SARIMA – Modeling Trend and Seasonality Together

In the previous lesson, ARIMA helped us solve one major problem: trend.

But real-world time series rarely stop there. Most of them also repeat patterns at fixed intervals.

This lesson answers one important question:

What if the data has both trend and seasonality?

Why ARIMA Still Fails for Seasonal Data

Consider these examples:

• Retail sales spike every December
• Electricity demand rises every summer
• Website traffic peaks every weekend

Even after removing trend using differencing, the seasonal pattern still remains.

ARIMA does not understand repeating cycles. It only sees short-term correlations.

The Idea Behind SARIMA

SARIMA simply extends ARIMA by adding:

Seasonal AR, Seasonal MA, and Seasonal Differencing

So instead of modeling only:

Past values and errors

We also model:

Past values and errors from previous seasons

SARIMA Model Structure

SARIMA is written as:

SARIMA(p, d, q)(P, D, Q, m)

• p, d, q → non-seasonal ARIMA
• P, D, Q → seasonal ARIMA
• m → length of season

Example:

SARIMA(1,1,1)(1,1,1,12)

This means:

• Monthly data
• Yearly seasonality
• Trend + seasonal differencing

Real-World Analogy

Imagine tracking gym attendance:

• Overall attendance grows year by year (trend)
• Attendance drops every weekend (seasonality)

ARIMA handles the growth. SARIMA handles the weekend pattern.

Both are required for accurate forecasting.

Step 1: Create Seasonal Time Series

Let’s simulate data with:

• Upward trend
• Clear repeating seasonality

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(2)
time = np.arange(120)
trend = time * 0.3
seasonal = 15 * np.sin(2 * np.pi * time / 12)
noise = np.random.normal(0, 3, 120)

series = trend + seasonal + noise

plt.plot(series)
plt.title("Seasonal Time Series")
plt.show()

This is what the data actually looks like:

Observation:

• Overall upward movement
• Repeating wave every 12 steps
• Random fluctuations

Step 2: Seasonal Differencing

Seasonal differencing removes repeating cycles.

If season length is 12:

value(t) − value(t−12)

season_diff = series[12:] - series[:-12]

plt.plot(season_diff)
plt.title("Seasonally Differenced Series")
plt.show()

Here is the seasonal differenced plot:

What changed?

• Seasonal wave disappears
• Series fluctuates around zero
• Much more stable

Step 3: Combine with Trend Differencing

Sometimes one seasonal difference is not enough. We may also apply regular differencing.

This removes:

• Long-term trend
• Seasonal repetition

At this stage, the series becomes suitable for AR and MA modeling.

Why SARIMA Works So Well

• AR → short-term memory
• MA → error correction
• d → trend removal
• D → seasonal removal

Each part solves a specific real-world behavior.

When Should You Use SARIMA?

• Clear repeating patterns
• Fixed seasonal interval
• Strong historical structure

Examples:

• Monthly sales forecasting
• Electricity demand
• Tourism and hotel bookings

Common Mistakes

• Using ARIMA when seasonality exists
• Wrong season length (m)
• Over-differencing the data

Always visualize before modeling.

Practice Questions

Key Takeaways

• ARIMA handles trend
• SARIMA handles trend + seasonality
• Visualization is critical before modeling

Next, we’ll see how to add external factors that influence the time series.