SARIMA and ETS

• Write and interpret the SARIMA seasonal terms (P,D,Q,m)
• Understand ETS models from the smoothing recursion
• Choose SARIMA vs ETS from AIC and residual quality

• Plain ARIMA leaks seasonality into residuals — the pattern is memory, not noise.
• ETS handles seasonality with interpretable component weights and direct interval forecasts.
• Auto-selection (auto_arima, ETS selector via AIC) gives reproducible picks without hand-tweaking.
• They're the honest baselines ML forecasters must beat before adding complexity.

SARIMA explicitly models 'what happened this season vs last' with lagged terms — a tide chart built from lunar lags. ETS assumes the season pattern evolves smoothly, like a pendulum — track the amplitude, correct slowly. Both tell you the ocean; the physics they assume differs.

• SARIMA: add (P,D,Q)_m — seasonal differencing removes repeat-trend, seasonal AR multiplies the series by (1 − Φ₁B^m − ...).
• Seasonal order m = the cycle length (12 monthly, 7 daily/weekly).
• ETS: level, trend and seasonal equations update with smoothing parameters (α, β, γ), additive or multiplicative.
• Both are fit by maximum likelihood (SARIMA uses Kalman/state space internally; ETS matches a state-space likelihood).
• Select among candidates with AIC/BIC, then verify residual white noise and out-of-sample backtest.

• A retailer picks SARIMA(2,1,1)(1,1,1)_12 for monthly demand — 12-month seasonal effects matter more than deep AR memory.
• An energy scheduler uses Holt-Winters (ETS) for weekly load because the αβγ smoothing tracks slow seasonal drift.
• Both are compared each month by auto-selection, and the winner feeds safety-stock via its prediction intervals.

• Set m from real cycles and confirm with spectral/ACF checks.
• Let auto_arima/AIC-driven ETS do the order search, but sanity-check by hand.
• Compare SARIMA vs ETS candidates on the same walk-forward backtest, not just in-sample AIC.
• Forecast intervals come from the fitted state space — never hack percentiles of one path.

• Writing SARIMA with the wrong m (monthly data but m=4 'quarterly') and wondering at the residual spikes.
• Differencing both d and D aggressively, then overdifferencing (stationary field says immune).
• Fitting ETS additive when the series needs multiplicative seasonal — spikes blow up.
• Selecting purely on in-sample AIC and skipping the backtest where both fail differently.

• Residual ACF spike at lag m → add seasonal MA(Q) term (or raise it).
• PACF spike at lag m → add seasonal AR(P).
• ETS forecast variance wrong → check the residual variance decomposition and Box-Cox choice.
• Turn summer factors stay 'constant' forever? The smoothing γ may be pinned to ~0 — season is fixed or data short.