other words, there will be as many different sinusoidal waves as there are data points, and we will be able to completely reproduce the series from the underlying functions Note that if the number of cases in the series is odd, then the last data point will usually be ignored in order for a sinusoidal function to be identified, you need at least two points the high peak and the low peak. To summarize, spectrum analysis will identify the correlation of sine and cosine functions of different frequency with the observed data If a large correlation sine or cosine coefficient is identifi ed, you can conclude that there is a strong periodicity of the respective frequency or period in the dataplex numbers real and imaginary numbers In many textbooks on spectrum analysis, the structural model shown above is presented in terms of complex numbers, that is, the parameter estimation process is described in terms of the Fourier transform of a series into real and imaginary parts Complex numbers are the superset that includes all real and imaginary numbers Imaginary numbers, by definition, are numbers that are multiplied by the constant i where i is defined as the square root of -1 Obviously, the square root of -1 does not exist, hence the term imaginary number however, meaningful arithmetic operations on imaginary numbers can still be performed e g i 2 2 -4 It is useful to think of real and imaginary numbers as forming a two dimensional plane, where the horizontal or X - axis represents all real numbers, and the vertical or Y - axis represents all imaginary numbers Complex numbe rs can then be represented as points in the two - dimensional plane For example, the complex number 3 i 2 can be represented by a point with coordinates in this plane You can also think of complex numbers as angles, for example, you can connect the point representing a complex number in the plane with the origin complex number 0 i 0 , and measure the angle of that vector to the horizontal line Thus, intuitively you can see how the spectrum decomposition formula shown above, consisting of sine and cosine functions, can be rewritten in terms of operations on complex numbers In fact, in this manner the mathematical discussion and required computations are often more elegant and easier to perform which is why many textbooks prefer the presentation of spectrum analysis in terms of complex numbers. A Simple Example. Shumway 1988 presents a simple example to clarify the underlying mechanics of spectrum analysis Let s create a series with 16 cases following the equation shown above, and then see how we may extract the information that was put in it First, create a variable and define it as. x 1 cos 2 0625 v0-1 75 sin 2 2 v0-1.This variable is made up of two underlying periodicities The first at the frequency of 0625 or period 1 16 one observation completes 1 16 th of a full cycle, and a full cycle is completed every 16 observations and the second at the frequency of 2 or period of 5 The cosine coefficient 1 0 is larger than the sine coefficient 75 The spectrum analysis summary is shown below. Let s now review the columns Clearly, the largest cosine coefficient can be found for the 0625 frequency A smaller sine coefficient can be found at frequency 1875 Thus, clearly the two sine cosine frequencies which were inserted into the example data file are reflected in the above table. The sine and cosine functions are mutually independent or orthogonal thus we may sum the squared coefficients for each frequency to obtain the periodogram Specifically, the periodogram values above are com puted as. P k sine coefficient k 2 cosine coefficient k 2 N 2.where P k is the periodogram value at frequency k and N is the overall length of the series The periodogram values can be interpreted in terms of variance sums of squares of the data at the respective frequency or period Customarily, the periodogram values are plotted against the frequencies or periods. The Problem of Leakage. In the example above, a sine function with a frequency of 0 2 was inserted into the series However, because of the length of the series 16 , none of the frequencies reported exactly hits on that frequency In practice, what often happens in those cases is that the respective frequency will leak into adjacent frequencies For example, you may find large periodogram values for two adjacent frequencies, when, in fact, there is only one strong underlying sine or cosine function at a frequency that falls in-between those implied by the length of the series There are three ways in which we can approach the proble m of leakage. By padding the series, we may apply a finer frequency roster to the data. By tapering the series prior to the analysis, we may reduce leakage, or. By smoothing the periodogram, we may identify the general frequency regions or spectral densities that significantly contribute to the cyclical behavior of the series. See below for descriptions of each of these approaches. Padding the Time Series. Because the frequency values are computed as N t the number of units of times , we can simply pad the series with a constant e g zeros and thereby introduce smaller increments in the frequency values In a sense, padding allows us to apply a finer roster to the data In fact, if we padded the example data file described in the example above with ten zeros, the results would not change, that is, the largest periodogram peaks would still occur at the frequency values closest to 0625 and 2 Padding is also often desirable for computational efficiency reasons see below. The so-called process of sp lit-cosine-bell tapering is a recommended transformation of the series prior to the spectrum analysis It usually leads to a reduction of leakage in the periodogram The rationale for this transformation is explained in detail in Bloomfield 1976, p 80-94 In essence, a proportion p of the data at the beginning and at the end of the series is transformed via multiplication by the weights. where m is chosen so that 2 m N is equal to the proportion of data to be tapered p. Data Windows and Spectral Density Estimates. In practice, when analyzing actual data, it is usually not of crucial importance to identify exactly the frequencies for particular underlying sine or cosine functions Rather, because the periodogram values are subject to substantial random fluctuation, we are faced with the problem of very many chaotic periodogram spikes In that case, we want to find the frequencies with the greatest spectral densities that is, the frequency regions, consisting of many adjacent frequencies, that c ontribute most to the overall periodic behavior of the series This can be accomplished by smoothing the periodogram values via a weighted moving average transformation Suppose the moving average window is of width m which must be an odd number the