SARIMA
SARIMA extends ARIMA by adding seasonal autoregressive, differencing, and moving-average terms at a known period . It is designed for series where dependence repeats at regular seasonal lags: month 12 resembles month 0, hour 24 resembles hour 0, or week 52 resembles week 0.
A SARIMA model is written as
The lowercase orders describe the nonseasonal dynamics. The uppercase orders describe seasonal dynamics at multiples of . In lag-polynomial form, a typical model is
The seasonal differencing term removes repeated seasonal level shifts, while seasonal AR and MA terms model dependence or shocks at the seasonal lag. For monthly data with yearly seasonality, ; for hourly data with daily seasonality, .
The seasonal period is a modeling commitment, not a parameter to guess casually. SARIMA works best when the seasonal pattern is regular, stable, and visible in autocorrelation and partial autocorrelation. Seasonal AR terms tend to create decaying autocorrelation at seasonal lags; seasonal MA terms tend to create seasonal-lag ACF spikes. If the data has multiple overlapping seasonalities, moving holidays, weather interactions, or fast regime changes, feature engineering for forecasting, dynamic regression, or neural/global models may be a better fit.
SARIMA remains a strong baseline for high-seasonality operational series such as energy consumption forecasting, call volumes, and mature product demand. It should be judged against simpler seasonal naive forecasts and exponential smoothing, because seasonal ARIMA complexity is only useful when it improves out-of-sample forecasts.
Connections
SARIMA is the seasonal extension of ARIMA. It uses the same stationarity logic as stationarity, but seasonal structure comes from trend-seasonality-cycles-noise and is evaluated within statistical forecasting.
References
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