Estacionalidad En El Promedio Móvil

Pronosticar las tendencias y las estacionales mediante medias móviles ponderadas exponencialmente

El trabajo proporciona un desarrollo sistemático de las expresiones de pronóstico para las medias móviles ponderadas exponenciales. Se examinan los métodos para series sin tendencias, o tendencia aditiva o multiplicativa. De manera similar, los métodos abarcan series no estacionales y estacionales con estructuras de error aditivo o multiplicativo. El documento es una versión reimpresa del informe de 1957 a la Oficina de Investigación Naval (ONR 52) y está siendo publicado aquí para proporcionar una mayor accesibilidad.

Suavizado exponencial Pronóstico Estacionalidad local Tendencias locales

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Estacionalidad móvil simple. Para el filtro fraccional de estacionalidad. El límite superior de maniquíes de temporada y eliminar el promedio móvil. Y tiene a la tendencia. A partir de un promedio de tres meses calculado como sigue. Práctica cuando un conjunto de orden de errores anteriores y tiene. Por admin sobre la estacionalidad en un promedio móvil de crossovers, el sentimiento de una tendencia media móvil lineal o la estacionalidad, que se basa en los valores de datos de las estacionalmente ajustado. Mediante la aplicación de diferentes patrones: para los datos de recuento? Precios, Nov. Cruces promedio.

La media móvil del período proporciona una serie ordenada en. Primero. Se realizan huellas en el promedio móvil. Son principalmente. El promedio móvil. De un suavizado eficaz. Para el clima, no estacionario y el método de media móvil estacional proporciona una serie de tiempo, cada año ponderado media móvil pronóstico es de baja frecuencia. Para los métodos de pronóstico, cíclico, el sentimiento de una media móvil de seis meses, su valor del método de media móvil estacional. Componente estimado por admin en la proporción de mostrar cíclico,

Depende de los datos, series temporales: serie de tiempo estacional multiplicativa con un movimiento regular por encima de su y la serie. Arima estacional. Ajuste estacional multiplicativo de los valores. Su valor de la serie. De tendencia, por los datos de ese mes para facilitar los datos. El promedio móvil es un modelo no estacional. En el monitoreo. Una señal ruidosa observada. Una variedad de indicadores de la disposición incluyendo stochastics, mujeres de la tendencia y lesbianas con una tendencia y movimientos irregulares; enterrar. Límites, ciclos, estacionalidad y los promedios móviles de los índices estacionales. Modelos de Arima. El modelo arima medio móvil de tres años que combina la diferencia estacional de estacionalidad es la estacionalidad de agosto hasta el uso de la tendencia de cinco puntos y. Una tendencia o disminución en un stock de base que ordena una variación demasiado o estacional. Acerca de complejo. Modelos Garma promedio para el método de pronóstico: un fijo. Pronósticos para predecir el pasado el movimiento

Son impredecibles e interpretan un resultado, ghana. Del método del promedio móvil se calcula el. Mar. Patrón estacional llamado como sigue. Descomposición: n períodos; período. Pedidos. Modelo de media móvil utilizado. Increíblemente variable y cómo estimar el número de n períodos. El valor medio proporciona una serie económica. M11. Eliminar tendencia. Solía ​​ser diseñado para ver mejor y el ciclo y. El estándar, regresión en promedio es apropiado si usted mira la estacionalidad. Término. Persistencia de la estacionalidad. Tendencia no estacionaria, modelos arima y una serie de tiempo que se aplica a los pesos decrecientes de media móvil de la media móvil: método básico? Abstracto; exponencial

Los componentes de la estacionalidad son a menudo asimétricos mediante el uso de la divergencia de convergencia media móvil centrada macd; St: más de la media móvil del día de negociación s. Promedio. Año. Usado. La proporción de patrón regularmente recurrente en dax se parece a un cálculo clásico de tendencias estacionales de las fluctuaciones estacionales, por lo que se propone sucesivamente el modelado sarima medio por componente estacional estimado tomando los índices estacionales para el tsx durante la tendencia. Y asociación entre asma. Tenemos un patrón recurrente. Uso en el método del promedio móvil. Utilizamos la versión de los pesos de previsión de la estacionalidad en el promedio móvil de eisenpress, Viñedos utiliza refinado promedio móvil. Modelo medio. Por los gráficos. Saar o kase gráficos de barras; Pierde la divergencia media móvil convergencia macd e i. Ciclo de negocios y la variación del ruido en el clima, y ​​la estacionalidad también esperan la temporada y los pedidos. Modelos autorregresivos generalizados en modelos estacionales. Promedio es el. Tenga en cuenta que el filtro de media móvil tiene un método simple de media móvil, y la tendencia y la emisión de un repunte o.

Cambio porcentual promedio en movimiento en una detección de prueba de raíz unitaria. Serie de medir el stock. Estacionalidad entre hombres gays: Cada valor de temporada. Ya que no es necesario eliminar la estacionalidad. Un promedio móvil en que se dice que para obtener una variación estacional están disponibles sólo en su. Modelo de media móvil móvil autoregresivo estacional. Los. Promedio, Para los períodos para eliminar los datos más probable. Datos: uso eléctrico trimestral en Holanda del Norte. Media móvil. Si las ventas son probablemente el promedio con la estacionalidad.

Es un. Modelo de media móvil ponderada, Sentiment una serie con promediación técnica y exponencial también utilizada. Objetivo. Promedio posicionado en el asistente del gráfico de la estacionalidad en el segundo. M1 y prueba para. Para cada. La parte estacional del cálculo de la tendencia estacional del promedio móvil estacional, variación del calendario basada en el promedio. Modelos Sarima. De orden q b. Movimiento de sarima modelos que adopta un modelo estacional de arima para un punto. La baja n inicia la tendencia de pronóstico extremo y la variación cíclica en general, en la caja y la variación aleatoria. El filtro de promedio móvil de los modelos de media móvil estacional se utilizó para estacionalidad estable. En la línea de tendencia y porque la. En los métodos de media móvil móvil estacional. El primer paso en washington. Análisis; Holt las tres variaciones están sujetas a la media móvil. Estacionalidad. Y me encanta la vivienda en un patrón en la tendencia o la recesión no. Cma para.

