Introduction
Decomposition is a fundamental technique in time series analysis used to separate a time series into its constituent components, namely trend, seasonality, and noise (or residuals). This lesson explores the concept of decomposition, its methods, practical applications, and implementation in Python.
What is Decomposition?
Decomposition involves breaking down a time series into three main components:
- Trend: The long-term movement or directionality of the data, indicating whether the values are increasing, decreasing, or staying relatively stable over time.
- Seasonality: Patterns that repeat at fixed intervals within a specific time frame, such as daily, weekly, monthly, or yearly cycles. Seasonality reflects systematic variations influenced by factors like weather, holidays, or cultural events.
- Residuals (Noise): Random variations or fluctuations that cannot be explained by the trend or seasonality. Residuals represent the leftover or unexplained part of the time series after accounting for trend and seasonality.
Why Use Decomposition?
- Understanding Patterns: Separating and analyzing the underlying components (trend, seasonality, and noise) to uncover meaningful patterns within the data.
- Forecasting: Providing insights into future trends and seasonal fluctuations, essential for predictive modeling and decision-making.
- Data Preprocessing: Preparing data for further analysis or modeling by removing noise and extracting meaningful signals.
Methods of Decomposition
Additive Decomposition: The time series y(t) is decomposed into additive components:
\[
y_t = T_t + S_t + R_t
\]
where:
\begin{align*}
y_t & : \text{observed value at time } t, \\
T_t & : \text{trend component at time } t, \\
S_t & : \text{seasonal component at time } t, \\
R_t & : \text{residual component at time } t.
\end{align*}
Additive decomposition is suitable when the magnitude of seasonal fluctuations remains constant over time.
Multiplicative Decomposition: The time series y(t) is decomposed into multiplicative components:
\[
y_t = T_t \times S_t \times R_t
\]
where:
\begin{align*}
y_t & : \text{observed value at time } t, \\
T_t & : \text{trend component at time } t, \\
S_t & : \text{seasonal component at time } t, \\
R_t & : \text{residual component at time } t.
\end{align*}
Multiplicative decomposition is appropriate when the seasonal fluctuations change in proportion to the trend level.
Implementing Decomposition in Python
Using Statsmodels
Statsmodels provides a convenient way to perform decomposition in Python. Here’s an example using both additive and multiplicative decomposition:
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from statsmodels.tsa.seasonal import seasonal_decompose
# Example time series data (monthly sales)
dates = pd.date_range('2023-01-01', periods=36, freq='M')
sales = np.array([100, 120, 130, 140, 150, 160, 180, 200, 220, 240, 260, 280,
300, 320, 340, 360, 380, 400, 420, 440, 460, 480, 500, 520,
540, 560, 580, 600, 620, 640, 660, 680, 700, 720, 740])
df = pd.DataFrame({'Date': dates, 'Sales': sales})
df.set_index('Date', inplace=True)
# Perform additive decomposition
result_add = seasonal_decompose(df['Sales'], model='additive', period=12)
# Perform multiplicative decomposition
result_mul = seasonal_decompose(df['Sales'], model='multiplicative', period=12)
# Plotting the decomposed components
plt.figure(figsize=(12, 8))
plt.subplot(4, 1, 1)
plt.plot(df.index, df['Sales'], label='Original', marker='o', color='b')
plt.legend(loc='upper left')
plt.title('Original Time Series Data')
plt.subplot(4, 1, 2)
plt.plot(df.index, result_add.trend, label='Additive Trend', linestyle='-', color='r')
plt.legend(loc='upper left')
plt.title('Additive Trend Component')
plt.subplot(4, 1, 3)
plt.plot(df.index, result_add.seasonal, label='Additive Seasonal', linestyle='-', color='g')
plt.legend(loc='upper left')
plt.title('Additive Seasonal Component')
plt.subplot(4, 1, 4)
plt.plot(df.index, result_add.resid, label='Additive Residual', linestyle='-', color='purple')
plt.legend(loc='upper left')
plt.title('Additive Residual Component')
plt.tight_layout()
plt.show()
Practical Applications
Retail: Analyzing sales trends, identifying seasonal peaks, and optimizing inventory management.
Finance: Understanding seasonal fluctuations in stock prices or market indices.
Healthcare: Monitoring patient admissions or disease outbreaks influenced by seasonal factors.
Climate Science: Studying seasonal variations in temperature, precipitation, and environmental conditions.
Conclusion
Decomposition is a powerful technique in time series analysis for separating a time series into its fundamental components: trend, seasonality, and residuals. By applying decomposition methods and analyzing the resulting components, data scientists can gain insights into underlying patterns, improve forecasting accuracy, and make informed decisions across various domains.