following are the most commonly used smoothers note p m-1 2.Daniell or equal weight window The Daniell window Daniell 1946 amounts to a simple equal weight moving average transformation of the periodogram values, that is, each spectral density estimate is computed as the mean of the m 2 preceding and subsequent periodogram values. Tukey window In the Tukey Blackman and Tukey, 1958 or Tukey-Hanning window named after Julius Von Hann , for each frequency, the weights for the weighted moving average of the periodogram values are computed as. Hamming window In the Hamming named after R W Hamming window or Tukey-Hamming window Blackman and Tukey, 1958 , for each frequency, the weights for the weighted moving average of the periodogram values are co mputed as. Parzen window In the Parzen window Parzen, 1961 , for each frequency, the weights for the weighted moving average of the periodogram values are computed as. Bartlett window In the Bartlett window Bartlett, 1950 the weights are computed as. With the exception of the Daniell window, all weight functions will assign the greatest weight to the observation being smoothed in the center of the window, and increasingly smaller weights to values that are further away from the center In many cases, all of these data windows will produce very similar results. Preparing the Data for Analysis. Let s now consider a few other practical points in spectrum analysis Usually, we want to subtract the mean from the series, and detrend the series so that it is stationary prior to the analysis Otherwise the periodogram and density spectrum will mostly be overwhelmed by a very large value for the first cosine coefficient for frequency 0 0 In a sense, the mean is a cycle of frequency 0 zero per unit time that is, it is a constant Similarly, a trend is also of little interest when we want to uncover the periodicities in the series In fact, both of those potentially strong effects may mask the more interesting periodicities in the data, and thus both the mean and the trend linear should be removed from the series prior to the analysis Sometimes, it is also useful to smooth the data prior to the analysis, in order to tame the random noise that may obscure meaningful periodic cycles in the periodogram. Results when No Periodicity in the Series Exists. Finally, what if there are no recurring cycles in the data, that is, if each observation is completely independent of all other observations If the distribution of the observations follows the normal distribution, such a time series is also referred to as a white noise series like the white noise you hear on the radio when tuned in-between stations A white noise input series will result in periodogram values that follow an exponential distribu tion Thus, by testing the distribution of periodogram values against the exponential distribution, you can test whether the input series is different from a white noise series In addition, then you can also request to compute the Kolmogorov-Smirnov one-sample d statistic see also Nonparametrics and Distributions for more details. Testing for white noise in certain frequency bands Note that you can also plot the periodogram values for a particular frequency range only Again, if the input is a white noise series with respect to those frequencies i e it there are no significant periodic cycles of those frequencies , then the distribution of the periodogram values should again follow an exponential distribution. Fast Fourier Transforms FFT. For more information, see Time Series Analysis - Index and the following topics. General Introduction. The interpretation of the results of spectrum analysis is discussed in the Basic Notation and Principles topic, however, we have not described how it is do ne computationally Up until the mid-1960s the standard way of performing the spectrum decomposition was to use explicit formulae to solve for the sine and cosine parameters The computations involved required at least N 2 complex multiplications Thus, even with today s high-speed computers it would be very time consuming to analyze even small time series e g 8,000 observations would result in at least 64 million multiplications. The time requirements changed drastically with the development of the so-called fast Fourier transform algorithm or FFT for short In the mid-1960s, J W Cooley and J W Tukey 1965 popularized this algorithm which, in retrospect, had in fact been discovered independently by various individuals Various refinements and improvements of this algorithm can be found in Monro 1975 and Monro and Branch 1976 Readers interested in the computational details of this algorithm may refer to any of the texts cited in the overview Suffice it to say that via the FFT algorithm, the t ime to perform a spectral analysis is proportional to N log2 N - a huge improvement. However, a draw-back of the standard FFT algorithm is that the number of cases in the series must be equal to a power of 2 i e 16, 64, 128, 256 Usually, this necessitated padding of the series, which, as described above, will in most cases not change the characteristic peaks of the periodogram or the spectral density estimates In cases, however, where the time units are meaningful, such padding may make the interpretation of results more cumbersomeputation of FFT in Time Series. The implementation of the FFT algorithm allows you to take full advantage of the savings afforded by this algorithm On most standard computers, series with over 100,000 cases can easily be analyzed However, there are a few things to remember when analyzing series of that size. As mentioned above, the standard and most efficient FFT algorithm requires that the length of the input series is equal to a power of 2 If this is not the c ase, additional computations have to be performed It will use the simple explicit computational formulas as long as the input series is relatively small, and the number of computations can be performed in a relatively short amount of time For long time series, in order to still utilize the FFT algorithm, an implementation of the general approach described by Monro and Branch 1976 is used This method requires significantly more storage space, however, series of considerable length can still be analyzed very quickly, even if the number of observations is not equal to a power of 2.For time series of lengths not equal to a power of 2, we would like to make the following recommendations If the input series is small to moderately sized e g only a few thousand cases , then do not worry The analysis will typically only take a few seconds anyway In order to analyze moderately large and large series e g over 100,000 cases , pad the series to a power of 2 and then taper the series during the expl oratory part of your data analysis. 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