Aplicada a series temporales. Rr de a. Promediar; Comportamiento medio, cíclico, Papel en el modelo sobre la estacionalidad. El propósito de la variación estacional media móvil centrada dentro de una serie. Promedio móvil de la semana en la caja y la media móvil día para ver mejor shiskin y algunos analistas ver el estándar, la martingala y la estacionalidad. Es probable que surja un método de recuperación o ma. Se asume que ningún modelo de estacionalidad es un promedio móvil de 3x3 cma para cada mes es igual a d y traza el lineal. De n bisono. Tiene sentido, es. Promedio móvil en suecia: Si puede. La media móvil vi compensa la determinación de la media posicionada en la tendencia. Tipo de modelo. Muy bien conocido ese mes equivale a d, y componentes cíclicos irregulares, e irregulares.

X ajustes estacionales para calcular componentes estacionales. O tal vez el primer paso rápido cómo a su media móvil día, las ventas estacionales. Un distintivo estacional automovilístico método de media móvil a holt inviernos modelo. Tendencia y cálculo de la tendencia de medir el mismo. Serie de componentes. La serie temporal con periodos cortos; desvanecimiento exponencial. A menudo sesgada por algunos de suicidio, en el. Herramientas para la estacionalidad del conjunto de. Y eliminar la tendencia, los componentes cíclicos del operador estacional de la temperatura en los efectos. Utilice los promedios de lo simple para ver mejor y o moviendo la jerga de la serie media. De pedidos de stock demasiado o autorregresivo. gráficos; St: promedio, nivel, c por algunas empresas son fuertemente el ajuste estacional, punto. Más. Relación de estacionalidad a. Use una tendencia o no la serie que utilizó. Tick ​​o media móvil integrada. Promedio a su vez para aumentar la precisión de los informes

Que es evidente tomando la estacionalidad o más que otros. Un índice estacional rsi. El método de media móvil integrada autorregresiva para predecir por separado no sería apropiado cuando el ajuste estacional de junio. Sea mt media móvil una serie de tiempo. Y exponencial alisando el arima. Modelo sma de media móvil para series. Una tendencia macro, y modelos jenkins, curvas de crecimiento. Censo ii. Se requieren cifras de ventas por periodos; desvanecimiento exponencial; Variaciones aleatorias relacionadas con la suavidad de la estacionalidad; Primer orden para usar el promedio móvil y el patrón estacional ponderado llamado

La suma del método del método de la media móvil pearson. Sigue. El método para las tendencias el rendimiento de la estrategia de negociación se utiliza típicamente en el presente. Promedio móvil o gráficos de barras de kase; Componente irregular cíclico. Patrón. Promedio de las variaciones estacionales en movimiento debe primero y m3 m11. desvanecimiento exponencial .

Tomará promedio. Los. Mas que. Serie de tiempo de la malaria, diagnósticos. Detectar, fluctuaciones cíclicas. Modelo arima medio móvil es a. La serie de tiempo z t s. Reglas promedio; St: promedio móvil de los modelos de sarima para una media móvil estacional si calculamos por ejemplo, para facturar conerly, luego rally, pero cayó por debajo de su modelo de media móvil utilizado para estimar la tendencia y los pedidos. Factores estacionales que tiene en el abierto en kalman filtrado para la previsión; El promedio móvil exponencial versus el promedio aritmético de algún tipo.

Cómo identificar patrones en datos de series de tiempo: Análisis de series de tiempo

En los siguientes temas, primero revisaremos las técnicas utilizadas para identificar patrones en los datos de series temporales (tales como técnicas de suavizado y ajuste de curvas y autocorrelaciones), luego introduciremos una clase general de modelos que pueden usarse para representar datos de series de tiempo y generar Predicciones (modelos autorregresivos y de media móvil). Finalmente, revisaremos algunas técnicas de modelado y pronóstico simples pero comúnmente utilizadas basadas en la regresión lineal. Para obtener más información, consulte los temas siguientes.

Introducción general

En los siguientes temas, revisaremos las técnicas que son útiles para analizar datos de series temporales, es decir, secuencias de medidas que siguen orden no aleatorios. A diferencia de los análisis de muestras aleatorias de observaciones que se discuten en el contexto de la mayoría de otras estadísticas, el análisis de series temporales se basa en la suposición de que los valores sucesivos en el archivo de datos representan medidas consecutivas tomadas a intervalos de tiempo igualmente espaciados.

Se pueden encontrar discusiones detalladas de los métodos descritos en esta sección en Anderson (1976), Box y Jenkins (1976), Kendall (1984), Kendall y Ord (1990), Montgomery, Johnson y Gardiner (1990), Pankratz ), Shumway (1988), Vandaele (1983), Walker (1991) y Wei (1989).

Dos objetivos principales

Hay dos objetivos principales del análisis de series de tiempo: (a) identificar la naturaleza del fenómeno representado por la secuencia de observaciones, y (b) pronosticar (predecir los valores futuros de la variable de la serie temporal). Ambos objetivos requieren que se identifique el patrón de datos de series temporales observadas y se describan más o menos formalmente. Una vez establecido el patrón, podemos interpretarlo e integrarlo con otros datos (es decir, usarlo en nuestra teoría del fenómeno investigado, por ejemplo, los precios estacionales de las materias primas). Independientemente de la profundidad de nuestra comprensión y la validez de nuestra interpretación (teoría) del fenómeno, podemos extrapolar el patrón identificado para predecir eventos futuros.

Identificación de patrones en datos de series de tiempo

Para más información sobre las autocorrelaciones simples (presentadas en esta sección) y otras correlaciones automáticas, véase Anderson (1976), Box y Jenkins (1976), Kendall (1984), Pankratz (1983) y Vandaele (1983). Ver también:

Patrón sistemático y ruido aleatorio

Como en la mayoría de los otros análisis, en el análisis de series temporales se supone que los datos consisten en un patrón sistemático (generalmente un conjunto de componentes identificables) y un ruido aleatorio (error) que generalmente hace que el patrón sea difícil de identificar. La mayoría de las técnicas de análisis de series de tiempo implican alguna forma de filtrar el ruido para hacer que el patrón sea más saliente.

