What is time series analysis? Time Series Analysis Branch of statistics where
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What is time series analysis? Branch of statistics where Time Series Analysis observations are collected sequentially in time the analysis relies (at least in part) on understanding or exploiting the dependence among observations. Petrutza Caragea Time Department of Statistics Time is continuous, but data are usually collected at discrete points in time. Discrete pcaragea@iastate.edu Fall 2009, Iowa State University ! ! regular (this is what we will study) irregular Continous P. Caragea (ISU) Stat 551 Fall 2009 1 / 27 Stat 551 Fall 2009 2 / 27 Fall 2009 4 / 27 Example: U.S. population 240 Possible responses P. Caragea (ISU) univariate (single series) or multivariate (two or more series) ! 200 ! continuous (e.g. temperature, volume) ! ! 140 ! ! 80 binary (e.g. success of failure) ! ! ! 100 Population (Millions) discrete (e.g. number of houses, number of subjects). Often approximate discrete responses with a continuous model. Summary of impact 60 ! Almost any area where statistics is applied. Here are some examples: ! 40 ! ! 0 20 ! ! ! ! ! ! ! ! 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 Decade P. Caragea (ISU) Stat 551 Fall 2009 3 / 27 P. Caragea (ISU) Stat 551 Example: Nr. of deaths/serious injuries on roads in UK ! Example: Austrian Industrial Quarterly production ! 2200 ! ! ! ! 2000 ! ! ! ! ! ! ! ! ! ! ! 1600 ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! !! ! ! ! ! !! ! ! ! ! !! ! ! ! ! 1400 ! ! ! ! ! ! ! ! 100 Industrial Production 1800 ! ! ! ! ! ! ! ! ! 1200 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 60 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 80 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 120 ! Total number of deaths ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1976 1978 1980 1982 ! 1984 1980 1985 Month P. Caragea (ISU) Stat 551 Fall 2009 5 / 27 P. Caragea (ISU) Fall 2009 8 / 27 7000 ! 6 / 27 Red White(dry) Sparkling 6000 Dry White 7000 7000 ! Fall 2009 More Australian Wine data – Multivariate Red ! 1995 Stat 551 7000 Some Australian Wine: Monthly sales Sparkling 1990 Quarter 5000 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1000 ! ! ! !! ! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! !! ! ! ! !! ! ! ! ! ! !! ! ! ! ! !! ! ! !! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !!!! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! ! !! ! ! ! ! ! !! !! !! !!! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! ! !! !! !!! ! ! ! ! ! !!! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! !! !! ! !! ! ! ! ! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! 1985 1990 Month of the year 1995 5000 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! !! ! ! ! !! ! ! ! ! !! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! !! !! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1980 1985 1990 Month of the year 1995 1980 1985 1990 1995 Month of the year 1000 1980 4000 5000 ! ! ! ! ! ! 0 ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! 3000 3000 ! ! 3000 ! ! ! Monthly sales (kiloliters) ! ! ! ! 2000 ! ! ! ! ! ! 1000 ! ! ! ! ! ! ! ! ! 0 4000 ! 4000 ! ! 3000 ! ! ! 2000 Monthly sales (kiloliters) ! 2000 ! 1000 5000 ! ! ! 0 Monthly sales (kiloliters) ! 4000 ! ! ! 2000 ! 6000 6000 6000 ! ! ! ! 1980 1985 1990 1995 Time P. Caragea (ISU) Stat 551 Fall 2009 7 / 27 P. Caragea (ISU) Stat 551 40 Example: Los Angeles Annual Rainfall Example: Hourly wind speed at Storm Lake, IA ! ! ! ! ! 15 ! ! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! 30 ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 10 ! ! ! ! ! ! ! ! ! 1880 1900 ! ! ! !!! ! !!! ! ! ! ! !! !!! !! ! !!! !! !! ! ! !! ! !! !!!! !! ! ! !!!!!!!! !! !! ! ! ! ! ! !! ! ! ! !!! !!! ! ! ! ! !!!! ! ! !! !! !!! ! ! !! ! !! ! !!!! ! ! 1960 ! 1980 2 ! 4 !!!!!!! 6 ! !! 8 Time (Days) Stat 551 7000 !! ! ! Fall 2009 9 / 27 Example: Canadian Lynx Trappings data P. Caragea (ISU) Stat 551 Fall 2009 10 / 27 Fall 2009 12 / 27 Medical Statistics ! ! 400 6000 ! 5000 ! 4000 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1000 ! ! ! ! 0 ! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !200 3000 ! ! ! ! 0 voltage(mcv) ! ! ! 200 ! 2000 Number Trapped ! ! ! Year P. Caragea (ISU) ! ! ! !! ! ! 1940 ! ! ! ! ! ! 1920 ! ! !!! ! ! ! ! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !!!! ! ! ! !!! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! 5 ! ! ! ! ! 0 ! ! ! ! ! ! 10 Wind speed (knots) ! ! 20 Total rain (inches) ! ! ! !! ! !! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !!!! ! !! !! ! ! !! 0 ! !!! 5000 10000 15000 20000 25000 Time 1820 1840 1860 1880 1900 1920 Year P. Caragea (ISU) Stat 551 Fall 2009 11 / 27 P. Caragea (ISU) Stat 551 Objectives Possible approaches to time series analysis Description Plot the data and obtain simple descriptive measures of the main properties of the series. Time domain approach Explanation Assume that correlation between adjacent points in time can be explained through dependence of the current value on past values. Find a model to describe the time dependence in data. Frequency domain approach Forecasting Characteristics of interest relate to periodic (systematic) sinusoidal variations in the data, often caused by biological, physical, or environmental phenomena. Given a finite sample from the series (observations), forecast the next value or the next several values. Control/Tuning After forecasting, considerations about adjusting various control parameters to make the series fit closer to a target. Of importance in many engineering and industrial applications. P. Caragea (ISU) Stat 551 Fall 2009 13 / 27 Plot the DATA vs. TIME and look for the following patterns: P. Caragea (ISU) Stat 551 Fall 2009 14 / 27 US Population example: trend: long-term increase or decrease in the data What do we see? seasonality: influenced by seasonal factors (e.g. quarter of the year, month, or day of the week) smooth, clear upward trend 240 periodicity: exact repetition in regular pattern (seasonal series often called periodic, although they do not exactly repeat themselves over time) slight change in structure around ”baby boom” ! 200 ! No seasonality ! 140 100 ! 240 dependence: positive(successive observations are similar) or negative (successive observations are dissimilar) ! ! ! 80 heteroskedasticity: changing variance, particularly with changing level Strong positive dependence between adjacent values ! ! 60 cyclical: data exhibit rises and falls that are not of a fixed period Population (Millions) ! ! ! ! ! 200 ! 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 Decade Many data series include combinations of the preceding patterns ! ! 140 ! ! ! 100 20 0 Analysis ! ! ! ! ! ! 80 Current Decade Population (in millions) missing data, structural breaks, outliers ! ! ! 60 40 ! ! ! 40 ! ! 20 ! ! ! ! 0 ! ! ! Need methods that are capable of distinguishing each of the patterns P. Caragea (ISU) Stat 551 Fall 2009 15 / 27 0 20 40 60 80 100 120 140 160 200 220 Previous Decade Population (in millions) P. Caragea (ISU) Stat 551 Fall 2009 16 / 27 Is time series analysis really necessary ?? LA annual rainfall Often times in TS analysis Scatterplot (vs. Lagged series) 40 40 series display time-varying local behavior ! ! r= !0.03 ! ! ! ! ! ! 