SARIMA

Seasonal ARIMA (SARIMA) extends the standard ARIMA model by incorporating seasonal components. It is expressed as:

$\text{SARIMA}(p,d,q)(P,D,Q{)}_s$

Components

Non-seasonal part $(p,d,q)$

(See ARIMA for details.)

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

These terms allow ARIMA to model repeating seasonal patterns.

The parameter $s$ denotes the seasonal period, how often the pattern repeats (e.g., $s=12$ for monthly data with yearly seasonality).

• $P$ - Seasonal Autoregressive Order Analogous to $p$, but applies to seasonal lags. Example: If $P=1$ and $s=12$, the model uses $y_{t-12}$ (the value from one year ago) to predict $y_t$.

• $D$ - Seasonal Differencing Order The number of times seasonal differencing is applied to remove seasonal trends. Example: $D=1$ with $s=12$ applies $y_t-y_{t-12}$.

• $Q$ - Seasonal Moving Average Order Analogous to $q$, but incorporates forecast errors from seasonal lags. Example: $Q=1$ and $s=12$ uses the forecast error from 12 months prior.

How SARIMA Works

1. Seasonal differencing ($D>0$)

   • Remove repeating seasonal patterns to stabilize the mean. $y_t^{\prime }=y_t-y_{t-s}$

2. Non-seasonal differencing ($d>0$)

   • Remove overall trends to make the series stationary.

3. Fit ARMA components

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

4. Combine predictions The final forecast combines seasonal and non-seasonal components.

Determining Parameters

Refer to: Tsang, G. (2020). A semi-auto way to determine parameters for SARIMA models

Practical Notes

• SARIMA is conceptually similar to Holt–Winters exponential smoothing, but relies on more formal statistical assumptions.
• Before estimating $P$ and $Q$, ensure the series is seasonally stationary by applying seasonal Differencing if required ($D>0$).

Identifying Seasonal Orders with ACF and PACF Plots

To choose $Q$ (Seasonal MA order):

• Examine the ACF plot for significant autocorrelations at seasonal lags (multiples of $s$).
• A sharp cut-off at a seasonal lag suggests the appropriate $Q$.

To choose $P$ (Seasonal AR order):

• Examine the PACF plot for significant spikes at seasonal lags.
• A sharp cut-off indicates the seasonal AR order $P$.

Related

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