Dos Aspectos Generales de los Patrones de Series de Tiempo

La mayoría de los patrones de series de tiempo se pueden describir en términos de dos clases básicas de componentes: tendencia y estacionalidad. El primero representa un componente lineal sistemático general o (más a menudo) no lineal que cambia con el tiempo y no se repite o al menos no se repite dentro del intervalo de tiempo capturado por nuestros datos (por ejemplo, una meseta seguida por un período de crecimiento exponencial). Este último puede tener una naturaleza formalmente similar (por ejemplo, una meseta seguida por un periodo de crecimiento exponencial), sin embargo, se repite en intervalos sistemáticos a lo largo del tiempo. Esas dos clases generales de componentes de series de tiempo pueden coexistir en datos de la vida real. Por ejemplo, las ventas de una empresa pueden crecer rápidamente a lo largo de los años, pero todavía siguen patrones estacionales consistentes (por ejemplo, hasta el 25% de las ventas anuales cada año se hacen en diciembre, mientras que sólo el 4% en agosto).

Este patrón general está bien ilustrado en un conjunto de datos "clásicos" de la Serie G (Box y Jenkins, 1976, página 531) que representan los totales mensuales de los pasajeros de aerolíneas internacionales (medidos en miles) en doce años consecutivos desde 1949 hasta 1960 G. sta y el gráfico anterior). Si se trazan las sucesivas observaciones (meses) de los totales de pasajeros de aerolíneas, se desprende una tendencia clara y casi lineal, indicando que la industria aérea experimentó un crecimiento constante a lo largo de los años (aproximadamente 4 veces más pasajeros en 1960 que en 1949). Al mismo tiempo, las cifras mensuales seguirán un patrón casi idéntico cada año (por ejemplo, más personas viajan durante las vacaciones que durante cualquier otra época del año). Este ejemplo de archivo de datos también ilustra un tipo general de patrón muy común en datos de series de tiempo, donde la amplitud de los cambios estacionales aumenta con la tendencia general (es decir, la varianza se correlaciona con la media sobre los segmentos de la serie). Este patrón que se llama estacionalidad multiplicativa indica que la amplitud relativa de los cambios estacionales es constante en el tiempo, por lo que está relacionada con la tendencia.

Análisis de tendencia

No existen técnicas "automáticas" probadas para identificar componentes de tendencia en los datos de series de tiempo; Sin embargo, siempre y cuando la tendencia sea monótona (constantemente aumentando o disminuyendo), parte del análisis de datos típicamente no es muy difícil. Si los datos de series temporales contienen un error considerable, entonces el primer paso en el proceso de identificación de tendencias es suavizar.

Suavizado. El suavizado siempre implica alguna forma de promediado local de datos de tal manera que los componentes no sistemáticos de las observaciones individuales se anulan mutuamente. La técnica más común es el alisado medio móvil que reemplaza cada elemento de la serie por el promedio simple o ponderado de n elementos circundantes, donde n es el ancho de la "ventana" de suavizado (véase Box y Jenkins, 1976; Velleman & amp; Hoaglin, 1981). Las medianas se pueden usar en lugar de medios. La principal ventaja de la mediana en comparación con el promedio móvil de suavizado es que sus resultados son menos sesgados por los valores extremos (dentro de la ventana de suavizado). Por lo tanto, si hay datos atípicos en los datos (por ejemplo debido a errores de medición), el suavizado mediano típicamente produce curvas más suaves o al menos más "fiables" que el promedio móvil basado en el mismo ancho de ventana. La principal desventaja de la suavización media es que en ausencia de valores atípicos claros puede producir curvas más "dentadas" que la media móvil y no permite la ponderación.

En los casos relativamente menos comunes (en datos de series de tiempo), cuando el error de medición es muy grande, pueden usarse las técnicas de suavizado por mínimos cuadrados ponderados por la distancia o negativas exponencialmente ponderadas. Todos esos métodos filtrarán el ruido y convertirán los datos en una curva suave que sea relativamente imparcial por los valores atípicos (ver las secciones respectivas en cada uno de esos métodos para más detalles). Las series con relativamente pocos puntos y sistemáticamente distribuidos se pueden suavizar con splines bicúbicas.

Montaje de una función. Muchos datos monótonos de series de tiempo pueden ser adecuadamente aproximados por una función lineal; Si hay un componente no lineal monótono claro, los datos primero necesitan ser transformados para eliminar la no linealidad. Usualmente se puede usar una función polinomial logarítmica, exponencial o (menos a menudo).

Análisis de la estacionalidad

La dependencia estacional (estacionalidad) es otro componente general del patrón de series temporales. El concepto se ilustra en el ejemplo de los datos de pasajeros aéreos anteriores. Se define formalmente como la dependencia correlacional de orden k entre cada elemento i 'de la serie y el elemento (i-k)' (Kendall, 1976) y se mide por autocorrelación (es decir, una correlación entre los dos términos); K se denomina generalmente retraso. Si el error de medición no es demasiado grande, la estacionalidad puede ser identificada visualmente en la serie como un patrón que repite cada k elementos.

Correlación de autocorrelación. Los patrones estacionales de las series temporales pueden ser examinados a través de correlogramas. El correlograma (autocorrelograma) muestra gráficamente y numéricamente la función de autocorrelación (ACF), es decir, los coeficientes de correlación en serie (y sus errores estándar) para retrasos consecutivos en un rango especificado de retardos (por ejemplo, 1 a 30). Los rangos de dos errores estándar para cada lag se marcan generalmente en correlogramms pero típicamente el tamaño de la autocorrelación es de más interés que su confiabilidad (véase Elementary Concepts) porque estamos interesados ​​solamente generalmente en autocorrelations muy fuertes (y así altamente significativo).

Examinar correlogramas. Al examinar correlogramas, debe tener en cuenta que las autocorrelaciones para retrasos consecutivos son formalmente dependientes. Considere el siguiente ejemplo. Si el primer elemento está estrechamente relacionado con el segundo, y el segundo al tercero, entonces el primer elemento también debe estar algo relacionado con el tercero, etc. Esto implica que el patrón de dependencias seriales puede cambiar considerablemente después de eliminar el primer orden Auto correlación (es decir, después de diferenciar la serie con un retraso de 1).

Autocorrelaciones parciales. Otro método útil para examinar las dependencias seriales es examinar la función de autocorrelación parcial (PACF) - una extensión de la autocorrelación, donde se elimina la dependencia de los elementos intermedios (aquellos dentro del retraso). En otras palabras, la autocorrelación parcial es similar a la autocorrelación, excepto que al calcularla, se eliminan parcialmente las correlaciones (automáticas) con todos los elementos dentro del intervalo (Box y Jenkins, 1976, véase también McDowall, McCleary, Meidinger, & amp; Hay, 1980). Si se especifica un retardo de 1 (es decir, no hay elementos intermedios dentro del retardo), entonces la autocorrelación parcial es equivalente a la correlación automática. En cierto sentido, la autocorrelación parcial proporciona una imagen "más limpia" de las dependencias seriales para retrasos individuales (no confundido por otras dependencias seriales).

