ARIMA – Handling Trend in Time Series

Until now, all our models had one big assumption:

The data must be stationary.

But real-world data almost never behaves that way. Sales grow, users increase, prices rise, demand changes.

This is where ARIMA becomes essential.

Why ARMA Is Not Enough

In the previous lesson, ARMA worked well only when:

• No trend
• No long-term growth or decline

But consider:

• Monthly revenue increasing year after year
• Population growth
• Energy consumption rising over decades

These series are non-stationary. ARMA will fail completely.

The Missing Piece: Differencing

ARIMA introduces one powerful idea:

Instead of modeling the data, model the change in data.

That change is called differencing.

Example:

• Today sales = 120
• Yesterday sales = 115
• Difference = +5

Differences often remove trends and make data stationary.

ARIMA(p, d, q) Explained Clearly

ARIMA has three parts:

• p → AR (past values)
• d → differencing (trend removal)
• q → MA (past errors)

Example:

ARIMA(1,1,1)

• 1 past value
• 1 difference
• 1 past error

Real-World Analogy

Think about driving uphill:

• Height = raw data
• Slope = difference

Instead of predicting your absolute height, you predict how steep the road is.

That makes prediction much easier.

Step 1: Non-Stationary Time Series

Let’s first create a series with a clear upward trend.

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(1)
time = np.arange(100)
trend = time * 0.4
noise = np.random.normal(0, 3, 100)
series = trend + noise

plt.plot(series)
plt.title("Non-Stationary Time Series")
plt.show()

This is what that data actually looks like:

Observation:

• Clear upward movement
• Mean changes over time
• Not stationary

Step 2: Apply Differencing

Now we subtract each value from the previous one.

diff_series = np.diff(series)

plt.plot(diff_series)
plt.title("Differenced Series")
plt.show()

Here is the differenced data:

What changed?

• Trend is removed
• Mean stays around zero
• Looks stationary

Why Differencing Works

Trend is slow movement.

Differencing removes slow movement and keeps short-term behavior.

AR and MA models work perfectly on this transformed data.

Putting It All Together: ARIMA Conceptually

ARIMA works like this:

1. Difference the data (d times)
2. Apply ARMA on the differenced series
3. Convert predictions back to original scale

This is why ARIMA handles trends but ARMA cannot.

Visual Comparison: Before vs After Differencing

Interpretation:

• Raw series → growing, unstable
• Differenced series → stable, predictable

When Should You Use ARIMA?

• Data has trend
• No strong seasonality
• Short-term forecasting

Examples:

• Sales forecasting
• Demand prediction
• Traffic growth modeling

Practice Questions

Big Picture

AR remembers the past. MA corrects mistakes. Differencing removes trend.

ARIMA combines all three.

But what if seasonality also exists?

That leads directly to the next model.