Exponential smoothing methods for forecasting bar diagram-valued time series

When a set of categories with related frequencies of the observed variable is available for each time point we have a bar diagram-valued time series. This paper introduces exponential smoothing methods to forecast bar diagram-valued time series data. The proposed method is inspired in the approach introduced by Maia and De Carvalho (2011) to deal with inteval-valued time series. The smoothing parameters are estimated by using techniques for non-linear optimization problems with bound constraints. The results are discussed based on two wellknown classical performance measurements, which have been adapted here for this particular type of data: the U of Theil statistics and average relative variance (ARV) in the framework of a Monte Carlo experiment. The synthetic data sets take into account differents aspects, e.g., sample size and forecast horizons among others. Applications using real bar diagram-valued time series also were considered to demonstrate the practicality of the methods. The results demonstrate that the proposed approaches are useful in forecasting bar diagram-valued times series.