Forecasting using R. Rob J Hyndman. 2.5 Seasonal ARIMA models. Forecasting using R 1

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1 Forecasting using R Rob J Hyndman 2.5 Seasonal ARIMA models Forecasting using R 1

2 Outline 1 Backshift notation reviewed 2 Seasonal ARIMA models 3 ARIMA vs ETS 4 Lab session 12 Forecasting using R Backshift notation reviewed 2

3 Backshift notation A very useful notational device is the backward shift operator, B, which is used as follows: By t = y t 1. In other words, B, operating on y t, has the effect of shifting the data back one period. Two applications of B to y t shifts the data back two periods: B(By t ) = B 2 y t = y t 2. For monthly data, if we wish to shift attention to the same month last year, then B 12 is used, and the notation is B 12 y t = y t 12. Forecasting using R Backshift notation reviewed 3

4 Backshift notation A very useful notational device is the backward shift operator, B, which is used as follows: By t = y t 1. In other words, B, operating on y t, has the effect of shifting the data back one period. Two applications of B to y t shifts the data back two periods: B(By t ) = B 2 y t = y t 2. For monthly data, if we wish to shift attention to the same month last year, then B 12 is used, and the notation is B 12 y t = y t 12. Forecasting using R Backshift notation reviewed 3

5 Backshift notation A very useful notational device is the backward shift operator, B, which is used as follows: By t = y t 1. In other words, B, operating on y t, has the effect of shifting the data back one period. Two applications of B to y t shifts the data back two periods: B(By t ) = B 2 y t = y t 2. For monthly data, if we wish to shift attention to the same month last year, then B 12 is used, and the notation is B 12 y t = y t 12. Forecasting using R Backshift notation reviewed 3

6 Backshift notation A very useful notational device is the backward shift operator, B, which is used as follows: By t = y t 1. In other words, B, operating on y t, has the effect of shifting the data back one period. Two applications of B to y t shifts the data back two periods: B(By t ) = B 2 y t = y t 2. For monthly data, if we wish to shift attention to the same month last year, then B 12 is used, and the notation is B 12 y t = y t 12. Forecasting using R Backshift notation reviewed 3

7 Backshift notation First difference: 1 B. Double difference: (1 B) 2. dth-order difference: (1 B) d y t. Seasonal difference: 1 B m. Seasonal difference followed by a first difference: (1 B)(1 B m ). Multiply terms together together to see the combined effect: (1 B)(1 B m )y t = (1 B B m + B m+1 )y t = y t y t 1 y t m + y t m 1. Forecasting using R Backshift notation reviewed 4

8 Backshift notation for ARIMA ARMA model: y t = c + φ 1 y t φ p y t p + e t + θ 1 e t θ q e t q = c + φ 1 By t + + φ p B p y t + e t + θ 1 Be t + + θ q B q e t φ(b)y t = c + θ(b)e t where φ(b) = 1 φ 1 B φ p B p and θ(b) = 1 + θ 1 B + + θ q B q. ARIMA(1,1,1) model: (1 φ 1 B) (1 B)y t = c + (1 + θ 1 B)e t AR(1) First MA(1) difference Forecasting using R Backshift notation reviewed 5

9 Backshift notation for ARIMA ARMA model: y t = c + φ 1 y t φ p y t p + e t + θ 1 e t θ q e t q = c + φ 1 By t + + φ p B p y t + e t + θ 1 Be t + + θ q B q e t φ(b)y t = c + θ(b)e t where φ(b) = 1 φ 1 B φ p B p and θ(b) = 1 + θ 1 B + + θ q B q. ARIMA(1,1,1) model: (1 φ 1 B) (1 B)y t = c + (1 + θ 1 B)e t AR(1) First MA(1) difference Forecasting using R Backshift notation reviewed 5

10 Backshift notation for ARIMA ARIMA(p, d, q) model: (1 φ 1 B φ p B p ) (1 B) d y t = c + (1 + θ 1 B + + θ q B q )e t AR(p) d differences MA(q) Forecasting using R Backshift notation reviewed 6

11 Outline 1 Backshift notation reviewed 2 Seasonal ARIMA models 3 ARIMA vs ETS 4 Lab session 12 Forecasting using R Seasonal ARIMA models 7

12 Seasonal ARIMA models ARIMA (p, d, q) }{{} Non-seasonal part of the model (P, D, Q) m }{{} Seasonal part of of the model where m = number of observations per year. Forecasting using R Seasonal ARIMA models 8

