DAT320 Forecasting - Seasonal ARIMA - PDF

DAT320: Forecasting Seasonal ARIMA Kristian Hovde Liland [email protected] Autumn 2024 Norwegian University of Life Sciences Extensions of ARIMA Seasonal ARIMA Model Advanced ARIMA models 1 Norwegian University of Life Sciences Extension...

DAT320: Forecasting Seasonal ARIMA Kristian Hovde Liland [email protected] Autumn 2024 Norwegian University of Life Sciences Extensions of ARIMA Seasonal ARIMA Model Advanced ARIMA models 1 Norwegian University of Life Sciences Extensions of ARIMA AR MA ARMA ARIMA ARIMAX & SARIMA FARIMA VARIMA dynamic regr. seasonal data exogenous inputs long-term dependencies multivariate data 2 Norwegian University of Life Sciences Seasonal ARIMA model (SARIMA) I SARIM A(k, d, q)(K, D, Q)p allows both seasonal and non-seasonal components: I Non-seasonal ARIM A(k, d, q) I Seasonal part with parameters K (autoregressive), D (seasonal differencing) and Q (moving average) I Seasonal period p I Special cases: I SAR(K) seasonal AR I SM A(Q) seasonal MA 3 Norwegian University of Life Sciences Seasonal ARIMA model (SARIMA) SARIM A(1, 1, 1)(1, 1, 1)p Multiplicative combination of terms: (1 − ϕ1 B)(1 − Φ1 B p )(1 − B)(1 − B p )xt = (1 + θ1 B)(1 + Θ1 B p )εt I (1 − ϕ1 B − · · · − ϕk B k )... AR(k) terms I (1 − Φ1 B p − · · · − ΦK B kp )... SAR(K) terms I (1 − B)d... differencing terms I (1 − B p )D... seasonal differencing terms I (1 + θ1 B + · · · + θq B q )... M A(q) terms I (1 + Θ1 B p + · · · + ΘQ B Qp )... SM A(Q) terms 4 Norwegian University of Life Sciences Seasonal ARIMA model (SARIMA) SARIM A(2, A(0, 0, 1)(2, A(1, 0)(1, 0, 1)p 1)(1, εt−2p... εt−p... εt−3 εt−2 εt−1 εt xt−2p... xt−p... xt−3 xt−2 xt−1 xt SM SAR(1) SAR(2) M A(1)pp A(2) AR(1) A(1) t 5 Norwegian University of Life Sciences Seasonal unit root test I Check for trends via KPSS or (Augmented) Dickey-Fuller test ("unit root tests") I How to determine whether data have seasonalities to determine the number of seasonal differences? I "seasonal unit root tests" I Hylleberg-Engle-Granger-Yoo test (HEGY) library ( datasets ) library ( tseries ) library ( uroot ) data ( " AirPassengers " ) kpss. test ( AirPassengers ) hegy. test ( AirPassengers ) 6 Norwegian University of Life Sciences Hyperparameter selection Rule of thumb for SARIMA models I Determine order of differencing d and seasonal differencing D via KPSS and HEGY tests I Plot ACF and PACF I Estimate AR or MA parameters from ACF and PACF as in ARIMA models I Estimate SAR or SMA parameters from seasonal spikes in ACF and PACF 7 Norwegian University of Life Sciences SARIMA model I SARIM A(2, 1, 1)(0, 1, 0)12 model Figure 1: Fitted values & residuals 8 Norwegian University of Life Sciences SARIMA model (a) Point prediction (b) Prediction interval 9 Norwegian University of Life Sciences SARIMA model library ( forecast ) library ( datasets ) library ( tseries ) library ( uroot ) data ( " AirPassengers " ) acf ( AirPassengers , lag. max = 36) pacf ( AirPassengers , lag. max = 36) ndiffs ( AirPassengers ) nsdiffs ( AirPassengers ) mod _ sar