📊 Exploratory Time Series Analysis (TSA)

Exploratory TSA is the first step in understanding the structure, behavior, and characteristics of your time series data. It includes visualizations, statistical summaries, and decomposition.

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1. 📈 Plotting Time Series Data

The first and most important step is to visualize the data.

🧠 Objective:

• Identify trend, seasonality, outliers, sudden shifts

✅ Code Example:

import pandas as pd
import matplotlib.pyplot as plt

# Load sample time series data
url = 'https://raw.githubusercontent.com/jbrownlee/Datasets/master/airline-passengers.csv'
df = pd.read_csv(url, parse_dates=['Month'], index_col='Month')

# Plot the raw time series
df.plot(title='Monthly Airline Passengers', figsize=(10, 5))
plt.ylabel("Number of Passengers")
plt.grid(True)
plt.show()

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2. 📉 Rolling Statistics (Moving Average and Std)

Rolling statistics smooth the time series by computing averages over a moving window.

🧠 Objective:

• Denoise the series
• Visualize local trends and variation

✅ Code Example:

# Calculate rolling mean and std
rolling_mean = df['Passengers'].rolling(window=12).mean()
rolling_std = df['Passengers'].rolling(window=12).std()

# Plot
plt.figure(figsize=(10, 5))
plt.plot(df['Passengers'], label='Original')
plt.plot(rolling_mean, label='12-Month Rolling Mean')
plt.plot(rolling_std, label='12-Month Rolling Std', linestyle='--')
plt.title('Rolling Mean & Standard Deviation')
plt.legend()
plt.show()

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3. 🔁 ACF and PACF (Autocorrelation & Partial Autocorrelation)

🧠 Objective:

• Detect seasonality, lags, and model order (e.g., ARIMA p and q)

• ACF shows correlation with all previous lags

• PACF shows direct correlation with specific lags, removing intermediate effects

✅ Code Example:

from statsmodels.graphics.tsaplots import plot_acf, plot_pacf

# ACF
plot_acf(df['Passengers'], lags=40)
plt.title("Autocorrelation Function (ACF)")
plt.show()

# PACF
plot_pacf(df['Passengers'], lags=40, method='ywm')
plt.title("Partial Autocorrelation Function (PACF)")
plt.show()

ℹ️ If your series is non-stationary, apply .diff() before plotting ACF/PACF.

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4. 🔍 Seasonal Decomposition (Additive vs Multiplicative)

Time series can be broken into Trend + Seasonality + Residual components.

✅ Additive Model:

\[ Y(t) = T(t) + S(t) + R(t) \]

✅ Multiplicative Model:

\[ Y(t) = T(t) \times S(t) \times R(t) \]

🧠 When to use which?

• Use additive if seasonal fluctuations remain constant over time
• Use multiplicative if seasonal fluctuations increase proportionally with trend

✅ Code Example:

from statsmodels.tsa.seasonal import seasonal_decompose

# Multiplicative Decomposition
result_mul = seasonal_decompose(df['Passengers'], model='multiplicative')
result_mul.plot()
plt.suptitle("Multiplicative Decomposition", fontsize=14)
plt.show()

# Additive Decomposition (may not fit well here)
result_add = seasonal_decompose(df['Passengers'], model='additive')
result_add.plot()
plt.suptitle("Additive Decomposition", fontsize=14)
plt.show()

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📌 Summary Table

Task	Purpose
Plot time series	Understand structure, spot patterns
Rolling mean/std	Visualize local trend and variance
ACF	Detect seasonality, lag correlation
PACF	Determine AR order in ARIMA
Decomposition	Extract trend, seasonality, residual

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