Seasonal Decomposition — STL and Classical

Statistics

Separating Trend, Seasonality, and Residual Components

Seasonal decomposition breaks a time series into interpretable components — underlying trend, repeating seasonal patterns, and random noise. STL and classical methods each offer advantages for different data characteristics.

Retail Planning — Isolate holiday shopping trends from underlying growth
Energy Forecasting — Separate daily and weekly cycles from long-term demand shifts
Healthcare — Detect flu season patterns from baseline hospital admission trends

Understanding the parts makes the whole pattern clear.

Seasonal decomposition separates a time series into trend, seasonal, and residual components, making patterns easier to identify and model.

DfAdditive Decomposition

A time series can be decomposed as:

Additive Model

Y_t = T_t + S_t + R_t

Here,

$Y_t$ =Observed value at time t
$T_t$ =Trend component
$S_t$ =Seasonal component
$R_t$ =Residual (remainder) component

DfMultiplicative Decomposition

When seasonal variation changes with the level of the series:

Multiplicative Model

Y_t = T_t \times S_t \times R_t

Here,

$T_t$ =Trend component
$S_t$ =Seasonal component (proportion)
$R_t$ =Residual component

Choosing Model Type

Use the additive model when seasonal fluctuations are constant. Use the multiplicative model when seasonal fluctuations increase with the level of the series. The log transform converts multiplicative to additive.

Classical Decomposition

Moving Average Method

For additive decomposition with period $s$ :

Centered Moving Average

MA_t = \frac{1}{s}\sum_{j=-(s-1)/2}^{(s-1)/2} Y_{t+j}

Here,

$s$ =Seasonal period (e.g., 12 for monthly data)

Steps:

Compute the centered moving average to estimate $T_t$
Detrend: $Y_t - T_t$ (additive) or $Y_t / T_t$ (multiplicative)
Average detrended values by season to get $S_t$
Residual: $R_t = Y_t - T_t - S_t$

Classical Method Limitations

Classical decomposition assumes the seasonal pattern is fixed and does not adapt to changes. It also loses endpoints (first and last observations). For modern data, prefer STL.

STL Decomposition

Seasonal and Trend decomposition using Loess (Cleveland et al., 1990) is a more robust and flexible method.

DfSTL Algorithm

STL iteratively applies Loess smoothing (local regression) to estimate trend and seasonal components. It handles any seasonal period and is robust to outliers.

STL Advantages

| Feature | Classical | STL |

|---------|-----------|-----|

| Handles any period | No (fixed s) | Yes |

| Robust to outliers | No | Yes |

| Seasonal pattern adapts | No | Yes |

| Handles missing data | Limited | Yes |

| Computational speed | Fast | Moderate |

Strength of Seasonal Component

The strength of the seasonal component measures how much of the variation is due to seasonality.

Seasonal Strength

F_s = 1 - \frac{\text{Var}(R_t)}{\text{Var}(S_t + R_t)}

Here,

$F_s$ =Seasonal strength (0 to 1)
$R_t$ =Residual component
$S_t$ =Seasonal component

| Value | Interpretation |

|-------|---------------|

| 0.6 - 1.0 | Strong seasonality |

| 0.3 - 0.6 | Moderate seasonality |

| 0.0 - 0.3 | Weak seasonality |

Python Implementation


import numpy as np

import pandas as pd

import matplotlib.pyplot as plt

from statsmodels.tsa.seasonal import seasonal_decompose, STL



# Simulate monthly data with trend and seasonality

np.random.seed(42)

n = 240

t = np.arange(n)

trend = 100 + 0.5 * t

seasonal = 10 * np.sin(2 * np.pi * t / 12)

noise = np.random.randn(n) * 2

y = trend + seasonal + noise



dates = pd.date_range('2005', periods=n, freq='M')

ts = pd.Series(y, index=dates)



# Classical decomposition

result_classical = seasonal_decompose(ts, model='additive', period=12)



# STL decomposition

stl = STL(ts, period=12, robust=True)

result_stl = stl.fit()



# Plot comparison

fig, axes = plt.subplots(4, 2, figsize=(14, 10))

for i, (comp, label) in enumerate(zip(['observed','trend','seasonal','resid'],

                                        ['Observed','Trend','Seasonal','Residual'])):

    axes[i, 0].plot(result_classical.observed if comp=='observed'

                     else getattr(result_classical, comp))

    axes[i, 0].set_title(f'Classical - {label}')

    axes[i, 1].plot(result_stl.observed if comp=='observed'

                     else getattr(result_stl, comp))

    axes[i, 1].set_title(f'STL - {label}')

plt.tight_layout()

plt.show()



# Seasonal strength

var_resid = np.var(result_stl.resid)

var_season_resid = np.var(result_stl.seasonal + result_stl.resid)

F_s = 1 - var_resid / var_season_resid

print(f"Seasonal Strength: {F_s:.3f}")

Worked Example

Example: Airline Passengers

Monthly airline passenger data (1949-1960) shows:

Increasing trend from 100 to 500+ passengers
Multiplicative seasonality — amplitude grows with level
Period: 12 months

Steps:

Apply log transform: $Z_t = \ln(Y_t)$ to convert multiplicative to additive
Run STL with period = 12
Trend: Smooth upward curve from 4.6 to 6.2 (log scale)
Seasonal: Regular pattern oscillating around 0
Residual: Random noise, no remaining patterns
Seasonal strength: $F_s = 0.87$ -> strong seasonality

Key Takeaways

Summary: Seasonal Decomposition

Decompose series into trend $T_t$ , seasonal $S_t$ , and residual $R_t$
Additive model: $Y_t = T_t + S_t + R_t$ (constant seasonal variation)
Multiplicative model: $Y_t = T_t \times S_t \times R_t$ (growing variation)
STL is preferred: handles any period, robust to outliers, adaptive
Seasonal strength $F_s$ quantifies how seasonal the data is
Use decomposition to check if residuals look like white noise

Seasonal Decomposition — STL and Classical

Seasonal Decomposition — STL and Classical

Separating Trend, Seasonality, and Residual Components

DfAdditive Decomposition

Additive Model

DfMultiplicative Decomposition

Multiplicative Model

Classical Decomposition

Moving Average Method

Centered Moving Average

STL Decomposition

DfSTL Algorithm

STL Advantages

Strength of Seasonal Component

Seasonal Strength

Python Implementation

Worked Example

Example: Airline Passengers

Key Takeaways

Summary: Seasonal Decomposition

Related Topics

Premium Content

Need Expert Statistics Help?