13 Seasonal ARIMA models E.g., ARIMA(1, 1, 1)(1, 1, 1) 4 model (without constant) (1 φ 1 B)(1 Φ 1 B 4 )(1 B)(1 B 4 )y t = (1+θ 1 B)(1+Θ 1 B 4 )e t. ) ( ) ( ) Non-seasonal Non-seasonal Non-seasonal AR(1) difference MA(1) ( ) Seasonal AR(1) ( ) Seasonal difference ( ) Seasonal MA(1) Forecasting using R Seasonal ARIMA models 9

14 Seasonal ARIMA models E.g., ARIMA(1, 1, 1)(1, 1, 1) 4 model (without constant) (1 φ 1 B)(1 Φ 1 B 4 )(1 B)(1 B 4 )y t = (1+θ 1 B)(1+Θ 1 B 4 )e t. ) ( ) ( ) Non-seasonal Non-seasonal Non-seasonal AR(1) difference MA(1) ( ) Seasonal AR(1) ( ) Seasonal difference ( ) Seasonal MA(1) Forecasting using R Seasonal ARIMA models 9

15 Seasonal ARIMA models E.g., ARIMA(1, 1, 1)(1, 1, 1) 4 model (without constant) (1 φ 1 B)(1 Φ 1 B 4 )(1 B)(1 B 4 )y t = (1+θ 1 B)(1+Θ 1 B 4 )e t. ) ( ) ( ) Non-seasonal Non-seasonal Non-seasonal AR(1) difference MA(1) ( ) Seasonal AR(1) ( ) Seasonal difference ( ) Seasonal MA(1) Forecasting using R Seasonal ARIMA models 9

16 Seasonal ARIMA models E.g., ARIMA(1, 1, 1)(1, 1, 1) 4 model (without constant) (1 φ 1 B)(1 Φ 1 B 4 )(1 B)(1 B 4 )y t = (1+θ 1 B)(1+Θ 1 B 4 )e t. All the factors can be multiplied out and the general model written as follows: y t = (1 + φ 1 )y t 1 φ 1 y t 2 + (1 + Φ 1 )y t 4 (1 + φ 1 + Φ 1 + φ 1 Φ 1 )y t 5 + (φ 1 + φ 1 Φ 1 )y t 6 Φ 1 y t 8 + (Φ 1 + φ 1 Φ 1 )y t 9 φ 1 Φ 1 y t 10 + e t + θ 1 e t 1 + Θ 1 e t 4 + θ 1 Θ 1 e t 5. Forecasting using R Seasonal ARIMA models 10

17 Common ARIMA models In the US Census Bureau uses the following models most often: ARIMA(0,1,1)(0,1,1) m ARIMA(0,1,2)(0,1,1) m ARIMA(2,1,0)(0,1,1) m ARIMA(0,2,2)(0,1,1) m ARIMA(2,1,2)(0,1,1) m with log transformation with log transformation with log transformation with log transformation with no transformation Forecasting using R Seasonal ARIMA models 11

18 Seasonal ARIMA models The seasonal part of an AR or MA model will be seen in the seasonal lags of the PACF and ACF. ARIMA(0,0,0)(0,0,1) 12 will show: a spike at lag 12 in the ACF but no other significant spikes. The PACF will show exponential decay in the seasonal lags; that is, at lags 12, 24, 36,... ARIMA(0,0,0)(1,0,0) 12 will show: exponential decay in the seasonal lags of the ACF a single significant spike at lag 12 in the PACF. Forecasting using R Seasonal ARIMA models 12

19 European quarterly retail trade autoplot(euretail) + xlab("year") + ylab("retai 100 Retail index Year Forecasting using R Seasonal ARIMA models 13

20 European quarterly retail trade ggtsdisplay(diff(euretail,4)) diff(euretail, 4) 2 x Time ACF PACF Lag Lag Forecasting using R Seasonal ARIMA models 14

21 European quarterly retail trade ggtsdisplay(diff(diff(euretail,4))) x diff(diff(euretail, 4)) Time ACF PACF Lag Lag Forecasting using R Seasonal ARIMA models 15

22 European quarterly retail trade d = 1 and D = 1 seems necessary. Significant spike at lag 1 in ACF suggests non-seasonal MA(1) component. Significant spike at lag 4 in ACF suggests seasonal MA(1) component. Initial candidate model: ARIMA(0,1,1)(0,1,1) 4. We could also have started with ARIMA(1,1,0)(1,1,0) 4. Forecasting using R Seasonal ARIMA models 16

