SARIMA

SARIMA extends ARIMA by adding seasonal components. It is written as: $\text{SARIMA}(p,d,q)(P,D,Q{)}_s$

Non-Seasonal ARIMA Part $(p,d,q)$ see ARIMA

Seasonal Part $(P,D,Q{)}_s$

These extend ARIMA to handle repeating seasonal patterns. Here, $s$ is the seasonal period (how often the pattern repeats, e.g., 12 for monthly yearly seasonality).

• $P$ – seasonal autoregressive order: Similar to $p$, but at the seasonal lag. Example: If $P=1$ and $s=12$, the model uses $y_{t-12}$ (value from one year ago) to predict $y_t$.

• $D$ – seasonal differencing order: Number of times seasonal differencing is applied. Removes seasonal trends. Example: $D=1$ with $s=12$: $y_t-y_{t-12}$.

• $Q$ – seasonal moving average order: Similar to $q$, but considers past seasonal forecast errors. Example: $Q=1$ and $s=12$ uses the error from 12 months ago.

How SARIMA Works (Step by Step)

1. Seasonal differencing first (if $D>0$): Remove repeating seasonal trends to stabilize the mean. $y_t^{\prime }=y_t-y_{t-s}$

2. Non-seasonal differencing (if $d>0$): Remove overall trend to make the series stationary.

3. Apply ARMA on transformed series:

   • Non-seasonal AR ($p$) and MA ($q$) model short-term dependencies.
   • Seasonal AR ($P$) and MA ($Q$) model dependencies at seasonal lags.

4. Combine predictions: The model combines seasonal and non-seasonal components to forecast the next time step.

Related:

• AIC in Model Evaluation
• Evolving Seasonality
• Differencing in Time Series

Determining parameters

https://tsanggeorge.medium.com/a-semi-auto-way-to-determine-parameters-for-sarima-model-74cdee853080

Notes

• Alternatively, SARIMA is conceptually similar to Holt-Winters but with more statistical rigor, so they can be compared side by side.