Remark

Please be aware that these lecture notes are accessible online in an ‘early access’ format. They are actively being developed, and certain sections will be further enriched to provide a comprehensive understanding of the subject matter.

3.3. Additive vs. Multiplicative Decomposition

3.3.1. Mathematical Framework

For a time series \(y_t\), two decomposition models exist [Cleveland et al., 1990, Wen et al., 2019, Wen et al., 2020]:

Additive model (appropriate when seasonal amplitude is constant):

(3.8)\[\begin{equation} y_t = T_t + S_t + R_t \end{equation}\]

Multiplicative model (appropriate when seasonal amplitude grows with trend):

(3.9)\[\begin{equation} y_t = T_t \times S_t \times R_t \end{equation}\]

When seasonal variations scale proportionally with the series level, a multiplicative model is more appropriate. STL, which assumes an additive structure, handles multiplicative data via logarithmic transformation:

(3.10)\[\begin{equation} \log(y_t) = \log(T_t \times S_t \times R_t) = \log(T_t) + \log(S_t) + \log(R_t) \end{equation}\]

After decomposing \(\log(y_t)\) additively, components are back-transformed:

(3.11)\[\begin{align} T_t &= \exp(\tilde{T}_t) \\ S_t &= \exp(\tilde{S}_t) \\ R_t &= \exp(\tilde{R}_t) \end{align}\]

where \(\tilde{T}_t\), \(\tilde{S}_t\), \(\tilde{R}_t\) are the additive components of \(\log(y_t)\).

3.3.2. Example: Air Passengers (Multiplicative Decomposition)

The classic Air Passengers dataset (monthly totals from 1949–1960) exhibits clear multiplicative seasonality: seasonal fluctuations grow proportionally with the increasing trend in air travel demand.

Fig. 3.13 Monthly airline passenger totals from January 1949 to December 1960. The series exhibits a strong upward trend with clear seasonal peaks during summer months (June–August). Crucially, the amplitude of seasonal fluctuations increases proportionally with the trend level, indicating multiplicative rather than additive seasonality.

3.3.3. Additive vs. Multiplicative: Visual Comparison

Fig. 3.14 Comparison of additive (left) versus multiplicative (right) STL decomposition for the Air Passengers series. Left panels: Additive decomposition incorrectly assumes constant seasonal amplitude, leading to increasing residual variance over time as the model fails to account for growing seasonal fluctuations. Right panels: Multiplicative decomposition (via log-transformation) correctly captures proportional seasonal growth, yielding residuals with stable variance throughout the series. The multiplicative model is appropriate for this dataset.

Key Observations from Fig. 3.14

Additive model (left)

• Trend: Captures overall growth but underestimates the acceleration.

• Seasonal: Fixed amplitude (~20–30 passengers) throughout, failing to account for larger seasonal swings in later years.

• Residual: Shows heteroscedasticity (increasing variance over time), indicating model misspecification.

Multiplicative model (right)

• Trend: Smooth exponential growth trajectory.

• Seasonal: Amplitude proportional to trend level (e.g., seasonal multiplier ranges from 0.85 to 1.15, meaning ±15% variation around trend).

• Residual: Homoscedastic (constant variance), confirming appropriate model choice.

Fig. 3.15 Variance decomposition for the multiplicative STL decomposition of Air Passengers (on log scale). The trend dominates, explaining approximately 80% of total log-variance, while seasonality accounts for ~8.5%, and residuals capture only ~2.5%, indicating an excellent fit.

Table 3.4 Variance decomposition for multiplicative STL (log scale). Trend captures the exponential growth, seasonality captures the annual cycle, and residuals represent unexplained variation.

Component	Variance (log scale)	Percent_of_total
Trend	0.1776	91.1518
Seasonal	0.0166	8.5292
Residual	0.0005	0.2507

Fig. 3.16 Multiplicative seasonal pattern for Air Passengers (1949–1960). Each bar represents the seasonal multiplier for that month (e.g., July ≈ 1.13 means +13% above trend). Summer months (June–August) show strong positive seasonality (blue bars), while winter months (November–February) show negative seasonality (orange bars), reflecting vacation travel patterns.

Table 3.5 Multiplicative seasonal pattern: monthly multipliers and percent changes relative to trend. Positive values indicate months with above-trend passenger counts; negative values indicate below-trend months.

Month	Seasonal_Multiplier	Percent_Change
January	0.9128	-8.7185
February	0.976	-2.4028
March	1.0882	8.818
April	1.0227	2.268
May	0.9699	-3.0139
June	1.0691	6.9107
July	1.1671	16.7145
August	1.1686	16.858
September	1.0663	6.6333
October	0.9206	-7.9434
November	0.7982	-20.1813
December	0.9189	-8.1087

Here is a version consistent with the new seasonal multipliers:

1. Exponential trend growth: Air travel still exhibits strong exponential growth over 1949–1960, with passenger volumes increasing several‑fold over the sample, consistent with post‑war economic expansion and rapid adoption of commercial aviation.

2. Proportional seasonality: Seasonal multipliers now range from about 0.80 in November (roughly 20% below trend) to about 1.17 in July–August (roughly 17% above trend). This confirms a strongly multiplicative seasonal structure: as the baseline travel volume rises over time, the absolute seasonal swings become larger (e.g., from tens of thousands of passengers early in the sample to many tens of thousands later), while the relative percentage variation by month remains roughly stable from year to year.

3. Stable residuals: After accounting for the exponential trend and these proportional seasonal effects, the remaining residuals continue to show approximately constant variance and no systematic temporal pattern, indicating that the multiplicative decomposition captures the main systematic features of the series.

3.3.4. Forecasting Implications

For forecasting, the multiplicative decomposition enables:

(3.12)\[\begin{equation} \hat{y}_{T+h} = \hat{T}_{T+h} \times \hat{S}_{(T+h \mod 12)} \end{equation}\]

where:

• \(\hat{T}_{T+h}\) is an extrapolated trend (e.g., via exponential smoothing or regression on log-trend)

• \(\hat{S}_{(T+h \mod 12)}\) is the seasonal multiplier for month \(h\)

This formulation ensures that forecasted seasonal fluctuations grow proportionally with the trend, unlike additive forecasts which would underestimate future seasonal peaks.

Show code cell source

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# Reconstruct the series to verify decomposition
reconstructed = mult_trend * mult_seasonal * mult_resid

# Reconstruction error
error = air_series - reconstructed

print(f"Mean absolute reconstruction error: {np.mean(np.abs(error)):.4f}")
print(f"Root mean squared error: {np.sqrt(np.mean(error**2)):.4f}")

Mean absolute reconstruction error: 0.0000
Root mean squared error: 0.0000

Fig. 3.17 Top panel: Original Air Passengers series (black) versus reconstructed series from multiplicative STL components \(T_t \times S_t \times R_t\) (red dashed). The two curves are nearly indistinguishable. Bottom panel: Reconstruction error, showing negligible deviations (< 1 passenger on average), confirming that the multiplicative decomposition accurately represents the observed data.