ARIMA

ARIMA (AutoRegressive Integrated Moving Average) is a popular method for Time Series Forecasting that models the autocorrelations within the data. It is particularly useful for datasets with trends and patterns that are not seasonal. However, it is not perfect; for example, it struggles with predicting stock trading data.

ARIMA Components

• AR (AutoRegressive): Utilizes the dependency between an observation and a number of lagged observations.
• I (Integrated): Involves differencing the data to achieve stationarity.
• MA (Moving Average): Models the dependency between an observation and a residual error from a moving average model applied to lagged observations.

ARIMA Explained

ARIMA stands for:

• AutoRegressive (AR): Uses past values to predict the current one.
• Integrated (I): Differencing to make the series stationary.
• MAving Average (MA): Uses past forecast errors for prediction.

A typical ARIMA(p,d,q) model has:

• $p$: Number of lagged values (AR terms)
• $d$: Number of differences needed to make the series stationary
• $q$: Number of lagged forecast errors (MA terms)

🔧 What ARIMA Does:

1. Checks for stationarity – if not, applies differencing ($d$ times).
2. Models relationships between current values and:
   • Past values (AR part)
   • Past errors (MA part)
3. Fits parameters by minimizing a loss (typically log-likelihood).
4. Forecasts future values using this learned structure.

Why ARIMA Isn’t Enough for Seasonal Data

Your quarterly data might show repeating patterns every 4 quarters (seasonality), and ARIMA has no built-in mechanism to model this periodic structure. This is where SARIMA comes in.

SARIMA: Seasonal ARIMA

SARIMA extends ARIMA by adding seasonal components. It is written as:

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

Where:

• $(p,d,q)$ are non-seasonal ARIMA parameters (as above).
• $(P,D,Q{)}_s$ are seasonal ARIMA parameters:
  • $P$: Seasonal autoregressive terms.
  • $D$: Seasonal differences.
  • $Q$: Seasonal moving average terms.
  • $s$: Seasonality period (e.g., $s=4$ for quarterly data).

What SARIMA Adds:

• Seasonal differencing ($D$): e.g., subtract the value from 4 quarters ago to remove annual cycles.
• Seasonal AR/MA terms: Model relationships from past seasonal lags and errors.

✅ Summary

Model	Handles Trend	Handles Seasonality	Use When…
ARIMA	✅	❌	Data has trend, but no regular cycles
SARIMA	✅	✅	Data has both trend and seasonality

Advanced Variants

SARIMAX (Seasonal ARIMA with Exogenous Variables)

• Incorporates external variables (e.g., interest rates, volume) into the forecasting model.
• Useful for Datasets where external factors influence the time series.

Related terms

In ML_Tools see:

• https://github.com/rhyslwells/ML_Tools/blob/main/Explorations/Build/TimeSeries/ARIMA.ipynb
• Forecasting_AutoArima.py
• pmdarima