23 European quarterly retail trade fit <- Arima(euretail, order=c(0,1,1), seasonal=c(0,1,1)) ggtsdisplay(residuals(fit)) x residuals(fit) Time ACF 0.0 PACF Forecasting using LagR Seasonal ARIMA models Lag 17

24 European quarterly retail trade ACF and PACF of residuals show significant spikes at lag 2, and maybe lag 3. AICc of ARIMA(0,1,2)(0,1,1) 4 model is AICc of ARIMA(0,1,3)(0,1,1) 4 model is fit <- Arima(euretail, order=c(0,1,3), seasonal=c(0,1,1)) ggtsdisplay(residuals(fit)) Forecasting using R Seasonal ARIMA models 18

25 European quarterly retail trade ACF and PACF of residuals show significant spikes at lag 2, and maybe lag 3. AICc of ARIMA(0,1,2)(0,1,1) 4 model is AICc of ARIMA(0,1,3)(0,1,1) 4 model is fit <- Arima(euretail, order=c(0,1,3), seasonal=c(0,1,1)) ggtsdisplay(residuals(fit)) Forecasting using R Seasonal ARIMA models 18

26 European quarterly retail trade residuals(fit) x Time ACF PACF Lag Lag Forecasting using R Seasonal ARIMA models 19

27 European quarterly retail trade res <- residuals(fit) Box.test(res, lag=16, fitdf=4, type="ljung") ## ## Box-Ljung test ## ## data: res ## X-squared = , df = 12, p-value = Forecasting using R Seasonal ARIMA models 20

28 European quarterly retail trade autoplot(forecast(fit, h=12)) Forecasts from ARIMA(0,1,3)(0,1,1)[4] 100 y 95 level Forecasting using R Time Seasonal ARIMA models 21

29 European quarterly retail trade auto.arima(euretail) ## Series: euretail ## ARIMA(1,1,2)(0,1,1)[4] ## ## Coefficients: ## ar1 ma1 ma2 sma1 ## ## s.e ## ## sigma^2 estimated as : log likelihood= ## AIC=69.37 AICc=70.51 BIC=79.76 Forecasting using R Seasonal ARIMA models 22

30 European quarterly retail trade auto.arima(euretail, stepwise=false, approximation=false) ## Series: euretail ## ARIMA(0,1,3)(0,1,1)[4] ## ## Coefficients: ## ma1 ma2 ma3 sma1 ## ## s.e ## ## sigma^2 estimated as : log likelihood=-28.7 ## AIC=67.4 AICc=68.53 BIC=77.78 Forecasting using R Seasonal ARIMA models 23

31 Cortecosteroid drug sales H02 sales (million scripts) Log H02 sales Year Forecasting using R Seasonal ARIMA models 24

32 Cortecosteroid drug sales 0.4 Seasonally differenced H02 scripts 0.2 x Year ACF 0.2 PACF Lag Lag Forecasting using R Seasonal ARIMA models 25

33 Cortecosteroid drug sales Choose D = 1 and d = 0. Spikes in PACF at lags 12 and 24 suggest seasonal AR(2) term. Spikes in PACF suggests possible non-seasonal AR(3) term. Initial candidate model: ARIMA(3,0,0)(2,1,0) 12. Forecasting using R Seasonal ARIMA models 26

34 Cortecosteroid drug sales Model AICc ARIMA(3,0,0)(2,1,0) ARIMA(3,0,1)(2,1,0) ARIMA(3,0,2)(2,1,0) ARIMA(3,0,1)(1,1,0) ARIMA(3,0,1)(0,1,1) ARIMA(3,0,1)(0,1,2) ARIMA(3,0,1)(1,1,1) Forecasting using R Seasonal ARIMA models 27

35 Cortecosteroid drug sales (fit <- Arima(h02, order=c(3,0,1), seasonal=c(0,1,2), lambda=0)) ## Series: h02 ## ARIMA(3,0,1)(0,1,2)[12] ## Box Cox transformation: lambda= 0 ## ## Coefficients: ## ar1 ar2 ar3 ma1 sma1 sma2 ## ## s.e ## ## sigma^2 estimated as : log likelihood= ## AIC= AICc= BIC= Forecasting using R Seasonal ARIMA models 28

36 Cortecosteroid drug sales ggtsdisplay(residuals(fit)) 0.2 residuals(fit) x Time ACF 0.0 PACF Lag Lag Forecasting using R Seasonal ARIMA models 29

