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Local and Global Trend Bayesian Exponential Smoothing Models

Slawek Smyl, Christoph Bergmeir, Alexander Dokumentov, Xueying Long, Erwin Wibowo, Daniel Schmidt

TL;DR

A family of seasonal and non-seasonal time series models that can be viewed as generalisations of additive and multiplicative exponential smoothing models, to model series that grow faster than linear but slower than exponential, which outperform the best algorithms in the competition as well as other benchmarks.

Abstract

This paper describes a family of seasonal and non-seasonal time series models that can be viewed as generalisations of additive and multiplicative exponential smoothing models, to model series that grow faster than linear but slower than exponential. Their development is motivated by fast-growing, volatile time series. In particular, our models have a global trend that can smoothly change from additive to multiplicative, and is combined with a linear local trend. Seasonality when used is multiplicative in our models, and the error is always additive but is heteroscedastic and can grow through a parameter sigma. We leverage state-of-the-art Bayesian fitting techniques to accurately fit these models that are more complex and flexible than standard exponential smoothing models. When applied to the M3 competition data set, our models outperform the best algorithms in the competition as well as other benchmarks, thus achieving to the best of our knowledge the best results of per-series univariate methods on this dataset in the literature. An open-source software package of our method is available.

Local and Global Trend Bayesian Exponential Smoothing Models

TL;DR

A family of seasonal and non-seasonal time series models that can be viewed as generalisations of additive and multiplicative exponential smoothing models, to model series that grow faster than linear but slower than exponential, which outperform the best algorithms in the competition as well as other benchmarks.

Abstract

This paper describes a family of seasonal and non-seasonal time series models that can be viewed as generalisations of additive and multiplicative exponential smoothing models, to model series that grow faster than linear but slower than exponential. Their development is motivated by fast-growing, volatile time series. In particular, our models have a global trend that can smoothly change from additive to multiplicative, and is combined with a linear local trend. Seasonality when used is multiplicative in our models, and the error is always additive but is heteroscedastic and can grow through a parameter sigma. We leverage state-of-the-art Bayesian fitting techniques to accurately fit these models that are more complex and flexible than standard exponential smoothing models. When applied to the M3 competition data set, our models outperform the best algorithms in the competition as well as other benchmarks, thus achieving to the best of our knowledge the best results of per-series univariate methods on this dataset in the literature. An open-source software package of our method is available.
Paper Structure (33 sections, 18 equations, 3 figures, 7 tables)

This paper contains 33 sections, 18 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Bayesian ETS
  3. A motivating example
  4. Our Contributions
  5. Exponential Smoothing
  6. ETS model family nomenclature
  7. Classical ETS model estimation
  8. Non-seasonal (damped) trend model
  9. Seasonal (damped) trend model
  10. Weaknesses of classical ETS models
  11. Limited functional form of trend
  12. Non-normal error distribution
  13. Heteroscedastic error distribution
  14. Bayesian estimation
  15. Monte Carlo Markov Chain (MCMC) Methods
  16. ...and 18 more sections

Figures (3)

  • Figure 1: a) Quarterly revenue of Amazon Web Services (in million USD, source: statista.com). Forecasts from ETS with linear and exponential trends show that the trend is either under- or over-estimated. LGT is able to capture the trend adequately. b) The logarithm of the series confirms that the series grows slower than exponential (as the log of the series doesn't grow linearly). c) Absolute differences of original data. Assuming errors are bigger than changes in trend, these values estimate absolute errors. Under this assumption, the graph shows that errors grow over time. d) Absolute differences divided by first value, i.e., $\frac{|y_{t+1} - y_t|}{y_t}$. We see that errors are not proportional to the trend.
  • Figure 2: Results of the statistical testing in a critical difference diagram. The x-axis is the average rank across time series, in terms of MASE performance. Black bars show methods that are not statistically significantly different from each other. We see that {L,S}GT obtains the best average rank and is statistically significantly better than all comparison methods.
  • Figure 3: Quarterly revenue of Amazon Web Services (in million USD, source: statista.com). LGT forecast 2 years ahead, ten examples of the forecasting trajectories (in light gray), and the 1st, 5th, 50th, 95th, and 99th percentiles (in blue). Actuals shown in black.