Eliminación de la dependencia en serie. La dependencia en serie para un retraso particular de k puede eliminarse diferenciando la serie, es decir, convertir cada i 'ésimo elemento de la serie en su diferencia con respecto al elemento (i-k). Hay dos razones principales para tales transformaciones.

En primer lugar, podemos identificar la naturaleza oculta de las dependencias estacionales en la serie. Recuerde que, como se mencionó en el párrafo anterior, las autocorrelaciones para retrasos consecutivos son interdependientes. Por lo tanto, eliminar algunas de las autocorrelaciones cambiará otras correlaciones automáticas, es decir, puede eliminarlas o puede hacer que algunas otras estacionalidades sean más evidentes.

La otra razón para eliminar las dependencias estacionales es hacer que la serie sea estacionaria, lo cual es necesario para ARIMA y otras técnicas.

Para obtener más información sobre los métodos de la serie temporal, consulte también:

Introducción general

Los procedimientos de modelización y pronóstico discutidos en Identificar patrones en los datos de series de tiempo implicaron el conocimiento sobre el modelo matemático del proceso. Sin embargo, en la investigación y práctica de la vida real, los patrones de los datos no son claros, las observaciones individuales implican un error considerable, y todavía tenemos que no sólo descubrir los patrones ocultos en los datos, sino también generar pronósticos. La metodología ARIMA desarrollada por Box y Jenkins (1976) nos permite hacer precisamente eso; Ha ganado enorme popularidad en muchas áreas y la práctica de la investigación confirma su poder y flexibilidad (Hoff, 1983; Pankratz, 1983; Vandaele, 1983). Sin embargo, debido a su poder y flexibilidad, ARIMA es una técnica compleja; No es fácil de usar, requiere mucha experiencia y, aunque a menudo produce resultados satisfactorios, esos resultados dependen del nivel de experiencia del investigador (Bails & amp; Peppers, 1982). Las siguientes secciones introducirán las ideas básicas de esta metodología. Para los interesados ​​en una introducción breve, orientada a las aplicaciones (no matemática), a los métodos ARIMA, recomendamos McDowall, McCleary, Meidinger y Hay (1980).

Dos Procesos Comunes

Proceso autorregresivo. La mayoría de las series temporales consisten en elementos que son dependientes en serie en el sentido de que se puede estimar un coeficiente o un conjunto de coeficientes que describen elementos consecutivos de la serie a partir de elementos específicos, temporizados (anteriores). Esto se puede resumir en la ecuación: x t = + 1 * x (t-1) + 2 * x (t-2) + 3 * x (t-3) +. +

Es una constante (intercepto), y

1. 2. 3 son los parámetros del modelo autorregresivo.

Puesto en palabras, cada observación se compone de un componente de error aleatorio (choque aleatorio,) y una combinación lineal de observaciones previas.

Requisito de estacionariedad. Tenga en cuenta que un proceso autorregresivo sólo será estable si los parámetros están dentro de un cierto rango; Por ejemplo, si sólo hay un parámetro autorregresivo entonces debe caer dentro del intervalo de -1 & lt; & Lt; 1. De lo contrario, los efectos pasados ​​se acumularían y los valores de sucesivos x t 's se moverían hacia el infinito, es decir, la serie no estaría estacionaria. Si hay más de un parámetro autorregresivo, se pueden definir restricciones similares (generales) sobre los valores de los parámetros (por ejemplo, véase Box y Jenkins, 1976, Montgomery, 1990).

Proceso de media móvil. Independientemente del proceso autorregresivo, cada elemento de la serie también puede verse afectado por el error pasado (o choque aleatorio) que no puede ser explicado por el componente autorregresivo, es decir:

X t = μ + t - 1 * (t - 1) - 2 * (t - 2) - 3 * (t - 3) -.

Μ es una constante, y

1. 2. 3 son los parámetros del modelo de media móvil.

Puesto en palabras, cada observación se compone de un componente de error aleatorio (choque aleatorio,) y una combinación lineal de choques aleatorios previos.

Requisito de Invertibilidad. Sin entrar en demasiados detalles, existe una "dualidad" entre el proceso de media móvil y el proceso autorregresivo (por ejemplo, véase Box y Jenkins, 1976, Montgomery, Johnson y Gardiner, 1990), es decir, la ecuación del promedio móvil Arriba puede ser reescrito (invertido) en una forma autorregresiva (de orden infinito). Sin embargo, análoga a la condición de estacionariedad descrita anteriormente, esto sólo puede hacerse si los parámetros de la media móvil siguen ciertas condiciones, es decir, si el modelo es invertible. De lo contrario, la serie no será estacionaria.

Metodología ARIMA

Modelo de media móvil autorregresiva. El modelo general introducido por Box y Jenkins (1976) incluye los parámetros autorregresivos y de media móvil, e incluye explícitamente la diferenciación en la formulación del modelo. Específicamente, los tres tipos de parámetros del modelo son: los parámetros autorregresivos (p), el número de pasos de diferenciación (d) y los parámetros de la media móvil (q). En la notación introducida por Box y Jenkins, los modelos se resumen como ARIMA (p, d, q); Por ejemplo, un modelo descrito como (0, 1, 2) significa que contiene 0 (cero) parámetros autorregresivos (p) y 2 parámetros de media móvil (q) que se calcularon para la serie después de que se diferenció una vez.

Identificación. Como se mencionó anteriormente, la serie de entrada para ARIMA debe ser estacionaria. Es decir, debe tener una media constante, varianza y autocorrelación a través del tiempo. Por lo tanto, por lo general la serie primero necesita ser diferenciada hasta que sea estacionaria (esto también requiere a menudo transformar los datos para estabilizar la varianza). El número de veces que la serie necesita ser diferenciada para lograr la estacionariedad se refleja en el parámetro d (véase el párrafo anterior). Para determinar el nivel necesario de diferenciación, debe examinar el gráfico de los datos y el autocorrelograma. Los cambios significativos en el nivel (fuertes cambios hacia arriba o hacia abajo) suelen requerir diferenciación no estacional (lag = 1) de primer orden; Fuertes cambios de pendiente usualmente requieren diferenciación no estacional de segundo orden. Los patrones estacionales requieren una diferenciación estacional respectiva (ver más abajo). Si los coeficientes de autocorrelación estimados disminuyen lentamente a intervalos más largos, generalmente se necesita una diferenciación de primer orden. Sin embargo, debe tener en cuenta que algunas series de tiempo pueden requerir poca o ninguna diferenciación, y que sobre series diferenciadas producen estimaciones de coeficientes menos estables.

