TBATS — Trigonometric Exponential Smoothing for Complex Seasonality
Also known as: trigonometric exponential smoothing, multiple seasonal exponential smoothing, complex seasonal exponential smoothing, TBATS — Çoklu Mevsimsel Üstel Düzleştirme
TBATS is an innovations state space forecasting model, introduced by De Livera, Hyndman and Snyder (2011), that combines a Box-Cox transformation, ARMA errors and trigonometric (Fourier) seasonal terms. It is built to handle continuous time series with several nested seasonal cycles at once — for example hourly data that also repeats daily, weekly and yearly.
Key highlights
- Handles several seasonal cycles simultaneously, including long and non-integer periods such as 365.25 days.
- Trigonometric representation keeps the model compact and lets seasonality evolve smoothly over time.
- Built-in Box-Cox transformation and ARMA error term address changing variance and residual autocorrelation automatically.
Intuition
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How it works
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When to use it
Use TBATS for forecasting a single continuous time series that shows complex seasonality — multiple seasonal periods that may be long or non-integer — and has enough history (at least about 100 observations). It is most appropriate when a Box-Cox transformation can stabilise the variance and when at least one clear seasonal cycle is present. It is less suited to very short series, to problems driven mainly by external regressors, or to settings where seasonal patterns shift sharply over time.
Strengths & limitations
- Handles several seasonal cycles simultaneously, including long and non-integer periods such as 365.25 days.
- Trigonometric representation keeps the model compact and lets seasonality evolve smoothly over time.
- Built-in Box-Cox transformation and ARMA error term address changing variance and residual autocorrelation automatically.
- Requires a reasonably long series (around 100 or more observations) to estimate the seasonal harmonics reliably.
- Assumes the seasonal patterns are stable enough to be summarised by a fixed set of harmonics; abrupt regime changes are not captured well.
- Does not natively incorporate external explanatory variables, so covariate-driven effects must be modelled elsewhere.
Common pitfalls
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Applications
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Frequently asked
What does the acronym TBATS stand for?
Trigonometric seasonality, Box-Cox transformation, ARMA errors, and Trend and Seasonal components — the four ingredients combined in the model.
How is TBATS different from SARIMA?
SARIMA typically handles a single integer seasonal period, whereas TBATS represents several seasonal cycles at once using trigonometric harmonics and can cope with long or non-integer periods such as 365.25 days.
Can TBATS handle non-integer seasonal periods?
Yes. Because each season is modelled with sine and cosine terms rather than one state per period, TBATS naturally accommodates fractional periods like 365.25, which dummy-based seasonal models struggle with.
How much data does TBATS need?
It needs a reasonably long history — roughly 100 observations or more — so that the trigonometric harmonics for each seasonal cycle can be estimated reliably; very short series risk overfitting.
Sources
- 1.De Livera, A. M., Hyndman, R. J. & Snyder, R. D. (2011). Forecasting Time Series with Complex Seasonal Patterns Using Exponential Smoothing. Journal of the American Statistical Association, 106(496), 1513-1527.
- 2.Hyndman, R. J. & Athanasopoulos, G. (2021). Forecasting: Principles and Practice (3rd ed.). OTexts.
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Cite this page
ScholarGate. (2026, June 1). TBATS. ScholarGate. https://scholargate.app/econometrics/tbats