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BATS: Box-Cox, ARMA errors, Trend, Seasonal Models

The BATS framework extends exponential smoothing to accommodate multiple, possibly long seasonal cycles together with Box–Cox variance stabilization and ARMA error correction. It was introduced by De Livera, Hyndman & Snyder (2011) as part of the innovation-state-space family and is the method implemented by Durbyn’s [bats] function.

This page summarizes the core equations, highlights limitations (and why TBATS was proposed in the paper), and shows how to use the Julia interface.

1. Box–Cox transformation

Each BATS model may apply a Box–Cox transformation to the observed series, which stabilizes variance prior to modeling:

\[y_t^{(\omega)} = \begin{cases} \dfrac{y_t^\omega - 1}{\omega}, & \omega \neq 0, \\ \ln y_t, & \omega = 0 . \end{cases}\]

The parameter $\omega$ (often denoted $\lambda$ in code) is estimated within the automated model search when the user permits Box–Cox transforms.

2. BATS state-space formulation

After optional transformation, BATS is written in innovations form with an ARMA error process.

Observation equation

\[y_t^{(\omega)} = \ell_{t-1} + \phi b_{t-1} + \sum_i s_{i,t-1} + d_t,\]

where $\ell_t$ is the level, $b_t$ the trend, $\phi$ the damping parameter, $s_{i,t}$ the seasonal state for seasonal period $m_i$, and $d_t$ the ARMA error term.

State equations

Level and trend:

\[\ell_t = \ell_{t-1} + \phi b_{t-1} + \alpha d_t, \qquad b_t = \phi b_{t-1} + \beta d_t.\]

Additive seasonality for each seasonal block $i$ (normalized form):

\[s_{i,t} = -\sum_{j=1}^{m_i-1} s_{i,t-j} + \gamma_i d_t.\]

ARMA error component

\[d_t = \varepsilon_t + \sum_{k=1}^p \varphi_k \varepsilon_{t-k} + \sum_{\ell=1}^q \theta_\ell d_{t-\ell}, \qquad \varepsilon_t \sim \mathcal{N}(0, \sigma^2).\]

Combining these pieces yields the descriptor BATS(ω, {p,q}, φ, {m₁,…,m_J}) that Durbyn prints for each fitted model.

3. Limitations and relation to TBATS

In the original paper, TBATS was introduced to address several BATS limitations:

  • Seasonal periods must be integers, and each requires storing $m_i$ state components, which becomes expensive for very long cycles.
  • Non-integer or dual-calendar seasonalities (e.g., Hijri and Gregorian) cannot be represented exactly.

TBATS replaces the seasonal states with Fourier (trigonometric) terms to overcome those issues. Durbyn provides both BATS and TBATS implementations. The bats function corresponds strictly to the BATS formulation above for integer seasonal periods, while tbats supports non-integer periods and more efficient handling of long seasonal cycles via Fourier representation.

TBATS documentation

For detailed information about TBATS, including non-integer seasonal periods, dual calendar effects, and computational efficiency, see the TBATS documentation.

4. Usage in Durbyn

Basic example

using Durbyn

# Hourly demand with weekly (24*7) and yearly (24*365) seasonality
m = [168, 8760]
fit = bats(load, m; use_box_cox = true, use_arma_errors = true)

println(string(fit))
fc = forecast(fit; h = 168)

Key keyword arguments

  • use_box_cox, use_trend, use_damped_trend: Bool, Vector{Bool}, or nothing to try both options; the best combination is chosen using AIC.
  • use_arma_errors: toggles fitting an ARMA(p, q) model to the residuals via [auto_arima]; if the selected ARMA orders are zero, the pure exponential-smoothing state-space model is retained.
  • bc_lower, bc_upper: bounds for the Box–Cox search when enabled.
  • biasadj: apply bias correction during inverse Box–Cox transformation.
  • model: pass a previous BATSModel to refit the same structure to new data without re-running the full model selection process.

The convenience method bats(y, m::Int; kwargs...) simply wraps the vector interface ([m]), making single-season calls ergonomic.

5. Reference