White Noise and Random Walk

Before learning forecasting models, we must clearly understand two very important concepts that often confuse beginners:

• White Noise
• Random Walk

These are not just theory terms. They directly decide whether forecasting is possible or meaningless.

Why This Lesson Matters

Many beginners try to forecast data without checking its nature. As a result:

• Models give nonsense predictions
• Accuracy looks random
• Results change every time

Understanding white noise and random walk helps you answer one key question:

“Is this data even forecastable?”

1. What Is White Noise?

White noise is a time series that contains pure randomness. There is:

• No trend
• No seasonality
• No structure
• No memory of the past

Each value is completely independent of previous values.

Real-world intuition:

• Sensor measurement errors
• Background electronic noise
• Daily tiny fluctuations with no pattern

Python Example: Generating White Noise

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(42)
white_noise = np.random.normal(0, 1, 200)

plt.figure(figsize=(8,4))
plt.plot(white_noise)
plt.title("White Noise Time Series")
plt.xlabel("Time")
plt.ylabel("Value")
plt.show()

Let’s not imagine the output. Here is the actual plot produced by this code:

What you should observe:

• Values jump randomly
• No visible direction
• No repeating pattern
• No dependency between points

This is white noise.

Can White Noise Be Forecasted?

No.

Because:

• There is no information in the past
• Future values are independent
• Best prediction is always the mean

If your data looks like white noise, forecasting models are useless.

2. What Is a Random Walk?

A random walk looks random — but it is very different from white noise.

In a random walk:

• Each value depends on the previous value
• Random changes accumulate over time

Simple definition:

Today = Yesterday + Random Change

Real-world intuition:

• Stock prices (short term)
• Cryptocurrency prices
• Uncontrolled drifting processes

Python Example: Random Walk

np.random.seed(42)
steps = np.random.normal(0, 1, 200)
random_walk = np.cumsum(steps)

plt.figure(figsize=(8,4))
plt.plot(random_walk)
plt.title("Random Walk Time Series")
plt.xlabel("Time")
plt.ylabel("Value")
plt.show()

Here is the real output from the code:

What you should notice:

• The line drifts up and down
• Shocks persist over time
• No clear mean level

Unlike white noise, random walk has memory.

White Noise vs Random Walk (Visual Comparison)

Why Random Walk Breaks Forecasting

Random walk violates stationarity.

That means:

• Mean keeps changing
• Variance grows over time
• Models like ARIMA fail directly

This is why we transform data before modeling.

Practice Questions

Key Takeaways

• White noise = pure randomness
• Random walk = accumulated randomness
• Neither should be modeled directly
• Transformations are mandatory

Next Lesson

In the next lesson, we will learn time series decomposition — how to separate trend, seasonality, and noise visually and mathematically.