A Bayesian Dirichlet auto-regressive moving average model for forecasting lead times

In the hospitality industry, lead time data are a form of compositional data that are crucial for business planning, resource allocation, and staffing. Hospitality businesses accrue fees daily, but recognition of these fees is often deferred. This paper presents a novel class of Bayesian time series models, the Bayesian Dirichlet auto-regressive moving average (B-DARMA) model, designed specifically for compositional time series. The model is motivated by the analysis of five years of daily fees data from Airbnb, with the aim of forecasting the proportion of future fees that will be recognized in 12 consecutive monthly intervals. Each day’s compositional data are modeled as Dirichlet distributed, given the mean and a scale parameter. The mean is modeled using a vector auto-regressive moving average process, which depends on previous compositional data, previous compositional parameters, and daily covariates. The B-DARMA model provides a robust solution for analyzing large compositional vectors and time series of varying lengths. It offers efficiency gains through the choice of priors, yields interpretable parameters for inference, and produces reasonable forecasts. The paper also explores the use of normal and horseshoe priors for the vector auto-regressive and vector moving average coefficients, and for regression coefficients. The efficacy of the B-DARMA model is demonstrated through simulation studies and an analysis of Airbnb data.