SARIMA

SARIMA, or Seasonal ARIMA, is a class of univariate time series models that extends the ARIMA framework to account for both non-seasonal dynamics and seasonal patterns. The model is denoted SARIMA(p,d,q)(P,D,Q)_s, where p, d, q are the orders of the non-seasonal AR, differencing, and MA parts, P, D, Q are the seasonal orders, and s is the seasonal period (for example, s = 12 for monthly data, s = 4 for quarterly data). The combination of non-seasonal and seasonal components enables the model to capture trend, short-run correlations, and repeating seasonal behavior.

Mathematically, the model can be written using backshift operators as φ(B) Φ(B^s) ∇^d ∇_s^D y_t = θ(B) Θ(B^s) ε_t, where φ and θ are polynomials in the backshift operator B for the non-seasonal part, Φ and Θ are seasonal polynomials in B^s, ∇^d represents non-seasonal differencing of order d, ∇_s^D represents seasonal differencing of order D, and ε_t is white noise. The seasonal period s determines how often the seasonal terms recur.

Estimation and model selection typically rely on maximum likelihood or conditional least squares, with orders chosen by information criteria such as AIC or BIC and guided by ACF/PACF analysis. Diagnostic checks on residuals (e.g., Ljung-Box tests) assess model adequacy. SARIMA is widely used for forecasting economic, inventory, energy, and demand data that exhibit regular seasonal patterns. It can be extended to include exogenous variables (SARIMAX) but remains a foundational tool for modeling seasonal time series in a neutral, data-driven manner.