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[A useful way to view time series data is as a combination of trend, season and noise. For data that are equally spaced over time (t = 1, …, n) an equation that splits the series into these three parts is 4.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${Y }_{t} = {\mu }_{t} + {s}_{t} + {\varepsilon }_{t},\qquad t = 1,\ldots ,n,$ \end{document} where μt is the trend, st is the seasonal pattern and ɛt is the random noise (also known as the error or residuals). We assume the residuals are uncorrelated, with a zero mean and constant variance, so \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\varepsilon }_{t} \sim \textrm{ N}(0,{\sigma }_{\varepsilon }^{2}),\qquad \textrm{ cov}({\varepsilon }_{ t},{\varepsilon }_{k}) = 0,\,t\neq k.$ \end{document}]
Published: Jan 4, 2010
Keywords: Markov Chain Monte Carlo; Seasonal Pattern; Linear Change; Markov Chain Monte Carlo Sample; Markov Chain Monte Carlo Chain
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