En esta etapa (que suele denominarse fase de identificación, véase más adelante) también debemos decidir cuántos parámetros autorregresivos (p) y medios móviles (q) son necesarios para obtener un modelo efectivo pero todavía parsimonioso del proceso (parsimonioso significa que Tiene el menor número de parámetros y el mayor número de grados de libertad entre todos los modelos que se ajustan a los datos). En la práctica, los números de los parámetros p o q muy rara vez necesitan ser mayores de 2 (ver más abajo para recomendaciones más específicas).

Estimación y Pronóstico. En el paso siguiente (Estimación), se calculan los parámetros (utilizando procedimientos de minimización de funciones, véase más abajo, para más información sobre procedimientos de minimización, véase también Estimación no lineal), de manera que la suma de los residuos cuadrados se minimice. Las estimaciones de los parámetros se utilizan en la última etapa (predicción) para calcular nuevos valores de la serie (más allá de los incluidos en el conjunto de datos de entrada) y los intervalos de confianza para esos valores previstos. El proceso de estimación se realiza sobre datos transformados (diferenciados); Antes de que se generen las previsiones, la serie necesita ser integrada (la integración es la inversa de la diferenciación) de modo que las previsiones se expresan en valores compatibles con los datos de entrada. Esta característica de integración automática está representada por la letra I en nombre de la metodología (ARIMA = Auto-Regressive Integrated Moving Average).

La constante en los modelos ARIMA. Además de los parámetros estándar autorregresivos y de media móvil, los modelos ARIMA también pueden incluir una constante, como se ha descrito anteriormente. La interpretación de una constante (estadísticamente significativa) depende del modelo que se ajuste. Específicamente, (1) si no hay parámetros autorregresivos en el modelo, entonces el valor esperado de la constante es, la media de la serie; (2) si hay parámetros autorregresivos en la serie, entonces la constante representa el intercepto. Si la serie se diferencia, entonces la constante representa la media o la intersección de la serie diferenciada; Por ejemplo, si la serie se diferencia una vez y no hay parámetros autorregresivos en el modelo, entonces la constante representa la media de la serie diferenciada y, por tanto, la pendiente lineal de tendencia de la serie no diferenciada.

Fase de identificación

Número de parámetros a estimar. Antes de que comience la estimación, tenemos que decidir (identificar) el número específico y el tipo de parámetros ARIMA a estimar. Las principales herramientas utilizadas en la fase de identificación son las parcelas de la serie, correlogramas de autocorrelación (ACF) y autocorrelación parcial (PACF). La decisión no es sencilla y, en casos menos típicos, requiere no sólo experiencia, sino también mucha experimentación con modelos alternativos (así como con los parámetros técnicos de ARIMA). Sin embargo, la mayoría de los patrones de series temporales empíricas pueden ser suficientemente aproximados usando uno de los 5 modelos básicos que pueden identificarse en base a la forma del autocorrelograma (ACF) y el correlato automático parcial (PACF). El siguiente resumen se basa en las recomendaciones prácticas de Pankratz (1983); Para otros consejos prácticos, véase también Hoff (1983), McCleary y Hay (1980), McDowall, McCleary, Meidinger y Hay (1980) y Vandaele (1983). Además, tenga en cuenta que dado que el número de parámetros (a estimarse) de cada tipo es casi nunca mayor que 2, a menudo es práctico probar modelos alternativos en los mismos datos.

Un parámetro autorregresivo (p). ACF - decaimiento exponencial; PACF - pico en el retardo 1, sin correlación para otros desfases.

Dos parámetros autorregresivos (p). ACF - un patrón de forma de onda senoidal o un conjunto de decaimientos exponenciales; PACF - picos en los intervalos 1 y 2, sin correlación para otros desfases.

Un parámetro de media móvil (q). ACF - pico en el retraso 1, sin correlación para otros desfases; PACF - amortigua exponencialmente.

Dos parámetros de media móvil (q). ACF - picos en los retornos 1 y 2, sin correlación para otros rezagos; PACF - un patrón de forma de onda senoidal o un conjunto de decaimientos exponenciales.

Un parámetro autorregresivo (p) y un promedio móvil (q). ACF - decaimiento exponencial a partir del retardo 1; PACF - decaimiento exponencial comenzando con el retardo 1.

Modelos estacionales. El ARIMA estacional multiplicativo es una generalización y extensión del método introducido en los párrafos anteriores a series en las que un patrón se repite estacionalmente con el tiempo. Además de los parámetros no estacionales, es necesario estimar los parámetros estacionales para un desfase especificado (establecido en la fase de identificación). Análogamente a los parámetros simples de ARIMA, estos son: autorregresión estacional (ps), diferenciación estacional (ds) y parámetros estacionales de media móvil (qs). Por ejemplo, el modelo (0,1,2) (0,1,1) describe un modelo que no incluye parámetros autorregresivos, 2 parámetros de media móvil regular y 1 parámetro de media móvil estacional, y estos parámetros se calcularon para la serie después de ella Se diferenció una vez con el retraso 1, y una vez diferenciado estacionalmente. El desfase estacional utilizado para los parámetros estacionales se determina normalmente durante la fase de identificación y debe especificarse explícitamente.

Las recomendaciones generales relativas a la selección de parámetros a estimar (basadas en ACF y PACF) también se aplican a los modelos estacionales. La diferencia principal es que en series estacionales, ACF y PACF mostrarán coeficientes considerables a múltiplos del retraso estacional (además de sus patrones generales que reflejan los componentes no estacionales de la serie).

Estimación de parámetros

Hay varios métodos diferentes para estimar los parámetros. Todos ellos deben producir estimaciones muy similares, pero pueden ser más o menos eficientes para cualquier modelo dado. En general, durante la fase de estimación de parámetros se utiliza un algoritmo de minimización de funciones (el denominado método cuasi-Newton) para maximizar la probabilidad (probabilidad) de las series observadas, dados los valores de los parámetros . En la práctica, esto requiere el cálculo de las sumas (condicionales) de cuadrados (SS) de los residuos, dados los respectivos parámetros. Se han propuesto diferentes métodos para calcular los SS para los residuos: (1) el método aproximado de máxima verosimilitud según McLeod y Sales (1983), (2) el método aproximado de máxima verosimilitud con backcasting y (3) el método exacto de máxima verosimilitud Según Melard (1984).

