Rob J Hyndman. Forecasting using. 3. Autocorrelation and seasonality OTexts.com/fpp/2/ OTexts.com/fpp/6/1. Forecasting using R 1

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1 Rob J Hyndman Forecasting using 3. Autocorrelation and seasonality OTexts.com/fpp/2/ OTexts.com/fpp/6/1 Forecasting using R 1

2 Outline 1 Time series graphics 2 Seasonal or cyclic? 3 Autocorrelation Forecasting using R Time series graphics 2

3 Time series graphics Time plots R command: plot or plot.ts Seasonal plots R command: seasonplot Seasonal subseries plots R command: monthplot Lag plots R command: lag.plot ACF plots R command: Acf Forecasting using R Time series graphics 3

4 Thousands Time series graphics Economy class passengers: Melbourne Sydney plot(melsyd[,"economy.class"]) Year Forecasting using R Time series graphics 4

5 Time series graphics $ million > plot(a10) Antidiabetic drug sales Year Forecasting using R Time series graphics 5

6 Time series graphics $ million Seasonal plot: antidiabetic drug sales Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Year Forecasting using R Time series graphics 6

7 Seasonal plots Data plotted against the individual seasons in which the data were observed. (In this case a season is a month.) Something like a time plot except that the data from each season are overlapped. Enables the underlying seasonal pattern to be seen more clearly, and also allows any substantial departures from the seasonal pattern to be easily identified. In R: seasonplot Forecasting using R Time series graphics 7

8 Seasonal plots Data plotted against the individual seasons in which the data were observed. (In this case a season is a month.) Something like a time plot except that the data from each season are overlapped. Enables the underlying seasonal pattern to be seen more clearly, and also allows any substantial departures from the seasonal pattern to be easily identified. In R: seasonplot Forecasting using R Time series graphics 7

9 Seasonal plots Data plotted against the individual seasons in which the data were observed. (In this case a season is a month.) Something like a time plot except that the data from each season are overlapped. Enables the underlying seasonal pattern to be seen more clearly, and also allows any substantial departures from the seasonal pattern to be easily identified. In R: seasonplot Forecasting using R Time series graphics 7

10 Seasonal plots Data plotted against the individual seasons in which the data were observed. (In this case a season is a month.) Something like a time plot except that the data from each season are overlapped. Enables the underlying seasonal pattern to be seen more clearly, and also allows any substantial departures from the seasonal pattern to be easily identified. In R: seasonplot Forecasting using R Time series graphics 7

11 Seasonal subseries plots $ million Seasonal subseries plot: antidiabetic drug sales > monthplot(a10) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month Forecasting using R Time series graphics 8

12 Seasonal subseries plots Data for each season collected together in time plot as separate time series. Enables the underlying seasonal pattern to be seen clearly, and changes in seasonality over time to be visualized. In R: monthplot Forecasting using R Time series graphics 9

13 Seasonal subseries plots Data for each season collected together in time plot as separate time series. Enables the underlying seasonal pattern to be seen clearly, and changes in seasonality over time to be visualized. In R: monthplot Forecasting using R Time series graphics 9

14 Seasonal subseries plots Data for each season collected together in time plot as separate time series. Enables the underlying seasonal pattern to be seen clearly, and changes in seasonality over time to be visualized. In R: monthplot Forecasting using R Time series graphics 9

15 Quarterly Australian Beer Production beer <- window(ausbeer,start=1992) plot(beer) seasonplot(beer,year.labels=true) monthplot(beer) Forecasting using R Time series graphics 10

16 Time series graphics Australian quarterly beer production megaliters Forecasting using R Time series graphics 11

17 megalitres Time series graphics Seasonal plot: quarterly beer production Q1 Q2 Q3 Q4 Quarter Forecasting using R Time series graphics 12

18 Time series graphics Seasonal subseries plot: quarterly beer production Megalitres Jan Apr Jul Oct Quarter Forecasting using R Time series graphics 13

19 Outline 1 Time series graphics 2 Seasonal or cyclic? 3 Autocorrelation Forecasting using R Seasonal or cyclic? 14

20 Time series patterns Trend pattern exists when there is a long-term increase or decrease in the data. Seasonal pattern exists when a series is influenced by seasonal factors (e.g., the quarter of the year, the month, or day of the week). Cyclic pattern exists when data exhibit rises and falls that are not of fixed period (duration usually of at least 2 years). Forecasting using R Seasonal or cyclic? 15

21 Time series patterns Australian electricity production GWh Year Forecasting using R Seasonal or cyclic? 16

22 Time series patterns million units Australian clay brick production Year Forecasting using R Seasonal or cyclic? 17

23 Time series patterns Total sales Sales of new one family houses, USA Forecasting using R Seasonal or cyclic? 18

24 Time series patterns US Treasury bill contracts price Day Forecasting using R Seasonal or cyclic? 19

25 Time series patterns Number trapped Annual Canadian Lynx trappings Time Forecasting using R Seasonal or cyclic? 20

26 Seasonal or cyclic? Differences between seasonal and cyclic patterns: seasonal pattern constant length; cyclic pattern variable length average length of cycle longer than length of seasonal pattern magnitude of cycle more variable than magnitude of seasonal pattern The timing of peaks and troughs is predictable with seasonal data, but unpredictable in the long term with cyclic data. Forecasting using R Seasonal or cyclic? 21