120 ! 10 20 40 60 80 ! ! inefficient estimates of regression parameters ! ! ! ! 0 ! ! ! ! 1800 1840 1880 1920 1960 !! ! ! ! ! ! ! ! ! ! ! ! 1920 1940 ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1980 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1960 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! 1900 ! ! ! ! ! 1880 ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 10 20 Year poor predictions Year ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! Ignoring dependence leads to ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 30 ! ! ! ! ! ! ! ! 20 160 ! Current Year Inches 30 dependence ! 10 200 ! ! ! ! ! 20 seasonality Total rain (inches) 240 ! ! ! Population (Millions) Original series trend functions are not easily fitted with global regression functions Quadratic fit to US Population 30 40 Previous Year Inches standard errors unrealistically small (too narrow CI) ⇒ improper inferences Stat 551 Fall 2009 17 / 27 Austrian Industrial Quarterly production P. Caragea (ISU) Stat 551 60 80 100 !! ! ! ! ! ! ! 120 100 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1980 1985 1990 1995 60 Quarter 80 100 120 Previous Quarter Production ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !!! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! !!! ! ! ! ! !! !! !! ! ! ! !!! ! !! ! ! ! Fall 2009 19 / 27 P. Caragea (ISU) Stat 551 Fall 2009 20 / 27 ! ! !! ! !! 60 Stat 551 ! ! ! ! ! ! lag 3 P. Caragea (ISU) ! ! ! ! ! ! ! ! ! 120 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! 60 60 ! ! ! ! ! ! ! ! lag 2 ! 100 ! ! ! lag 1 ! ! ! austrProd ! !! !! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! ! ! ! !! ! !! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! 80 !! lag 4 100 80 80 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 60 ! ! ! ! ! ! ! 80 100 ! ! ! Current Quarter Production ! ! ! ! ! ! ! ! ! ! ! ! ! 60 120 ! ! ! Industrial Production ! ! ! ! ! ! ! ! ! ! 80 ! ! ! ! ! ! !! ! ! ! austrProd ! 100 austrProd ! ! ! ! ! ! ! ! ! ! ! 120 Scatterplot (vs. Lagged series) ! 18 / 27 120 ! ! Original series Fall 2009 Austrian Industrial Quarterly production (cntd.) austrProd P. Caragea (ISU) 120 Austrian Industrial Quarterly production (cntd.) Australian Red Wine data Seasonal Plot: Austrian Industrial Production 1989 ! ! ! ! ! 1988 ! 1987 1986 ! ! 1985 1984 ! ! ! ! ! ! ! ! ! ! 3500 3000 ! ! ! !! !! ! !! ! ! ! ! ! ! !! !! !! !!! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! ! !! !! !!! ! ! ! ! ! !!! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! 1982 1981 1980 ! ! ! ! ! ! 2000 1983 ! ! ! ! ! ! ! ! ! 1000 ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! !! ! !! ! ! ! ! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! 1980 1995 ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! !! ! ! !! ! ! !! ! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! ! ! 1990 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1985 ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 0 80 ! ! ! ! 60 ! ! ! ! ! Current Month Sales ! Monthly sales (kiloliters) 100 ! ! 4000 1990 ! 2500 ! ! ! r= 0.74 2000 1991 1500 ! ! ! ! 1000 ! ! ! ! ! ! 500 ! Scatterplot (vs. Lagged series) 6000 ! ! Original series 5000 ! 1995 4000 ! 1993 1994 1992 ! 7000 ! ! ! ! 3000 120 ! ! ! 