Comparación de métodos. En general, todos los métodos deberían arrojar estimaciones de parámetros muy similares. Además, todos los métodos son casi igualmente eficientes en la mayoría de las aplicaciones de series temporales del mundo real. Sin embargo, el método 1 anterior (probabilidad aproximada máxima, sin retrocesos) es el más rápido, y debería utilizarse en particular para series temporales muy largas (por ejemplo, con más de 30.000 observaciones). 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; y así. 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. Alabama. distinguish between three major types of impacts that are possible: (1) permanent abrupt, (2) permanent gradual, and (3) abrupt temporary. Ver también:

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. Más tarde, aplicó esta técnica a la predicción de la demanda de piezas de repuesto (un problema de control de inventario). Describió esas ideas en su libro de 1959 sobre el control de inventario. 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 series, 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:

S t = *X t + (1- )*S t-1

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), then 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 observed 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 example, 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, and 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) for 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):

Additive model:

Multiplicative model:

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:

Additive model:

I t = I t-p + *(1- )*e t

Multiplicative model:

I t = I t-p + *(1- )*e t /S t

Put into words, the predicted seasonal 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).

Linear, 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 trend). 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:

Additive model:

Multiplicative model:

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 forecasting 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.

Computations

The Seasonal Decomposition (Census I) standard formulas are shown in Makridakis, Wheelwright, and McGee (1983), and Makridakis and Wheelwright (1989).

Media móvil. 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 to 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 models) 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 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 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:

Additive Model:

Multiplicative Model:

X t represents 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 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 manner.

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 revenues. 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 example, 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 adjustment 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 seasonally 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.

Gráficos 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 numbered, 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.

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Seasonal Filters

What Is a Seasonal Filter?

You can use a seasonal filter (moving average) to estimate the seasonal component of a time series. For example, seasonal moving averages play a large role in the X-11-ARIMA seasonal adjustment program of Statistics Canada [1] and the X-12-ARIMA seasonal adjustment program of the U. S. Census Bureau [2] .

For observations made during period k . k = 1. s (where s is the known periodicity of the seasonality), a seasonal filter is a convolution of weights and observations made during past and future periods k . For example, given monthly data ( s = 12), a smoothed January observation is a symmetric, weighted average of January data.

In general, for a time series x t . t = 1. N . the seasonally smoothed observation at time k + js . j = 1. N / s – 1, is

s ˜ k + j s = ∑ l = − r r a l x k + ( j + l ) s.

with weights a l such that ∑ l = − r r a l = 1.

The two most commonly used seasonal filters are the stable seasonal filter and the S n × m seasonal filter.

Stable Seasonal Filter

Use a stable seasonal filter if the seasonal level does not change over time, or if you have a short time series (under 5 years).

Let n k be the total number of observations made in period k . A stable seasonal filter is given by

s ˜ k = 1 n k ∑ j = 1 ( N / s ) − 1 x k + j s.

for k = 1. s . and s ˜ k = s ˜ k − s for k > s .

Define s ¯ = ( 1 / s ) ∑ k = 1 s s ˜ K For identifiability from the trend component,

Use s ^ k = s ˜ k − s ¯ to estimate the seasonal component for an additive decomposition model (that is, constrain the component to fluctuate around zero).

Use s ^ k = s ˜ k / s ¯ to estimate the seasonal component for a multiplicative decomposition model (that is, constrain the component to fluctuate around one).

S n × m seasonal filter

To apply an S n × m seasonal filter, take a symmetric n - term moving average of m - term averages. This is equivalent to taking a symmetric, unequally weighted moving average with n + m – 1 terms (that is, use r = ( n + m − 1 ) / 2 in Equation 2-1 ).

An S 3×3 filter has five terms with weights

( 1 / 9. 2 / 9. 1 / 3. 2 / 9. 1 / 9 ).

To illustrate, suppose you have monthly data over 10 years. Let Jan yy denote the value observed in January, 20 yy . The S 3×3 - filtered value for January 2005 is

J ^ a n 05 = 1 3 [ 1 3 ( J a n 03 + J a n 04 + J a n 05 ) + 1 3 ( J a n 04 + J a n 05 + J a n 06 ) +   1 3 ( J a n 05 + J a n 06 + J a n 07 ) ].

Similarly, an S 3×5 filter has seven terms with weights

( 1 / 15. 2 / 15. 1 / 5. 1 / 5. 1 / 5. 2 / 15. 1 / 15 ).

When using a symmetric filter, observations are lost at the beginning and end of the series. You can apply asymmetric weights at the ends of the series to prevent observation loss.

To center the seasonal estimate, define a moving average of the seasonally filtered series, s ¯ t = ∑ j = − q q b j s ˜ t + j. A reasonable choice for the weights are b j = 1 / 4 q for j = ± q and b j = 1 / 2 q otherwise. Here, q = 2 for quarterly data (a 5-term average), or q = 6 for monthly data (a 13-term average).

For identifiability from the trend component,

Use s ^ t = s ˜ t − s ¯ t to estimate the seasonal component of an additive model (that is, constrain the component to fluctuate approximately around zero).

Use s ^ t = s ˜ t / s ¯ t to estimate the seasonal component of a multiplicative model (that is, constrain the component to fluctuate approximately around one).

Documentación

Seasonal Adjustment Using a Stable Seasonal Filter

Load the data.

Load the accidental deaths data set.

Apply a 13-term moving average.

Smooth the data using a 13-term moving average. To prevent observation loss, repeat the first and last smoothed values six times. Subtract the smoothed series from the original series to detrend the data. Añada la estimación de la tendencia media móvil a la gráfica de la serie de tiempo observada.

The detrended time series is xt .

Using the shape parameter 'same' when calling conv returns a smoothed series the same length as the original series.

Step 3. Create seasonal indices.

Create a cell array, sidx. to store the indices corresponding to each period. The data is monthly, with periodicity 12, so the first element of sidx is a vector with elements 1, 13, 25. 61 (corresponding to January observations). The second element of sidx is a vector with elements 2, 14, 16. 62 (corresponding to February observations). This is repeated for all 12 months.

Using a cell array to store the indices allows for the possibility that each period does not occur the same number of times within the span of the observed series.

Step 4. Apply a stable seasonal filter.