27 Seasonal or cyclic? Differences between seasonal and cyclic patterns: seasonal pattern constant length; cyclic pattern variable length average length of cycle longer than length of seasonal pattern magnitude of cycle more variable than magnitude of seasonal pattern The timing of peaks and troughs is predictable with seasonal data, but unpredictable in the long term with cyclic data. Forecasting using R Seasonal or cyclic? 21

28 Seasonal or cyclic? Differences between seasonal and cyclic patterns: seasonal pattern constant length; cyclic pattern variable length average length of cycle longer than length of seasonal pattern magnitude of cycle more variable than magnitude of seasonal pattern The timing of peaks and troughs is predictable with seasonal data, but unpredictable in the long term with cyclic data. Forecasting using R Seasonal or cyclic? 21

29 Seasonal or cyclic? Differences between seasonal and cyclic patterns: seasonal pattern constant length; cyclic pattern variable length average length of cycle longer than length of seasonal pattern magnitude of cycle more variable than magnitude of seasonal pattern The timing of peaks and troughs is predictable with seasonal data, but unpredictable in the long term with cyclic data. Forecasting using R Seasonal or cyclic? 21

30 Seasonal or cyclic? Differences between seasonal and cyclic patterns: seasonal pattern constant length; cyclic pattern variable length average length of cycle longer than length of seasonal pattern magnitude of cycle more variable than magnitude of seasonal pattern The timing of peaks and troughs is predictable with seasonal data, but unpredictable in the long term with cyclic data. Forecasting using R Seasonal or cyclic? 21

31 Outline 1 Time series graphics 2 Seasonal or cyclic? 3 Autocorrelation Forecasting using R Autocorrelation 22

32 Autocorrelation Covariance and correlation: measure extent of linear relationship between two variables (y and X). Autocovariance and autocorrelation: measure linear relationship between lagged values of a time series y. We measure the relationship between: y t and y t 1 y t and y t 2 y t and y t 3 etc. Forecasting using R Autocorrelation 23

33 Autocorrelation Covariance and correlation: measure extent of linear relationship between two variables (y and X). Autocovariance and autocorrelation: measure linear relationship between lagged values of a time series y. We measure the relationship between: y t and y t 1 y t and y t 2 y t and y t 3 etc. Forecasting using R Autocorrelation 23

34 Autocorrelation Covariance and correlation: measure extent of linear relationship between two variables (y and X). Autocovariance and autocorrelation: measure linear relationship between lagged values of a time series y. We measure the relationship between: y t and y t 1 y t and y t 2 y t and y t 3 etc. Forecasting using R Autocorrelation 23

35 Example: Beer production Forecasting using R Autocorrelation 24 lag 1 beer lag 2 beer lag 3 beer lag 4 beer lag 5 beer lag 6 beer lag 7 beer lag 8 beer lag 9 beer > lag.plot(beer,lags=9)

36 Example: Beer production Forecasting using R Autocorrelation 25 lag 1 beer lag 2 beer lag 3 beer lag 4 beer lag 5 beer lag 6 beer lag 7 beer lag 8 beer lag 9 beer > lag.plot(beer,lags=9,do.lines=false)

37 Lagged scatterplots Each graph shows y t plotted against y t k for different values of k. The autocorrelations are the correlations associated with these scatterplots. Forecasting using R Autocorrelation 26

38 Lagged scatterplots Each graph shows y t plotted against y t k for different values of k. The autocorrelations are the correlations associated with these scatterplots. Forecasting using R Autocorrelation 26

39 Autocorrelation We denote the sample autocovariance at lag k by c k and the sample autocorrelation at lag k by r k. Then define c k = 1 T T (y t ȳ)(y t k ȳ) t=k+1 and r k = c k /c 0 r 1 indicates how successive values of y relate to each other r 2 indicates how y values two periods apart relate to each other r k is almost the same as the sample correlation between y t and y t k. Forecasting using R Autocorrelation 27

40 Autocorrelation We denote the sample autocovariance at lag k by c k and the sample autocorrelation at lag k by r k. Then define c k = 1 T T (y t ȳ)(y t k ȳ) t=k+1 and r k = c k /c 0 r 1 indicates how successive values of y relate to each other r 2 indicates how y values two periods apart relate to each other r k is almost the same as the sample correlation between y t and y t k. Forecasting using R Autocorrelation 27

