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.

Forecasting has always been at the forefront of decision making and planning. The uncertainty that surrounds the future is both exciting and challenging, with individuals and organisations seeking to minimise risks and maximise utilities. The large number of forecasting applications calls for a diverse set of forecasting methods to tackle real-life challenges. This article provides a non-systematic review of the theory and the practice of forecasting. We provide an overview of a wide range of theoretical, state-of-the-art models, methods, principles, and approaches to prepare, produce, organise, and evaluate forecasts. We then demonstrate how such theoretical concepts are applied in a variety of real-life contexts. We do not claim that this review is an exhaustive list of methods and applications. However, we wish that our encyclopedic presentation will offer a point of reference for the rich work that has been undertaken over the last decades, with some key insights for the future of forecasting theory and practice. Given its encyclopedic nature, the intended mode of reading is non-linear. We offer cross-references to allow the readers to navigate through the various topics. We complement the theoretical concepts and applications covered by large lists of free or open-source software implementations and publicly-available databases.