chap1 - Introduction to Time Series (Chapter 1) 1.1...

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1 1 1.1 Examples of time series Ex 1.1.1 (Australia red wine sales; WINE.DAT ) Introduction to Time Series (Chapter 1) x t = monthly sales of red wine (1000 litres) t = (Jan, 1980), (Feb, 1980), . . . , (Oct, 1991) or t=1, 2, . . . , 142. 2 Figure 1.1: Australian red wine sales (thousands) 1980 1982 1984 1986 1988 1990 1992 0.5 1.0 1.5 2.0 2.5 3.0 Features: upward trend seasonal pattern (peak in July, trough in Jan) increase in variability
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2 3 Time Series Plots: Examine plot for: trend over time (does the series increase or decrease with time) regular seasonal (or cyclical) components constant variability over time other systematic features of the data 4 Ex 1.1.2 (All-star baseball games, 1933-1995) x t = 1940 1950 1960 1970 1980 1990 -2 -1 0 1 2 1 1 if the National League won in year if the American League won in year t t
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3 5 Ex 1.1.3 (Accidental deaths, USA; DEATHS.DAT ) (thousands) 1973 1974 1975 1976 1977 1978 1979 7891 0 1 Features: slight trend seasonal component (peak in July) Figure 1.3 : Monthly accidental deaths 6 {} s. rv' N(0,.25) of sequence IID an is N where 2,. ..,200 1, = t , N ) 10 cos(t/ t t + = t X Ex. 1.1.4 (Signal Detection; SIGNAL.DAT ) Model: Figure 1.4: red = estimated signal black= true signal 0 5 10 15 20 -2 -1 2 3
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4 7 Ex. 1.1.5 (Population of USA.; USPOP.DAT ) Figure 1.5. (Millions) 1790 1820 1850 1880 1910 1940 1970 0 40 80 120 160 200 240 8 1.2 Objectives of Time Series Analysis Modelling paradigm : set up family of probability models to represent data estimate parameters of model check model for goodness of fit Applications of models: provides a compact description of the data interpretation prediction hypothesis testing
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5 9 DEFINITION 1.3.1. A time series model for the observed data { x t } is a specification of the joint distributions of a sequence of random variables { X t } of which { x t } is postulated to be a realization. 1.3 Simple Time Series Models 2nd Order Properties. means: E( X t ) 2nd -order moments: E( X t+h X t ) 10 Ex 1.3.2 (Binary Process). { X t } ~ IID P[ X t = 1] = p , P[ X t = -1] = 1- p, where p= .5. (Model for All Star baseball games??) 1.3.1 Zero-mean Models Ex 1.3.1 (IID NOISE). { X t }~IID(0, σ 2 ) if { X t } is IID sequence with mean 0 and variance σ 2 .
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6 11 1.3.2 Models with trend and seasonality Model with no seasonal component. X t =m t +Y t , where m t is a slowly varying function called the trend function.
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This note was uploaded on 06/16/2009 for the course STAT 525 taught by Professor Brockwell during the Spring '09 term at Colorado State.

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chap1 - Introduction to Time Series (Chapter 1) 1.1...

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