Local Linear Trend and Seasonality

Summary

The BSTS model decomposes trend as a local linear trend with a stochastic slope (random walk with drift) and seasonality as a sum-to-zero constraint over $S$ seasons. Both are modular state-space components that can be combined independently. The local linear trend adapts quickly to short-term variation but can produce implausibly wide uncertainty for long-horizon predictions.

Local Linear Trend

Definition: Local Linear Trend

The level ${\mu }_t$ and slope ${\delta }_t$ evolve as:

$${\mu }_{t+1}={\mu }_t+{\delta }_t+{\eta }_{\mu ,t},{\eta }_{\mu ,t}\sim \mathcal{N}(0,{\sigma }_{\mu }^2)\text{\hspace{1em}(2.3)}$$ $${\delta }_{t+1}={\delta }_t+{\eta }_{\delta ,t},{\eta }_{\delta ,t}\sim \mathcal{N}(0,{\sigma }_{\delta }^2)$$

• ${\mu }_t$: current level (value of trend at time $t$)
• ${\delta }_t$: slope (expected increment in $\mu$ per time step)

Both the level and slope evolve as random walks.

Interpretation:

• ${\sigma }_{\mu }^2$ controls level noise — how quickly the trend level jumps
• ${\sigma }_{\delta }^2$ controls slope noise — how quickly the trend slope changes

Trade-off: Very flexible (adapts to local variation), but produces wide prediction intervals for long-horizon forecasting since the slope itself drifts.

Generalization: AR(1) Slope

A more stable variant allows the slope to exhibit AR(1) variation around a long-term slope $D$:

$${\mu }_{t+1}={\mu }_t+{\delta }_t+{\eta }_{\mu ,t}$$ $${\delta }_{t+1}=D+\rho ({\delta }_t-D)+{\eta }_{\delta ,t}\text{\hspace{1em}(2.4)}$$

where $|\rho |<1$ is the learning rate. This model balances short-term local variation with a long-term slope $D$, preventing the slope from drifting without bound.

Seasonality Component

Definition: Seasonal State Component

For $S$ seasons, the seasonal effect ${\gamma }_t$ evolves as:

$${\gamma }_{t+1}=-\sum _{s=0}^{S-2}{\gamma }_{t-s}+{\eta }_{\gamma ,t}\text{\hspace{1em}(2.5)}$$

where $S$ is the number of seasons per period and ${\gamma }_t$ denotes their joint contribution to the observed response.

Key property: The mean of ${\gamma }_{t+1}$ is zero when summed over $S$ seasons — this is the sum-to-zero seasonal constraint.

Example: For monthly seasonality ($S=12$), each new month’s seasonal effect is minus the sum of the previous 11 months’ effects, plus noise.

Transition matrix $T_t$ for seasonality: An $(S-1)\times (S-1)$ matrix with $-1$‘s along the top row, $1$‘s along the subdiagonal, and $0$‘s elsewhere.

Multiple seasonal periods: For daily data, one might include $S=7$ (day-of-week) and $S=52$ (annual). The $S=52$ cycle: set $T_t=I_{S-1}$ when not a new week, standard seasonal matrix when starting a new week.

Combining Components

The full state vector ${\alpha }_t$ concatenates all components:

$${\alpha }_t=({\mu }_t,{\delta }_t,{\gamma }_t,{\gamma }_{t-1},\dots ,{\beta }_t,\dots {)}^{\top }$$

The overall matrices $T_t$, $R_t$, $Q_t$ become block-diagonal with one block per component (assuming independent component errors).

Choosing Components in Practice

• Always include: Local level (at minimum ${\sigma }_{\delta }^2=0$ gives random walk)
• Seasonality: Include when seasonal pattern is obvious by inspection
• Trend: Include local linear trend for data with visible trend; use AR(1) slope for longer-range forecasting
• Static vs. dynamic regression: Static when treatment-control relationship is stable; dynamic when relationship changes over time

Connections

• Components of Bayesian Structural Time-Series Model
• Fits are used in MCMC Inference for CausalImpact (Kalman filter/smoother)
• The resulting counterfactual prediction drives Counterfactual Impact Estimation

See Also

• Bayesian Structural Time-Series Model — full state-space setup
• MCMC Inference for CausalImpact — how these components are estimated
• Single Marketing Time Series — ARIMA and classical time series models that these state-space components complement
• Linear-Gaussian State-Space Models — these are state components of the general state-space form