Seasonal Decomposition of Time Series (STL, Classical Decomposition)

1. Introduction to Seasonal Decomposition

Seasonal Decomposition of Time Series (SDTS) is a statistical technique used to break down a time series into its fundamental components:

Trend (T) – The long-term movement in the data.
Seasonality (S) – Periodic fluctuations that occur at regular intervals (e.g., daily, monthly, yearly).
Residual/Irregular Component (R) – The random noise or unexplained variations in the data.

This decomposition helps in analyzing patterns and making forecasts.

2. Why Use Seasonal Decomposition?

✅ Identifies trends and seasonal patterns in data.
✅ Removes seasonality for better forecasting.
✅ Enhances model interpretability.
✅ Useful for pre-processing time series data before applying forecasting models (like ARIMA, Exponential Smoothing, LSTMs).

3. Types of Seasonal Decomposition Methods

There are two main approaches:

1️⃣ Classical Additive and Multiplicative Decomposition
2️⃣ STL Decomposition (Seasonal-Trend Decomposition using LOESS)

4. Classical Seasonal Decomposition (Additive & Multiplicative Models)

4.1. Additive Model

Used when the seasonal fluctuations remain constant over time. Yt=Tt+St+RtY_t = T_t + S_t + R_t

Y_t = Observed time series value at time t.
T_t = Trend component.
S_t = Seasonal component.
R_t = Residual component.

✅ Suitable for linear trends and fixed seasonal variations.

4.2. Multiplicative Model

Used when seasonal variations increase or decrease over time (non-constant seasonal fluctuations). Yt=Tt×St×RtY_t = T_t \times S_t \times R_t

✅ Suitable for exponential growth trends and changing seasonality effects.

📌 When to Use Which?

Use Additive → If variations remain constant.
Use Multiplicative → If variations increase/decrease over time.

5. Implementing Classical Seasonal Decomposition in Python

Step 1: Import Required Libraries

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from statsmodels.tsa.seasonal import seasonal_decompose

Step 2: Load and Visualize the Time Series Data

# Load dataset (ensure it has a datetime index)
df = pd.read_csv("time_series_data.csv", parse_dates=["Date"], index_col="Date")

# Plot the original data
plt.figure(figsize=(12,6))
plt.plot(df.index, df['Value'], label="Original Time Series", color='blue')
plt.xlabel("Time")
plt.ylabel("Value")
plt.title("Time Series Data")
plt.legend()
plt.show()

Step 3: Apply Seasonal Decomposition

# Perform decomposition
result = seasonal_decompose(df['Value'], model='multiplicative', period=12)  # Change 'additive' if needed

# Plot the decomposed components
result.plot()
plt.show()

✅ The above code breaks down the time series into Trend, Seasonality, and Residual components.

6. STL Decomposition (Seasonal-Trend Decomposition using LOESS)

📌 Why Use STL?
STL (Seasonal and Trend decomposition using LOESS) is an advanced method that:
✔ Provides more flexible seasonality handling.
✔ Works well for non-stationary data.
✔ Handles missing data better than classical decomposition.

Step 1: Install Necessary Library

from statsmodels.tsa.seasonal import STL

Step 2: Apply STL Decomposition

stl = STL(df['Value'], seasonal=13)  # Adjust seasonality period if needed
result_stl = stl.fit()

# Plot STL decomposition
fig = result_stl.plot()
plt.show()

✅ STL is more robust than classical decomposition and works well with complex seasonality.

7. Analyzing the Components

After decomposition, we get:

1️⃣ Trend Component

Shows the underlying direction of the time series (upward or downward movement).
Extracted using a moving average or smoothing function.

2️⃣ Seasonal Component

Represents repeating patterns (e.g., daily, weekly, yearly).
Helps detect recurring fluctuations in the data.

3️⃣ Residual Component

Represents random noise in the data after removing trend and seasonality.
If residuals appear random, the decomposition is good.
If patterns remain, another model may be needed.

8. Model Selection Based on Decomposition

Once we analyze the components:

✅ If trend is strong → Use Holt’s Exponential Smoothing or ARIMA.
✅ If seasonality is strong → Use Holt-Winters, SARIMA, or Prophet.
✅ If residuals are high → Data may need further pre-processing.

9. Forecasting with Decomposed Components

Step 1: Extract Components for Forecasting

trend = result.trend
seasonal = result.seasonal
residual = result.resid

Step 2: Forecast Using Trend + Seasonality

future_trend = trend[-12:].mean()  # Assuming trend is stable
future_seasonality = seasonal[-12:]  # Assuming seasonality repeats

future_forecast = future_trend + future_seasonality

# Plot Forecast
plt.figure(figsize=(12,6))
plt.plot(df.index, df['Value'], label="Original Data")
plt.plot(pd.date_range(start=df.index[-1], periods=12, freq='M'), future_forecast, label="Forecast", color='red')
plt.legend()
plt.show()

✅ Combining trend and seasonality helps generate forecasts.