Seasonality Decomposition Attribution

The foundational distinction in decomposition logic. Additive models assume the seasonal amplitude is constant over time (y = Trend + Seasonality + Residual), while multiplicative models assume the amplitude scales with the trend level (y = Trend × Seasonality × Residual).

• Additive: Best for series with stable variance, like daily temperature anomalies
• Multiplicative: Best for series with exponential growth, like e-commerce revenue
• Pseudo-additive: A hybrid for series with zero values where multiplicative fails

The choice directly impacts how attribution scores are distributed across components.

Seasonal-Trend decomposition using LOESS is a robust, iterative algorithm that handles any seasonal periodicity and is resilient to outliers. Unlike classical methods, STL allows the seasonal component to evolve over time.

• Inner loop: Updates trend and seasonal components via LOESS smoothing
• Outer loop: Computes robustness weights to downweight outliers
• Attribution: Each time step's prediction is partitioned into trend, seasonal, and residual contributions

STL is the default choice for complex real-world series where seasonality is not strictly fixed.

The seasonal component captures periodic fluctuations of a fixed and known frequency—daily, weekly, monthly, or quarterly cycles. Attribution to this component answers: 'How much of this forecast is due to recurring calendar effects?'

• Fourier decomposition: Represents seasonality as a sum of sine/cosine waves
• Dummy variable regression: Encodes seasons as binary indicators
• Periodogram analysis: Identifies dominant frequencies in the spectrum

In forecasting models like Prophet or SARIMA, the seasonal attribution is explicitly parameterized and directly extractable.

The trend-cycle component represents the long-term directional movement of the series after removing seasonal and irregular effects. It captures structural changes, economic expansions, or technology adoption curves.

• Henderson filters: Symmetric weighted moving averages that preserve local trends
• Hodrick-Prescott filter: Separates trend from cycle via a smoothing penalty
• Breakpoint detection: Identifies structural shifts where the trend changes slope

Attribution to the trend reveals whether a model's forecast is extrapolating a genuine long-run signal or overfitting to transient drift.

The residual component captures stochastic noise—the unexplained variance after extracting trend and seasonality. High attribution to the residual signals either genuine randomness or model misspecification.

• White noise testing: Ljung-Box test checks if residuals are independently distributed
• Heteroskedasticity: Non-constant residual variance indicates unmodeled volatility
• Outlier attribution: Isolated spikes in residuals can be traced to specific events

A well-specified model should have residuals that are stationary with zero mean. Persistent structure in residuals indicates an omitted component.

In hierarchical forecasting, decomposition attribution ensures coherence across aggregation levels. The sum of component attributions at a lower level must equal the attribution at the higher level.

• Bottom-up: Forecast components at the granular level and aggregate
• Top-down: Decompose the aggregate forecast and distribute to sub-series
• Optimal reconciliation: Minimizes a loss function to balance both approaches

This is critical for supply chain and retail forecasts where SKU-level, category-level, and regional predictions must be mutually consistent.

The foundational statistical process that separates a time series into trend, seasonal, and residual components using moving averages or LOESS smoothing.

• Additive Model: Assumes components sum together (Y = T + S + R), used when seasonal magnitude is constant.
• Multiplicative Model: Assumes components multiply (Y = T × S × R), used when seasonal magnitude scales with trend.
• Forms the preprocessing backbone for many attribution techniques by providing structured input features to the model.

Seasonal-Trend decomposition using LOESS is a robust, iterative filtering method that handles any seasonal periodicity and is resilient to outliers.

• Uses locally weighted regression to extract smooth trend and seasonal curves.
• Unlike classical methods, STL can model time-varying seasonality where the seasonal pattern evolves.
• Often paired with forecasting models like Prophet or ETS; attribution then maps prediction variance back to the decomposed STL components.

Meta's Prophet library natively decomposes forecasts into trend, weekly/yearly seasonality, and holiday effects using a generalized additive model.

• Each component is a separate, interpretable function with its own parameters.
• Attribution is direct: the model's prediction is the sum of component outputs, so the contribution of a holiday or seasonal cycle is explicitly quantified.
• Enables business users to ask 'How much of this spike is due to Black Friday versus the underlying growth trend?'

The systematic examination of the irregular component remaining after trend and seasonality are removed, used to validate model fit and diagnose unexplained variance.

• White Noise Check: Residuals should have zero mean, constant variance, and no autocorrelation if the decomposition is complete.
• Attribution Signal: Large residuals at specific time steps indicate events the decomposition missed—external shocks, outliers, or structural breaks—that the model cannot explain.
• Drives feature engineering by identifying missing exogenous variables.

A mathematical approach that models seasonal patterns as a sum of sine and cosine waves with different frequencies, rather than discrete dummy variables.

• Captures smooth, continuous cyclical behavior with far fewer parameters than one-hot encoded periods.
• Harmonic Order: Controls the complexity of the seasonal curve; higher orders fit more intricate patterns.
• Attribution involves decomposing the model's prediction into the amplitude and phase contribution of each Fourier harmonic, revealing which frequencies dominate the forecast.

A mathematical tool that extracts the long-term trend component by solving an optimization problem that balances fit to the data against smoothness of the trend.

• Controlled by a smoothing parameter (λ) that penalizes acceleration in the trend; higher λ produces a more linear trend.
• Commonly used in macroeconomics to separate cyclical fluctuations from the structural growth path.
• In attribution workflows, the HP trend serves as a baseline counterfactual: 'What would the model predict if only the smooth trend existed?'