ARIMA and SARIMA Time Series Models

A clear, concise explanation of ARIMA, SARIMA, and related models — followed by minimal R examples..
statistics
time-series
R
Author

Abdullah Al Mahmud

Published

October 6, 2025

Here’s a clear, concise explanation of ARIMA, SARIMA, and related models — followed by minimal R examples.

1. ARIMA (AutoRegressive Integrated Moving Average)

ARIMA is a time-series forecasting model used for non-seasonal, univariate data. It is written as:

\(\text{ARIMA}(p,d,q)\)

Meaning of the parameters

  • p = number of autoregressive (AR) terms
  • d = degree of differencing (to remove trend)
  • q = number of moving average (MA) terms

When to use ARIMA

Use ARIMA when your data:

  • has trend, but
  • no seasonality.

Minimal ARIMA Example in R

library(forecast)

# Use an example built-in dataset
ts_data <- AirPassengers[1:60]   # Monthly passengers (first 5 years)

# Fit a simple ARIMA model
fit <- auto.arima(ts_data)

# Print model
fit

# Forecast next 12 periods
forecast(fit, h = 12)

auto.arima() automatically selects (p, d, q).

2. SARIMA (Seasonal ARIMA)

SARIMA extends ARIMA by adding seasonal components.

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

Where:

  • P = seasonal autoregressive order

  • D = seasonal differencing

  • Q = seasonal moving average

  • m = seasonal period

    • Example: monthly data → m = 12
    • Quarterly data → m = 4

When to use SARIMA

Use SARIMA when your time series has:

  • trend,
  • and clear seasonality (repeating yearly, quarterly, weekly pattern).

Minimal SARIMA Example in R

library(forecast)

ts_data <- AirPassengers  # strong yearly seasonality

# Fit SARIMA
fit <- auto.arima(ts_data, seasonal = TRUE)

fit
forecast(fit, h = 12)

3. ARIMAX / SARIMAX (ARIMA with regressors)

These models include external variables (e.g., temperature, ads, price). Useful when other predictors help forecast the main series.

x <- rnorm(length(ts_data))   # Example external regressor
fit <- auto.arima(ts_data, xreg = x)

4. Seasonal decomposition (not a model)

Often used to inspect data before ARIMA.

plot(stl(AirPassengers, s.window = "periodic"))

Shows:

  • trend,
  • seasonal,
  • remainder components.

⭐ Summary Table

Model Handles Trend Handles Seasonality Includes External Variables
ARIMA ✔️ ❌ ✔️ (ARIMAX)
SARIMA ✔️ ✔️ ✔️ (SARIMAX)
ETS ✔️ ✔️ ❌
Prophet ✔️ ✔️ ✔️

Very Small Example for Conceptual Understanding

Suppose monthly sales:

Month Sales
Jan 100
Feb 110
Mar 120
Apr 130
May 140
Jun 150

This has trend but no seasonality → ARIMA.

If you have:

Month Sales
Jan 100
Feb 120
Mar 150
… …
Jan (next year) 110
Feb (next year) 130

This has trend + yearly cycle → SARIMA.