41 Autocorrelation We denote the sample autocovariance at lag k by c k and the sample autocorrelation at lag k by r k. Then define c k = 1 T T (y t ȳ)(y t k ȳ) t=k+1 and r k = c k /c 0 r 1 indicates how successive values of y relate to each other r 2 indicates how y values two periods apart relate to each other r k is almost the same as the sample correlation between y t and y t k. Forecasting using R Autocorrelation 27

42 Autocorrelation We denote the sample autocovariance at lag k by c k and the sample autocorrelation at lag k by r k. Then define c k = 1 T T (y t ȳ)(y t k ȳ) t=k+1 and r k = c k /c 0 r 1 indicates how successive values of y relate to each other r 2 indicates how y values two periods apart relate to each other r k is almost the same as the sample correlation between y t and y t k. Forecasting using R Autocorrelation 27

43 Autocorrelation Results for first 9 lags for beer data: r 1 r 2 r 3 r 4 r 5 r 6 r 7 r 8 r Forecasting using R Autocorrelation 28

44 Autocorrelation Results for first 9 lags for beer data: r 1 r 2 r 3 r 4 r 5 r 6 r 7 r 8 r ACF Lag Forecasting using R Autocorrelation 28

45 Autocorrelation r 4 higher than for the other lags. This is due to the seasonal pattern in the data: the peaks tend to be 4 quarters apart and the troughs tend to be 2 quarters apart. r 2 is more negative than for the other lags because troughs tend to be 2 quarters behind peaks. Together, the autocorrelations at lags 1, 2,..., make up the autocorrelation or ACF. The plot is known as a correlogram Forecasting using R Autocorrelation 29

46 Autocorrelation r 4 higher than for the other lags. This is due to the seasonal pattern in the data: the peaks tend to be 4 quarters apart and the troughs tend to be 2 quarters apart. r 2 is more negative than for the other lags because troughs tend to be 2 quarters behind peaks. Together, the autocorrelations at lags 1, 2,..., make up the autocorrelation or ACF. The plot is known as a correlogram Forecasting using R Autocorrelation 29

47 Autocorrelation r 4 higher than for the other lags. This is due to the seasonal pattern in the data: the peaks tend to be 4 quarters apart and the troughs tend to be 2 quarters apart. r 2 is more negative than for the other lags because troughs tend to be 2 quarters behind peaks. Together, the autocorrelations at lags 1, 2,..., make up the autocorrelation or ACF. The plot is known as a correlogram Forecasting using R Autocorrelation 29

48 Autocorrelation r 4 higher than for the other lags. This is due to the seasonal pattern in the data: the peaks tend to be 4 quarters apart and the troughs tend to be 2 quarters apart. r 2 is more negative than for the other lags because troughs tend to be 2 quarters behind peaks. Together, the autocorrelations at lags 1, 2,..., make up the autocorrelation or ACF. The plot is known as a correlogram Forecasting using R Autocorrelation 29

49 ACF ACF Acf(beer) Lag Forecasting using R Autocorrelation 30

50 ACF ACF Acf(beer) Lag Forecasting using R Autocorrelation 30

51 Recognizing seasonality in a time series If there is seasonality, the ACF at the seasonal lag (e.g., 12 for monthly data) will be large and positive. For seasonal monthly data, a large ACF value will be seen at lag 12 and possibly also at lags 24, 36,... For seasonal quarterly data, a large ACF value will be seen at lag 4 and possibly also at lags 8, 12,... Forecasting using R Autocorrelation 31

52 Recognizing seasonality in a time series If there is seasonality, the ACF at the seasonal lag (e.g., 12 for monthly data) will be large and positive. For seasonal monthly data, a large ACF value will be seen at lag 12 and possibly also at lags 24, 36,... For seasonal quarterly data, a large ACF value will be seen at lag 4 and possibly also at lags 8, 12,... Forecasting using R Autocorrelation 31

53 Australian monthly electricity production Australian electricity production GWh Year Forecasting using R Autocorrelation 32

54 Australian monthly electricity production ACF Lag Forecasting using R Autocorrelation 33

55 Australian monthly electricity production Time plot shows clear trend and seasonality. The same features are reflected in the ACF. The slowly decaying ACF indicates trend. The ACF peaks at lags 12, 24, 36,..., indicate seasonality of length 12. Forecasting using R Autocorrelation 34

56 Australian monthly electricity production Time plot shows clear trend and seasonality. The same features are reflected in the ACF. The slowly decaying ACF indicates trend. The ACF peaks at lags 12, 24, 36,..., indicate seasonality of length 12. Forecasting using R Autocorrelation 34

57 Which is which? 1. Daily morning temperature of a cow 2. Accidental deaths in USA (monthly) chirps per minute thousands International airline passengers 4. Annual mink trappings (Canada) thousands thousands A B ACF ACF C D ACF ACF

58 Time series graphics Time plots R command: plot.ts Seasonal plots R command: seasonplot Seasonal subseries plots R command: monthplot Lag plots R command: lag.plot ACF plots R command: Acf Forecasting using R Autocorrelation 36