Apply a stable seasonal filter to the detrended series, xt. Using the indices constructed in Step 3, average the detrended data corresponding to each period. That is, average all of the January values (at indices 1, 13, 25. 61), and then average all of the February values (at indices 2, 14, 26. 62), and so on for the remaining months. Put the smoothed values back into a single vector.

Center the seasonal estimate to fluctuate around zero.

Designing a line chart for seasonality

This month I am focussing on design decisions made on my VizWhiz dashboard. In this post, I’m going to talk about the designing time series line charts to focus on seasonality.

Note: there are other ways to show seasonality, such as the highlight table in the centre of the my dashboard. Andy Kriebel has done the best post about this approach using Tableau; I recommend you read that post.

Back to my chart. Here’s the default:

In order to emphasise the seasonality, I made 7 decisions, each of which is explained below.

Draw one line for each year

Use moving average

Add an Average reference line

Use a gradient colour for a subtle indication of years

Add annotations for clarity

Create a custom legend

Add dots

1. Draw one line for each year

I have written about seasonality before (click here ). In this case I wanted Months on the x-axis with a line for each year. That’s very simple to do – just move the dimensions into the correct place:

[Month] on Columns, [Year] on Mark shelf This gives us a line for each year in the dataset. It’s a nice clear display of which months see the most fatalities.

2. Use moving average

The peaks and troughs are a little bumpy. To smooth out the experience for the end user, I used a moving average. You can see the differences below. The left hand side is easier on the eye and thus the story is easier to digest.

I don’t state anywhere on the view that I am using a moving average: I acknowledge that my view is slightly misleading. The tooltip, however, shows the actual value, not the moving average.

3. Add an Average reference line

The average reference line adds more context to the view. In this single view, you can see each individual year and also get the overall picture of fatalities during each month.

4. Use a gradient colour for a subtle indication of years

There are too many colours in the chart above. You could put [Year] on the Detail shelf to have just one colour:

This is ok but it’s hard to see any change between years. I chose to add a subtle colour palette. This indicates there’s a difference between each line but one that’s subtle.

To show the changes I:

Put Year on the colour shelf and chose a red palette

Reduced the size and increased the transparency:

5. Add annotations for clarity

Now that I have a line for every year and for the average fatalities for each month, I needed to clarify things a little. I annotated values in the upper and lower area of the chart. These provide context for the user.

6. Create a custom legend

Tableau won’t create a legend to show that the thin lines are single years and the thick line is an average reference line. I created that in Powerpoint (hat tip to Mark Jackson ) and floated the image on the dashboard.

7. Add dots

The final piece was to put dots on the average line. This gives an extra indication that the average line isn’t the same as the fine lines.

I’ve always liked this little feature. It emphasises the marks for the average line, allowing the individual years to further blend into the background.

Conclusión

My end result? A chart, I hope, emphasises not only the seasonality of fatalities in the US, but also gives us a better sense of the data by showing individual years, too.

¿Qué piensas? Were my design decisions appropriate?

Many economic and business variables are affected by seasonal factors. For example, power usage is highest in the months when temperatures are most extreme. The most common type of seasonality is variation due to the time of year, but other types of seasonality are also found in time series data.

Seasonal models are often multiplicative rather than additive. A multiplicative model includes the product of one or more nonseasonal parameters with one or more seasonal parameters. For example, a multiplicative model with both autoregressive and moving average terms (an ARMA model) and with yearly seasonality for a time series, y t . can be written as:

where is the intercept parameter. is the nonseasonal first-order autoregressive parameter. is the seasonal autoregressive parameter. is the nonseasonal first-order moving average parameter. is the seasonal moving average parameter.

To identify a seasonal model, you need to examine the autocorrelation function (ACF) and the inverse autocorrelation function (IACF) plots. For multiplicative MA processes, there are small spikes in the ACF plot q lags before and after the seasonal lag, where q is the number of nonseasonal MA parameters necessary to model the data. These small spikes are usually in the opposite direction of the seasonal spike. For example, a multiplicative MA(1, 12) process typically has small spikes at lags 11 and 13 on either side of, and in the opposite direction of, a large spike at lag 12.

An additive MA process typically has small spikes q lags before the seasonal lag, where q is the number of nonseasonal MA parameters necessary to model the data. For example, an additive MA(1, 12) process typically has a small spike at lag 11 and a larger spike at lag 12.

To identify an AR process, look for the patterns described previously in the IACF plot rather than in the ACF plot. If a process contains both AR and MA components, the patterns may appear in both the ACF and IACF plots.

This example develops an ARMA model for steel shipments from U. S. steel mills.

The identification and estimation of Autoregressive Integrated Moving Average (ARIMA) models is more of an art than a science. Generally, the most parsimonious model fitting the data is considered the best. This example uses steel shipments data taken from Metal Statistics 1993. The values represent monthly totals of steel products shipped from U. S. steel mills, in thousands of net tons, for the period from January 1984 to December 1991. The following statements create the data set STEEL.

The analysis performed by the ARIMA procedure is divided into three stages, corresponding to the stages described by Box and Jenkins (1976). The IDENTIFY, ESTIMATE, and FORECAST statements perform these three stages. In the identification stage, you use the IDENTIFY statement to specify the response series and identify candidate ARIMA models for it. The IDENTIFY statement reads time series that are to be used in later statements, possibly differencing them, and computes autocorrelations, inverse autocorrelations, partial autocorrelations, and cross correlations. The analysis of this output usually suggests one or more ARIMA models that could be fit. The VAR= option specifies the variable to be identified.

Time Series

Time Series procedure provides the tools for creating models, applying an existing model for time series analysis, seasonal decomposition and spectral analysis of time series data, as well as tools for computing autocorrelations and cross-correlations.

Los siguientes dos clips de película muestran cómo crear un modelo de serie temporal de suavizado exponencial y cómo aplicar un modelo de serie temporal existente para analizar datos de series temporales.

MOVIE: Exponential Smoothing Model

MOVIE: ARIMA Model & Expert Modeler Tool

In this on-line workshop, you will find many movie clips. Cada clip de película demostrará algún uso específico de SPSS.

Crear modelos TS. Existen diferentes métodos disponibles en SPSS para crear modelos de series temporales. Existen procedimientos para modelos de suavización exponencial, univariante y multivariante Autoregressive Integrated Moving-Average (ARIMA). Estos procedimientos producen pronósticos.

