2.4 Decomposition methods (additive and multiplicative)

Time Series Decomposition

Time series decomposition breaks a complex series into simpler, interpretable parts: trend, seasonality, and irregularity. By isolating each component, you can understand what's driving changes in your data and build better forecasts. The two main approaches, additive and multiplicative, differ in how they assume these components combine.

Purpose of Time Series Decomposition

Decomposition serves three core goals:

• Isolate distinct patterns. A raw time series mixes together long-term growth, seasonal cycles, and random noise. Decomposition separates these so you can study each one independently.
• Understand underlying structure. Once separated, you can see why a series behaves the way it does. For example, are retail sales rising because of a genuine growth trend, or just because the holiday season inflates the numbers every December?
• Improve forecasting. With components isolated, you can apply specialized models to each one (or use the decomposition to inform models like ARIMA or exponential smoothing), rather than trying to model everything at once.

Additive vs. Multiplicative Decomposition Models

The key question is: does the size of seasonal swings stay constant, or does it grow as the series level increases?

Additive decomposition assumes the components simply add together:

$Y_t = T_t + S_t + I_t$

where $Y_t$ is the observed value, $T_t$ is the trend, $S_t$ is the seasonal component, and $I_t$ is the irregular (residual) component.

Use additive decomposition when seasonal fluctuations remain roughly the same size regardless of the series level. Think of monthly ice cream sales in a city with stable population: summer bumps up sales by about the same dollar amount each year.

Multiplicative decomposition assumes the components multiply together:

$Y_t = T_t \times S_t \times I_t$

Use this when seasonal swings grow (or shrink) proportionally with the trend. A fast-growing tech startup might see Q4 revenue spikes that get larger each year as the company's baseline revenue increases. The seasonal effect here is a ratio, not a fixed amount.

Application of Additive Decomposition

Here's the step-by-step process:

1. Estimate the trend ($T_t$) using a smoothing method. A common choice is a centered moving average. For monthly data, you'd use a 12-month moving average to smooth out seasonal cycles. Alternatively, you can fit a polynomial regression (quadratic, cubic) if the trend has curvature.

2. Detrend the series by subtracting the trend from the original values: $Y_t - T_t = S_t + I_t$ This removes the long-term direction and leaves only seasonal patterns and noise.

3. Estimate the seasonal component ($S_t$) by averaging the detrended values for each season. For monthly data, average all the January detrended values together, all the February values together, and so on. This gives you 12 seasonal indices. These indices are typically adjusted so they sum to zero across a full cycle.

4. Calculate the irregular component ($I_t$) by subtracting both the trend and seasonal estimates from the original series: $I_t = Y_t - T_t - S_t$ This residual captures random fluctuations, one-off events, and anything the trend and seasonal components didn't explain.

5. Verify the decomposition by reconstructing the series: $T_t + S_t + I_t$ should equal $Y_t$. If it doesn't match closely, something went wrong in your estimation.

Process of Multiplicative Decomposition

The logic mirrors additive decomposition, but you use division instead of subtraction:

1. Estimate the trend ($T_t$) using a moving average or regression, same as before.

2. Detrend the series by dividing the original values by the trend: $\frac{Y_t}{T_t} = S_t \times I_t$ Division preserves the proportional relationship between components, which is the whole point of the multiplicative model.

3. Estimate the seasonal component ($S_t$) by averaging the detrended ratios for each season. For quarterly data, average all Q1 ratios, all Q2 ratios, etc. These seasonal indices are typically adjusted so they average to 1.0 across a full cycle (rather than summing to zero, as in additive).

4. Calculate the irregular component ($I_t$) by dividing the original series by the product of trend and seasonal estimates: $I_t = \frac{Y_t}{T_t \times S_t}$

5. Verify by reconstructing: $T_t \times S_t \times I_t$ should reproduce $Y_t$.

Interpretation of Decomposition Components

Trend ($T_t$) shows the long-term direction of the series.

• A positive trend means the series is generally increasing (e.g., population growth, rising GDP).
• A negative trend means overall decline (e.g., shrinking market share for a legacy product).
• Shifts or bends in the trend can reveal structural changes, like the onset of a recession or a new competitor entering the market.

Seasonal component ($S_t$) captures regular, calendar-driven patterns that repeat within a fixed period (usually a year).

• In the additive model, seasonal values are expressed in the same units as the data (e.g., "+3,000 units in December").
• In the multiplicative model, seasonal values are ratios (e.g., "1.15 in December" means 15% above the trend).
• These patterns help with resource planning: a retailer can use seasonal indices to decide how much extra inventory to stock before the holidays.

Irregular component ($I_t$) is whatever's left after removing trend and seasonality.

• Small, random-looking residuals suggest your decomposition captured the data's structure well.
• Large or clustered residuals may signal outliers, unexpected events (natural disasters, policy changes, supply shocks), or a missing component your model didn't account for.
• Examining the irregular component is also useful for assessing how predictable a series really is. A series with large irregular swings carries more inherent uncertainty for forecasting.