Seasonality & SARIMA Model¶

1. Types of Seasonality¶

Seasonality can manifest in different ways depending on how the periodic component behaves over time:

Deterministic Seasonality: The seasonal effect is constant and repeats exactly every $s$ periods, such that $S_{t}^{(s)} = S_{t+ks}^{(s)}$ for $k = \pm 1, \pm 2, \dots$. Since the pattern simply repeats, the future seasonal component can be determined exactly.
Stationary Evolving Seasonality: The seasonal factor oscillates around a deterministic mean $\mu^{(s)}$, expressed as $S_{t}^{(s)} = \mu^{(s)} + v_{t}$, where $v_{t}$ is a stationary process. Here, $v_{t}$ introduces time-dependent variability around the seasonal effect.
Non-Stationary Evolving Seasonality: The seasonal component $S_{t}^{(s)}$ follows a non-stationary process, such as a random walk: $S_{t}^{(s)} = S_{t-s}^{(s)} + v_{t}$.

Universal Correction

Applying a seasonal difference is effective in correcting for seasonality in all three of the cases mentioned above.

2. The SARIMA Model¶

The Seasonal AutoRegressive Integrated Moving Average (SARIMA) model extends ARIMA by explicitly supporting seasonal components through seasonal differencing and seasonal operators.

Mathematical Structure¶

The model is defined by the following equation:
$$\Phi_{P}(B^S)\phi_{p}(B)(1-B^S)^D(1-B)^dY_{t} = \Theta_{Q}(B^S)\theta_{q}(B)e_{t}$$

This structure consists of four distinct operational parts:
* $\Phi_{P}(B^S)$: Seasonal Autoregressive (AR) operator of order $P$.
* $\phi_{p}(B)$: Regular Autoregressive (AR) operator of order $p$.
* $\Theta_{Q}(B^S)$: Seasonal Moving Average (MA) operator of order $Q$.
* $\theta_{q}(B)$: Regular Moving Average (MA) operator of order $q$.

Notation¶

A SARIMA model is typically denoted as:
$$\text{SARIMA}(p,d,q) \times (P,D,Q)_{S}$$
* $(p,d,q)$: Represents the non-seasonal part of the model.
* $(P,D,Q)_{S}$: Represents the seasonal part of the model with a period $S$.
* $D$: Number of seasonal differences (usually 0 or 1).
* $d$: Number of regular differences.

3. Pros & Cons of SARIMA¶

Pros	Cons
Simplicity: The model is relatively easy to understand and interpret.	Time Complexity: Computational requirements can grow exponentially as orders $p$ and $q$ increase.
Parsimony: Requires a limited number of variables and parameters to estimate.	Optimization Difficulty: Finding an optimal solution for $p$ and $q$ can be difficult with complex data.
	Data Intensity: A considerable amount of historical data is required for effective modeling.

4. Use Cases¶

SARIMA is used when both a trend and seasonality are present in the data:
1. Epidemiology: Forecasting the dynamics of the COVID-19 epidemic in India or bed requirements in Singapore.
2. Renewable Energy: Predicting daily and monthly average global solar radiation (e.g., in South Korea).
3. Supply Chain: Forecasting demand for food companies to manage inventory.

Model Selection

Use ARIMA if you only need to capture the trend component; use SARIMA if seasonality is also present.