500 1000 1500 Month of the year 2000 2500 3000 3500 Previous Month Sales ! January April July October Quarter Stat 551 Fall 2009 21 / 27 Australian Red Wine data (cntd.) 0 lag 7 lag 8 ! ! !!! !! ! ! !! ! ! ! ! !! ! !! ! !!! ! ! ! ! ! !! !!!!! !! ! !! ! !! ! ! ! !! !! ! !! !! !! !! ! ! !!!! ! ! !! ! !! ! !!!!! !!!! !!!!!!!!! ! ! ! !! ! ! ! !! ! ! !! ! !! ! ! !! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! !! !! ! !! !! !!! !!!! !! ! ! ! ! ! !!! ! ! ! !! redwine ! lag 10 0 1000 lag 11 2000 3000 Stat 551 3500 ! ! ! ! D 2500 ! ! 2000 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1500 JASDA D NM J A M J N N D AD AS MJ MM O A M S J D AS A D J ASN N JA MJ O A J J J N SN MM O M OM O SD S OM A D NM O MJ OA M F F A J A A F M D A JA F MJSN MJ O M NA J A F J J M J D S O F JA J A M JO NA F M J D S F J SN A M M F J MS M J A SD MJS J DA M F F J NA O N N J J O D O O M J A F J M M F J J F J M F F J J J J A 1000 J ! 500 1500 1000 J ! J J J ! 3000 A 4000 4000 3500 3000 redwine 2500 3500 2500 1500 ! ! ! ! J ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1992 1994 1993 ! ! ! 1989 1990 1987 1991 ! ! 1988 1984 ! 1986 ! 1985 1983 1981 1982 1980 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! Oct Nov Dec lag 9 ! ! ! ! !! ! ! ! !!!!!! ! ! ! ! !! !! ! ! ! ! ! ! ! !! !!!! ! ! !! ! !! ! !! !! ! !! ! !! !!! ! ! ! !! ! ! ! !! !! ! ! !! !! !!! !!!! !!! ! ! !!!!! ! !! ! !! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !!! ! ! ! !!!! !! ! ! ! !!! ! ! ! ! ! ! !! ! ! !!! ! !!! ! ! ! !! !! ! ! !!! ! ! ! 500 ! ! ! ! !! ! ! !! ! ! ! !! ! ! !! ! !! ! !! ! ! !! !!! ! !! ! ! ! !!! !! ! !!! ! ! ! ! !!!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !!! !! ! ! !!! !! !! ! ! ! ! !! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !! !!! !!! !!! !! !! !!!!!!! ! !!! ! !!! ! ! ! ! ! ! ! !! ! ! !! ! ! ! !!! ! ! ! ! ! ! !! ! ! ! ! ! Seasonal plot J A ! ! ! !! ! ! ! ! ! ! ! ! !! !! ! !! ! ! ! ! !! ! ! !!! ! !! ! !!!!! ! ! ! !! ! ! ! !! ! !! !! ! ! ! ! ! ! ! ! !! ! ! ! !!! !! ! !!! !! !! ! !! ! ! ! ! ! !!! ! ! ! !! !! !! ! ! ! !! !!! ! ! !! !!!!! ! ! ! !! !! !! !! !! !! ! !! ! !! !! !!! ! ! !! ! !! ! ! ! ! !! !! ! ! redwine ! ! lag 6 ! ! ! 500 ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! !! ! ! ! !! ! ! ! ! !!!! !! !! ! ! !! ! !! ! !!! !!! !! ! !! !! ! !! !!! ! !!! ! ! ! ! ! ! !! !!! ! !!! !! !! !!! ! ! !!!!! !! !!!! ! !! ! ! ! ! ! ! ! ! ! ! !!! !! ! ! ! !! ! ! ! ! ! !! !! ! ! !! !! ! ! !!! ! ! !! ! !! ! !!! ! ! ! ! ! ! ! ! ! ! ! !! !! !! ! ! !! !! !! ! ! ! !! !! ! ! !!!! ! ! ! ! ! ! ! ! ! ! !! !! ! ! ! !!! !! ! !! ! ! !!! ! !!! ! ! ! ! ! ! ! !! !! ! !! ! !! ! !! !! ! !! ! ! ! !! ! ! ! ! ! !! ! ! !! ! ! ! ! ! ! !! ! !! ! ! ! ! ! ! ! !! !! ! ! ! ! !! ! ! ! !! !! !!! ! lag 12 1980 3500 3500 ! ! !! lag 5 ! ! !!! ! !! ! ! ! ! ! ! !! ! ! ! ! !!! !! ! ! !! ! ! !! !! !! !! !! ! !!! !!! ! ! ! ! !! ! !! !! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! !!!! ! !! !! ! !!! !! ! ! ! ! ! !! ! ! !! ! ! !! !! ! ! ! ! !!!!! !!!! !! ! ! ! ! ! ! ! ! ! ! !!! ! !! ! !! !! ! ! !! ! ! ! !!!! !! ! ! ! !! ! Original series ! 1985 1990 1995 Jan Time ! Feb Mar Apr May Jun Jul Aug Sep Month 2500 ! ! lag 3 ! ! ! ! ! ! ! ! !! ! !! ! ! ! !!!! !!! ! !!!! ! !! ! !! ! !! ! ! !!!!! ! !! ! ! ! !! !!!! !! ! ! !!! ! ! !! !!! ! ! ! !! ! ! ! ! ! !! ! ! !! ! ! !! !! ! !!! !! ! ! ! ! !! ! !! !! ! ! ! !! ! ! !!!! ! ! !! !!! ! !! ! ! !! ! ! ! !!! !! !! !!! ! ! ! ! !! ! ! ! ! ! !!! ! ! !! redwine redwine ! lag 4 1500 2500 redwine ! ! ! !! ! !! ! ! ! ! ! ! ! ! !! !! !!! !! ! !!!! ! !! ! ! !!!! ! ! ! !! ! ! ! ! !! ! !! ! ! ! ! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !!!! ! !!! !! ! !! ! ! ! ! ! !! ! !!! !! ! !! ! ! ! !! !! ! ! !!! !! !! !!!! ! ! ! ! ! ! !! ! ! !! ! !! ! !! ! ! !! !! !! ! !! ! !!! ! redwine ! ! ! !! ! ! ! ! ! !! !! ! !!! ! !! ! ! ! !! ! !! !! !!! ! !! ! !!! ! !! ! ! ! ! ! ! ! !! ! !!! !!! ! !!!!!! ! ! ! !! !!!! !! ! ! ! ! !!! ! !! ! ! !! !! !! ! ! ! !! !! ! !!!! ! ! ! ! ! ! ! ! ! !!! ! !! ! ! !!! !! ! ! ! !! !!! ! ! !! !! ! !!!! !! ! !!! ! ! !! !! !! ! ! !! 4000 ! ! 500 ! ! ! 3000 ! ! ! ! ! redwine 2000 lag 2 ! ! 22 / 27 1500 redwine 1500 2500 redwine 500 lag 1 P. Caragea (ISU) 1000 ! ! ! ! !! ! ! ! !! ! ! ! ! !! ! !! !! ! ! ! !!! !!!! !!! ! ! ! ! ! ! !! !! ! !! ! !! !!! !!!!! ! ! ! ! !!! ! ! !!!! ! ! !! ! ! !! ! ! !! !! ! ! !! !! ! ! ! ! !! ! ! ! ! !! ! ! ! ! ! ! ! !! !! ! ! ! ! ! !! !! !! !! !! ! ! !! ! ! ! ! ! ! ! !! ! !! !! ! ! !!!! ! ! !! !! ! !!! !!! !!! ! ! Fall 2009 Australian Red Wine data (cntd.) 4000 ! ! ! !! ! ! ! ! !! ! ! !!! !! ! !! ! ! !! ! ! !!! !! !!! ! ! ! ! ! !! ! ! ! ! !!! !! ! ! ! ! !! ! !!! ! !! ! ! ! ! ! ! ! ! ! !! ! !! !! !!!! ! ! ! ! ! ! ! !!! ! !! ! ! ! ! ! ! !! ! !!!! ! !! ! !! ! ! ! ! ! ! ! ! ! ! ! !!! !! ! ! !! !! ! ! ! ! !!! !! !! ! ! ! ! ! !! !!! ! !! ! !! Stat 551 500 3000 redwine 2000 redwine 1000 redwine 3500 0 P. Caragea (ISU) 2000 P. Caragea (ISU) 4000 Fall 2009 23 / 27 P. Caragea (ISU) Stat 551 Fall 2009 24 / 27 Canadian Lynx Trappings data Dependence without trend or seasonality?? 7000 An Industrial Chemical Process ! Original series ! Scatterplot (vs. Lagged series) 6000 ! ! ! r= 0.55 85 ! ! ! 85 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 0 ! ! ! ! ! !! ! ! ! !!!! ! 1840 1860 1880 ! ! 70 ! !! ! !! ! !!! !! ! ! ! ! 1900 ! ! ! ! ! ! ! 0 1820 ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 65 1000 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 70 ! ! ! ! ! ! ! ! ! ! 75 ! ! ! ! ! ! ! Color Property ! ! ! 75 ! ! ! ! ! 65 3000 ! ! ! ! ! ! ! 80 ! ! ! ! 80 ! ! Current Batch Color 4000 ! ! 2000 Number Trapped 5000 ! ! ! 5 10 1920 15 20 25 30 35 Batch 65 70 75 80 85 Previous Batch Color Year P. Caragea (ISU) Stat 551 Fall 2009 25 / 27 Gaussian white noise Original series Scatterplot (vs. Lagged series) 3 ! 3 ! ! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !1 ! !! ! !2 ! !! ! ! ! ! ! !! ! !! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! !! ! ! ! ! !! ! !! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! !! ! ! ! ! !! !! ! ! ! ! ! !!! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! !! ! ! ! !! ! ! ! ! ! !!! !! ! ! !! ! ! !! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! !! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! !! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! !! !! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !!! ! ! !! ! !! !! ! ! ! ! ! ! ! ! ! !!!! ! !! ! !! ! !! ! !!! ! ! !! !! !! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! !! ! !! ! !! ! ! ! !! ! ! ! ! ! ! !!! !! ! !!! ! ! ! ! ! ! ! ! ! !! !! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !!! ! ! ! ! ! ! ! !! !! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! !! !! ! ! ! !!! !! ! ! ! ! ! ! ! ! ! ! ! !! !! ! !! ! ! ! !! !! ! ! ! ! !! ! ! ! ! ! ! ! !!! ! ! ! !! !! ! !! ! ! ! !! ! ! ! ! ! !!!! !! !! ! !! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! !! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! 0 0 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! Current Time ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! !1 1 ! ! ! ! ! ! ! ! ! ! ! ! ! 1 ! ! ! ! !2 2 ! ! ! ! ! ! ! 2 ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !3 ! !3 ! ! 0 100 200 300 400 ! 500 !3 Time !2 !1 0 1 2 3 Previous Time w ∼ i.i.d N(0, 1) P. Caragea (ISU) Stat 551 Fall 2009 27 / 27 P. Caragea (ISU) Stat 551 Fall 2009 26 / 27
