Muitas das aplicações mais intensivas e sofisticadas dos métodos de
séries temporais têm sido problemas nas ciências físicas e ambientais.
Este fato explica muito da linguagem da análise de séries temporais. Uma
das primeiras séries gravadas é o números mensais de manchas solares
estudados por Schuster em 1906. Investigações mais modernas podem se
concentrar em saber se o aquecimento está presente nas medições da
temperatura global ou se os níveis de poluição podem influenciar a
mortalidade diária. A modelagem de séries de fala é um importante
problema relacionado à transmissão eficiente de gravações de voz.
Características comuns das séries temporais, conhecidas como o espectro
de energia são usadas para ajudar os computadores a reconhecer e
traduzir a fala. Séries temporais geofísicas, como aquelas produzidas
por deposições anuais de vários tipos, podem fornecer proxies de longo
alcance para temperatura e precipitação. Gravações sísmicas podem
auxiliar no mapeamento de falhas geográficas ou na distinção entre
terremotos e explosões nucleares.
As séries acima são apenas exemplos de bancos de dados experimentais
que podem ser usados para ilustrar o processo pelo qual a metodologia
estatística clássica pode ser aplicada na estrutura de séries temporais
correlacionadas. O primeiro passo em qualquer investigação de séries
temporais sempre envolve um exame minucioso dos dados registrados
plotados ao longo do tempo. Esse escrutínio muitas vezes sugere o método
de análise, bem como estatísticas que serão úteis para resumir as
informações nos dados. Antes de examinar mais de perto os métodos
estatísticos específicos, é apropriado mencionar que existem duas
abordagens separadas, mas não necessariamente mutuamente exclusivas,
para análise de séries temporais, comumente identificadas como a
abordagem no domínio do tempo e a abordagem do domínio da
frequência.
A abordagem do domínio do tempo é geralmente motivada pela suposição
de que a correlação entre pontos adjacentes no tempo é melhor explicada
em termos de uma dependência do valor atual de valores passados. A
abordagem no domínio do tempo se concentra na modelagem de algum valor
futuro de uma série temporal como uma função paramétrica dos valores
atual e passado. Neste cenário, começamos com regressões lineares do
valor presente de uma série temporal sobre seus próprios valores
passados e sobre os valores passados de outras séries. Essa modelagem
leva a usar os resultados da abordagem no domínio do tempo como uma
ferramenta de previsão e é particularmente popular entre os economistas
por esse motivo.
Uma abordagem, defendida no livro de Box e Jenkins (1970) e também em
Box et al., (1994), desenvolve uma classe sistemática de modelos
denominados modelos de média móvel integrada autorregressiva (ARIMA)
para lidar com a modelagem e previsão correlacionadas. A abordagem
inclui uma provisão para o tratamento de mais de uma série de entradas
através de ARIMA multivariada ou através de modelagem de função de
transferência. A característica de definição desses modelos é que eles
são modelos multiplicativos, o que significa que os dados observados são
supostos como resultantes de produtos de fatores que envolvem operadores
de equações diferenciais ou diferenciais que respondem a uma entrada de
ruído branco.
Uma abordagem mais recente do mesmo problema usa modelos aditivos
mais familiares aos estatísticos. Nesta abordagem, presume-se que os
dados observados resultem de somas de séries, cada uma com uma estrutura
de séries temporais especificadas. Por exemplo, em economia, suponha que
uma série seja gerada como a soma da tendência, um efeito sazonal e um
erro. O modelo de espaço de estados resultante é então tratado com o uso
criterioso dos célebres filtros e suavizadores de Kalman, desenvolvidos
originalmente para estimação e controle em aplicações espaciais. Duas
apresentações relativamente completas deste ponto de vista estão em
Harvey (1991) e Kitagawa e Gersch (1996). A regressãão das séries
temporais é apresentado no Capítulo II e os modelos ARIMA e os modelos
de domínios de tempo relacionados são estudados na Capítulo III, com
ênfase na regressão linear clássica. Tópicos especiais sobre análise no
domínio do tempo são abordados no Capítulo V. Esses tópicos incluem
tratamentos modernos de, por exemplo, séries temporais com memória longa
e modelos GARCH para a análise de volatilidade. O modelo de espaço de
estados, filtragem e suavização de Kalman e tópicos relacionados são
desenvolvidos no Capítulo VI.
Por outro lado, a abordagem do domínio da frequência assume que as
características primárias de interesse em análises de séries temporais
estão relacionadas a variações senoidais periódicas ou sistemáticas
encontradas naturalmente na maioria dos dados. Essas variações
periódicas são frequentemente causadas por fenômenos biológicos, físicos
ou ambientais de interesse. Uma série de choques periódicos pode
influenciar certas áreas do cérebro; o vento pode afetar as vibrações de
uma asa de avião, as temperaturas da superfície do mar causadas pelas
oscilações do El Niño podem afetar o número de peixes no oceano. O
estudo da periodicidade estende-se à economia e às ciências sociais,
onde se pode estar interessado em periodicidades anuais em séries como
desemprego mensal ou taxas de natalidade mensais.
O objetivo destas páginas é fornecer uma exposição unificada e
razoavelmente completa dos métodos estatísticos usados na análise de
séries temporais, considerando seriamente as abordagens de domínio de
tempo e frequência. Como uma miríade de métodos possíveis para analisar
qualquer série experimental específica pode existir, nós integramos
dados reais de vários campos de estudo na exposição e sugerimos métodos
para analisar esses dados.
Como suporte computacional utilizamos a linguagem de programação e
ambiente de desenvolvimento integrado para cálculos estatísticos e
gráficos R, última versão 3.5.2, Eggshell Igloo de 20 de dezembro de
2018, em especial o pacote astsa.
Apêndice B. Teoria no domínio do tempo (em
elaboração)
B.1. Espaço de Hilbert e o Teorema da Projeção
B.2. Condições causais para modelos ARMA
B.3. Grande distribuição de amostra do mínimo condicional de AR
Estimadores de quadrados
B.4. A decomposição Wold
Apêndice C. Teoria no domínio espectral (em
elaboração)
C.1. Teoremas de representação espectral
C.2. Distribuição em grandes amostras do periodograma suavizado
C.3. A distribuição Normal multivariada complexa
C.4. Integração
C.5. Análise espectral como Análise de Componentes Principais
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