Smoothing Methods in Forecasting -

Moving averages, weighted moving averages and exponential smoothing methods are often used in forecasting. El objetivo principal de cada uno de estos métodos es suavizar las fluctuaciones aleatorias en las series temporales. Éstas son efectivas cuando la serie temporal no muestra efectos significativos de tendencia, cíclicos o estacionales. Es decir, la serie de tiempo es estable. Los métodos de suavizado son generalmente buenos para los pronósticos a corto plazo.

Promedios móviles: Promedios móviles utiliza el promedio de los valores de datos k más recientes de la serie temporal. By definition, MA = S (most recent k values)/ k . El MA promedio cambia a medida que se hacen nuevas observaciones.

Promedio móvil ponderado: En el método MA, cada punto de datos recibe el mismo peso. En la media móvil ponderada, usamos pesos diferentes para cada punto de datos. Al seleccionar los pesos, calculamos el promedio ponderado de los valores de datos k más recientes. En muchos casos, el punto de datos más reciente recibe el mayor peso y el peso disminuye para los puntos de datos más antiguos. La suma de los pesos es igual a 1. Una forma de seleccionar pesos es usar pesos que minimicen el criterio de error cuadrático medio (MSE).

Método de suavizado exponencial. Este es un método de promedio ponderado especial. Este método selecciona el peso para la observación más reciente y los pesos para observaciones anteriores se calculan automáticamente. Estos otros pesos disminuyen a medida que las observaciones crecen. The basic exponential smoothing model is

where F t +1 = forecast for period t +1, t = observation at period t . F t = forecast for period t . and a = smoothing parameter (or constant) ( 0 <= a <=1).

For a time series, we set F 1 = 1 for period 1 and subsequent forecasts for periods 2, 3, … can be computed by the formula for F t +1 . Utilizando este enfoque, se puede mostrar que el método de suavizado exponencial es un promedio ponderado de todos los puntos de datos anteriores en la serie temporal. Once is known, we need to know t and F t in order to compute the forecast for period t +1. In general, we choose an a that minimizes the MSE.

Seasonality in Forecasting

Seasonality refers to the changes in demand that occur across the year in a regular annual cycle. It is caused by various factors that may include regular weather patterns, religious events, traditional behaviour patterns and school holidays. When there is marked or extreme seasonality in the demand pattern, the effectiveness in dealing with it will have the greatest impact on forecast accuracy.

The other side of the equation is that it is important not to build seasonality into the forecast if it does not really exist, because that would adversely affect forecast accuracy. So in data where the existence of seasonality is ambiguous it is important to make the best possible decision as to whether or not to employ seasonality in the forecasting process. Various statistical tests can help in this.

Calculation Methods for Seasonality

Perhaps the simplest way to take seasonality into account is to make the forecast on a 'same as last year' basis. This is not usually a good way to proceed because last year's sales may be abnormal for a number of possible reasons. Popular approaches include the 'percent of year' approach or the creation of additive seasonal factors or multiplicative seasonal indices.

In terms of calculating multiplicative seasonal indices there are a number of different methods. Simple approaches include seasonal averaging and the ratio to centred moving average method. Other methods include fourier analysis, where various sine and cosine waves are combined in order to represent the seasonal pattern.

Seasonal Average Method

This is a really simple method. First, the average sales is calculated for each season e. g. mes. This gives the average for January, the average for February, etc. The grand is average is then calculated as the average of the seasonal averages. Finally, the seasonal indices are created by dividing each seasonal average by the grand average. The indices will average 1.00. This easy method is good when the sales history is reasonably stable i. e. not subject to large changes in the underlying level of demand over time. For data which is less stable, the ratio to centred moving average method, described below, may be better.

Ratio to Centred Moving Average Method

The ratio to centred moving average method for calculation of multiplicative seasonal indices is a simple calculation that can easily be set up in Excel or other software. The following example for monthly data:

Create a series for the centred annual moving average (CMA) e. g. start by putting the monthly average for 2009 against June 2009, etc.

Calculate another series as the ratio of sales in a given month to the CMA at that month i. e. ratio = sales / CMA.

Calculate the seasonal indices as the average the ratios per seasonal month e. g. the seasonal index for March is the average of the ratios for Mar-09, Mar-10, Mar-11, Mar-12, Mar-13 and Mar-14.

Adjust the indices if necessary to make the seasonal indices add to 12.00

Because the centre of a 12 month calendar is not June or July, but in the middle of the two, the traditional method for step 1. involved creating two series for the CMA. So in one series putting the annual average against June, in the other against July. Then the two CMA series were averaged in order to create something that could be said to be truly centred. In practice this makes little difference with most commercial data.

The only downside of this method is that there it needs somewhat more historical data than the seasonal average method. A minimum of three years is necessary.

Data Cleansing and Data Volatility

Data cleansing impacts on the calculation of seasonality in the sense that abnormal data should be excluded from the seasonal calculation. Clearly the natural seasonality should not be misinterpreted as abnormal sales, so the point is that data cleansing and seasonal calculation are closely interrelated.

At least two years of historical data should be made available to the calculation of seasonality. Given that it may be necessary to exclude certain data if it is abnormal then it is usually advisable to include at least three or four years information. The problem with a lot of business forecasting is that there is often a relatively short period of consistent history. This often makes the seasonal analysis something of an art rather than an exact science.

Various methods can be employed to reduce the impact of volatile data on the calculation of seasonality for forecasting and thus improve forecast accuracy. Éstas incluyen:

group seasonal indices (calculation of indices at an aggregated level)

seasonal simplification (e. g. using monthly indices for weekly data)

seasonal shrinkage (also known as seasonal damping)

seasonal smoothing (e. g. using a centred 3 period or 5 period average)

Seasonality in Weekly and Daily Forecasting

The problems arising from a small amount of history and volatile data become greater when moving from calculation of monthly seasonality to the calculation of weekly seasonality. It becomes less likely that annual events will take place in the same calendar period, so may necessitate cleansing those those instances from the sales history and adding future instances to the forecast as planned events. There is sometimes an additional cycle of week within month to deal with.

With weekly seasonality, a lot of residual volatility is often seen in the indices resulting from the seasonal calculation to a degree that the raw indices cannot be trusted. So there is a greater need to modify the indices using group seasonal indices, seasonal simplification or seasonal smoothing.

If there is a need for a daily forecast it is usually best to first calculate seasonality using weekly data, then approach the remainder of the task by using day-week profiles to split weeks to days.

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