Comparison and Contrast of ARIMA, SARIMA, and ETS Models

ARIMA, SARIMA, and ETS are all popular models used for time-series forecasting, each with its own strengths and appropriate use cases. Here is a detailed comparison and contrast of these models:

ARIMA (AutoRegressive Integrated Moving Average)

Components:

  1. AutoRegressive (AR): Uses the dependency between an observation and a number of lagged observations.
  2. Integrated (I): Uses differencing of observations to make the time series stationary.
  3. Moving Average (MA): Uses dependency between an observation and a residual error from a moving average model applied to lagged observations.

Model Formulation:

  • ARIMA(p, d, q) where:
    • $( p )$ = number of lag observations (autoregressive part).
    • $( d)$ = number of times the raw observations are differenced (integrated part).
    • $( q )$ = size of the moving average window (moving average part).

Key Characteristics:

  • Best suited for time series without strong seasonal components.
  • Requires the time series to be stationary, hence the need for differencing.
  • Well-suited for univariate time series forecasting.

SARIMA (Seasonal ARIMA)

Components:

  1. Seasonal ARIMA: Extends ARIMA by adding seasonal components.
  2. Seasonal Differencing: Accounts for seasonality by differencing at lag $( s )$.

Model Formulation:

  • SARIMA(p, d, q)(P, D, Q, s) where:
    • $( p, d, q )$ = parameters for the non-seasonal part.
    • $( P, D, Q, s )$ = parameters for the seasonal part.
    • $( s )$ = length of the seasonal cycle.

Key Characteristics:

  • Suitable for time series with strong seasonal patterns.
  • Combines seasonal differencing with the ARIMA model framework.
  • More complex due to additional seasonal parameters.

ETS (Error, Trend, Seasonal)

Components:

  1. Error (E): The nature of the error component (Additive or Multiplicative).
  2. Trend (T): The nature of the trend component (None, Additive, or Multiplicative).
  3. Seasonal (S): The nature of the seasonal component (None, Additive, or Multiplicative).

Model Formulation:

  • Represented as ETS(A, A, A), ETS(M, A, M), etc.
  • Components interact additively or multiplicatively depending on their specified type.

Key Characteristics:

  • Explicitly models trend and seasonality components.
  • Flexible in handling different combinations of error, trend, and seasonal components.
  • Highly interpretable due to its decomposition.

Comparison and Contrast

1. Handling Seasonality:

  • ARIMA: Does not handle seasonality directly; requires seasonal differencing or external preprocessing.
  • SARIMA: Specifically designed to handle seasonality by incorporating seasonal parameters.
  • ETS: Handles seasonality explicitly through the seasonal component in the model.

2. Stationarity Requirement:

  • ARIMA: Requires the series to be stationary, often achieved through differencing.
  • SARIMA: Also requires stationarity but includes seasonal differencing.
  • ETS: Does not require stationarity and can handle non-stationary data through its trend component.

3. Model Complexity:

  • ARIMA: Simpler in terms of parameters, focusing on lagged values and differencing.
  • SARIMA: More complex due to the addition of seasonal parameters.
  • ETS: Complexity varies with the chosen model, but offers clear decomposition of components.

4. Interpretability:

  • ARIMA: Moderate interpretability; parameters indicate lags and differencing needed.
  • SARIMA: Slightly less interpretable due to additional seasonal parameters.
  • ETS: Highly interpretable with clear components for error, trend, and seasonality.

5. Use Cases:

  • ARIMA: Suitable for non-seasonal univariate time series, or when seasonal patterns are weak.
  • SARIMA: Ideal for univariate time series with strong seasonal patterns.
  • ETS: Suitable for time series with distinct trend and seasonal patterns, offering a flexible and interpretable framework.

6. Parameter Selection:

  • ARIMA: Parameters $( p, d, q )$ selected based on autocorrelation and partial autocorrelation functions.
  • SARIMA: Parameters $( p, d, q, P, D, Q, s )$ selected considering both seasonal and non-seasonal autocorrelations.
  • ETS: Model selection involves choosing the best combination of error, trend, and seasonal components based on fit.

Summary

  • ARIMA is best for non-seasonal, stationary time series and is relatively simple but powerful for such data.
  • SARIMA extends ARIMA to handle seasonality, making it suitable for seasonal time series.
  • ETS explicitly models trend and seasonal components, offering flexibility and high interpretability, suitable for time series with distinct patterns.

Choosing between these models depends on the specific characteristics of the time series data and the forecasting requirements.