Forecasting Compositional Time Series: A State Space Approach

A method is proposed for forecasting composite time series such as the market shares for multiple brands. Its novel feature is that it relies on multi-series adaptations of exponential smoothing combined with the log-ratio transformation for the conversion of proportions onto the real line. It is designed to produce forecasts that are both non-negative and sum to one; are invariant to the choice of the base series in the log-ratio transformation; recognized and exploit features such as serial dependence and non-stationary movements in the data; allow for the possibility of non-uniform interactions between the series; and contend with series that start late, finish early, or which have values close to zero. Relying on an appropriate multivariate innovations state space mode, it can be used to generate prediction distributions in addition to point forecasts and to compute the probabilities of market share increases together with prediction intervals. A shared structure between the series in the multivariate model is used to ensure that the usual proliferation of parameter is avoided. The forecasting method is illustrated using data on the annual market shares of the major (groups of) brands in the U.S. automobile market, over the period 1961-2013.