Seasonality and Decomposition

Retail sales rise every December. Whether that December bump is 40,000 units or 12% of the year's level is not a detail; it determines the entire model structure. If the bump is a fixed quantity, the decomposition is additive. If it scales with the business, it is multiplicative, and treating it as additive will under-forecast peaks in growing years and over-forecast them in shrinking ones.

Classical decomposition splits a series into three components:

Taking logs converts the multiplicative form into the additive one, which is why a log transform so often appears before any time series modelling and why it is the first thing to try when seasonal amplitude visibly grows with the level.

What each component is, and how it is estimated

Trend-cycle is the long-run direction plus any slow cyclical movement. The classical estimator is a centred moving average of the seasonal period length, which is why the first and last \(m/2\) observations have no trend estimate, a real limitation at the point you care about most.

Seasonal is the repeating within-period pattern. Classical decomposition estimates it as the average detrended value per season, which forces it to be constant across the whole series.

Remainder is what is left, and inspecting it is the diagnostic. Visible structure in the remainder means the decomposition missed something.

STL improves on this substantially: it uses loess smoothing, allows the seasonal component to change over time at a controllable rate, is robust to outliers through an optional robustness weighting, and handles any seasonal period (Cleveland et al., 1990, STL: A Seasonal-Trend Decomposition Procedure Based on Loess, Journal of Official Statistics 6(1)). Allowing seasonality to evolve matters for any series spanning years of a changing business.

Multiple and non-integer seasonality

Real high-frequency series have several seasonal periods at once. Hourly electricity demand has a daily cycle of 24, a weekly cycle of 168, and an annual cycle of 8,766, which is not an integer number of days and shifts with leap years. Classical and STL decomposition handle one period each.

Two approaches are standard. MSTL applies STL iteratively for each period. Fourier terms represent each cycle with a small number of sine and cosine pairs at the relevant frequencies, which handles non-integer periods naturally and lets a regression or ARIMA absorb seasonality as regressors, at the cost of assuming a smooth periodic shape.

Fourier terms are the more practical choice for long periods: an annual cycle in daily data needs 365 seasonal dummies or four to six Fourier pairs.

When it breaks

Calendar effects are not seasonality. Easter moves between March and April, Ramadan moves through the solar year, and trading-day counts vary by month. A fixed seasonal index cannot express any of them, and they must enter as explicit regressors. This is the most common source of residual structure in an otherwise reasonable decomposition.

Outliers contaminate both components. A one-off spike inflates the seasonal index for that period in every year, and pulls the trend. Robust STL and explicit outlier handling before decomposition are the defences, and simply removing the spike before fitting is often the right move.

Decomposition is descriptive, not a forecasting method. Forecasting a decomposed series means forecasting the seasonally adjusted component and adding back a seasonal component projected forward, usually by assuming it repeats. That last assumption is doing real work and is rarely stated.

Seasonal adjustment can mislead. A seasonally adjusted series showing a decline may reflect a seasonal pattern that has genuinely shifted rather than a real decline, since the adjustment applies a historical pattern. This is a recurring problem in economic reporting and it appears identically in business metrics.