37 Cortecosteroid drug sales Box.test(residuals(fit), lag=36, fitdf=6, type="ljung") ## ## Box-Ljung test ## ## data: residuals(fit) ## X-squared = , df = 30, p-value = Forecasting using R Seasonal ARIMA models 30

38 Cortecosteroid drug sales fit <- auto.arima(h02, lambda=0, d=0, D=1, max.order=9, stepwise=false, approximation=false) ggtsdisplay(residuals(fit)) residuals(fit) x ACF Time PACF Lag Lag Forecasting using R Seasonal ARIMA models 31

39 Cortecosteroid drug sales Box.test(residuals(fit), lag=36, fitdf=8, type="ljung") ## ## Box-Ljung test ## ## data: residuals(fit) ## X-squared = , df = 28, p-value = Forecasting using R Seasonal ARIMA models 32

40 Cortecosteroid drug sales Model RMSE ARIMA(3,0,0)(2,1,0)[12] ARIMA(3,0,1)(2,1,0)[12] ARIMA(3,0,2)(2,1,0)[12] ARIMA(3,0,1)(1,1,0)[12] ARIMA(3,0,1)(0,1,1)[12] ARIMA(3,0,1)(0,1,2)[12] ARIMA(3,0,1)(1,1,1)[12] ARIMA(4,0,3)(0,1,1)[12] ARIMA(3,0,3)(0,1,1)[12] ARIMA(4,0,2)(0,1,1)[12] ARIMA(3,0,2)(0,1,1)[12] ARIMA(2,1,3)(0,1,1)[12] ARIMA(2,1,4)(0,1,1)[12] ARIMA(2,1,5)(0,1,1)[12] Forecasting using R Seasonal ARIMA models 33

41 Cortecosteroid drug sales Models with lowest AICc values tend to give slightly better results than the other models. AICc comparisons must have the same orders of differencing. But RMSE test set comparisons can involve any models. No model passes all the residual tests. Use the best model available, even if it does not pass all tests. In this case, the ARIMA(3,0,1)(0,1,2) 12 has the lowest RMSE value and the best AICc value for models with fewer than 6 parameters. Forecasting using R Seasonal ARIMA models 34

42 Cortecosteroid drug sales fit <- Arima(h02, order=c(3,0,1), seasonal=c(0,1,2), lambda=0) autoplot(forecast(fit)) + ylab("h02 sales (million scripts)") + xlab("year") Forecasts from ARIMA(3,0,1)(0,1,2)[12] H02 sales (million scripts) level Year Forecasting using R Seasonal ARIMA models 35

43 Outline 1 Backshift notation reviewed 2 Seasonal ARIMA models 3 ARIMA vs ETS 4 Lab session 12 Forecasting using R ARIMA vs ETS 36

44 ARIMA vs ETS Myth that ARIMA models are more general than exponential smoothing. Linear exponential smoothing models all special cases of ARIMA models. Non-linear exponential smoothing models have no equivalent ARIMA counterparts. Many ARIMA models have no exponential smoothing counterparts. ETS models all non-stationary. Models with seasonality or non-damped trend (or both) have two unit roots; all other models have one unit root. Forecasting using R ARIMA vs ETS 37

45 Equivalences ETS model ARIMA model Parameters ETS(A,N,N) ARIMA(0,1,1) θ 1 = α 1 ETS(A,A,N) ARIMA(0,2,2) θ 1 = α + β 2 θ 2 = 1 α ETS(A,A,N) ARIMA(1,1,2) φ 1 = φ θ 1 = α + φβ 1 φ θ 2 = (1 α)φ ETS(A,N,A) ARIMA(0,0,m)(0,1,0) m ETS(A,A,A) ARIMA(0,1,m + 1)(0,1,0) m ETS(A,A,A) ARIMA(1,0,m + 1)(0,1,0) m Forecasting using R ARIMA vs ETS 38

46 Outline 1 Backshift notation reviewed 2 Seasonal ARIMA models 3 ARIMA vs ETS 4 Lab session 12 Forecasting using R Lab session 12 39

47 Lab Session 12 Forecasting using R Lab session 12 40

Forecasting using R. Rob J Hyndman. 2.4 Non-seasonal ARIMA models. Forecasting using R 1

Forecasting using R. Rob J Hyndman. 2.4 Non-seasonal ARIMA models. Forecasting using R 1 Forecasting using R Rob J Hyndman 2.4 Non-seasonal ARIMA models Forecasting using R 1 Outline 1 Autoregressive models 2 Moving average models 3 Non-seasonal ARIMA models 4 Partial autocorrelations 5 Estimation