Summary  
This chapter explains how to assess and distinguish stationary from non-stationary data by comparing and visualizing their statistical properties—such as constant mean and variance versus trends and shifting variance—using generated sample series.  

General domain of usage  
Time series forecasting

Understanding whether a time series is **stationary** or **non-stationary** is a critical first step in time series analysis. A stationary time series has statistical properties – such as **mean**, **variance**, and **autocorrelation** – that do not change over time. This means the overall behavior of the series remains consistent, making it easier to model and predict. In contrast, a non-stationary time series shows **trends**, **cycles**, or **changing variance**, meaning its statistical properties evolve over time. Stationarity matters because many time series modeling techniques, such as `ARIMA`, assume the underlying data is stationary. Non-stationary data can lead to unreliable forecasts and misleading interpretations, so recognizing and transforming non-stationary series is often essential for accurate analysis.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

np.random.seed(0)

# Generate stationary time series: white noise
stationary_series = np.random.normal(loc=0, scale=1, size=100)

# Generate non-stationary time series: random walk
random_steps = np.random.normal(loc=0, scale=1, size=100)
non_stationary_series = np.cumsum(random_steps)

# Create a DataFrame for plotting
df = pd.DataFrame({
    "Stationary": stationary_series,
    "Non-Stationary": non_stationary_series
})

plt.figure(figsize=(12, 5))
plt.subplot(1, 2, 1)
plt.plot(df["Stationary"])
plt.title("Stationary Series (White Noise)")
plt.xlabel("Time")
plt.ylabel("Value")

plt.subplot(1, 2, 2)
plt.plot(df["Non-Stationary"])
plt.title("Non-Stationary Series (Random Walk)")
plt.xlabel("Time")
plt.ylabel("Value")

plt.tight_layout()
plt.show()

When you compare **stationary** and **non-stationary** time series visually, certain cues help you distinguish between them. A stationary series, such as white noise, fluctuates around a constant mean and maintains a consistent variance throughout the time span. In contrast, a non-stationary series – like a random walk – often displays trends, drifts, or changing variability, making the average level or spread change over time.

**Key visual cues for stationarity:**
- Fluctuates around a constant mean;
- Maintains consistent variance;
- No obvious trend or seasonality is present.

**Key visual cues for non-stationarity:**
- Shows trends or drifts over time;
- Variance may increase or decrease;
- Mean level may shift.

These visual cues are important because non-stationary data can mask underlying patterns or relationships, making statistical analysis and forecasting more challenging. Identifying and addressing non-stationarity is a foundational step before applying most time series models, as it ensures that the insights and predictions you derive are both valid and reliable.

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Stationarity in Time Series