Bayesian Forecasting of Many Count-Valued Time Series

Abstract

We develop and exemplify application of new classes of dynamic models for time series of nonnegative counts. Our novel univariate models combine dynamic generalized linear models for binary and conditionally Poisson time series, with dynamic random effects for over-dispersion. These models estimate dynamic regression coefficients in both binary and nonzero count components. Sequential Bayesian analysis allows fast, parallel analysis of sets of decoupled time series. New multivariate models then enable information sharing in contexts when data at a more highly aggregated level provide more incisive inferences on shared patterns such as trends and seasonality. A novel multiscale approach—one new example of the concept of decouple/recouple in time series—enables information sharing across series. This incorporates cross-series linkages while insulating parallel estimation of univariate models, and hence enables scalability in the number of series. The major motivating context is supermarket sales forecasting. Detailed examples drawn from a case study in multistep forecasting of sales of a number of related items showcase forecasting of multiple series, with discussion of forecast accuracy metrics, comparisons with existing methods, and broader questions of probabilistic forecast assessment.

Key Words:

• Decouple/recouple
• Forecast assessment
• Multiscale dynamic model
• Multistep forecast
• State-space model
• Supermarket sales forecasting

ACKNOWLEDGMENTS

The authors also acknowledge the constructive comments of the editor, associate editor, and two anonymous referees on the original version of the article. The research reported here was developed while Lindsay Berry was a PhD student in Statistical Science at Duke University. The authors acknowledge the input and vision of Dr. Paul Helman, Chief Science Officer at $84.51${}^{°}$ and contributions of data sets provided by 84.51${}^{°}$ Andrew Cron and Natalia Connolly provided useful input on early stages of the reported R&D. The authors also acknowledge the constructive comments of the editor, associate editor and two anonymous referees on the original version of the article.

FUNDING

The research of Lindsay Berry was partly supported by $84.51${}^{°}$, 100 West 5th Street, Cincinnati, OH 45202.