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 $(p,d,q)$

These are the standard ARIMA components:

• $p$ – autoregressive (AR) order: The number of lagged observations included in the model. AR terms capture how past values influence current values. Example: If $p=2$, the model uses the previous 2 time points to predict the current one.

• $d$ – differencing order: Number of times the data is differenced to make it stationary (remove trends). Example: $d=1$ means we model the difference: $y_t-y_{t-1}$.

• $q$ – moving average (MA) order: Number of lagged forecast errors in the prediction equation. MA terms capture short-term shocks or noise in the series. Example: $q=1$ means the model uses the previous time step’s error to adjust the prediction.

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.

Example

$\text{SARIMA}(1,1,1)(1,1,1{)}_{12}$

• $p=1$: depends on previous value
• $d=1$: difference once to remove trend
• $q=1$: uses previous forecast error
• $P=1$: depends on value 12 months ago
• $D=1$: seasonal difference to remove annual seasonality
• $Q=1$: adjusts using error from 12 months ago
• $s=12$: yearly seasonality

This model is good for monthly data with annual seasonal cycles.

Related:

• AIC in Model Evaluation