Set-valued and interval-valued stationary time series

Stationarity is a key tool in classical time series. In order to analyze the set-valued time series, it must be extended to the set-valued case. In this paper, stationary set-valued time series is defined via Dp metric of set-valued random variables. Then, estimation methods of expectation and auto-covariance function of stationary set-valued time series are proposed. Unbiasedness and consistency of the expectation estimator and asymptotic unbiasedness of the auto-covariance function estimator are justified. After that, a special case of the set-valued time series, known as interval-valued time series, is considered. Two forecast methods of the stationary interval-valued time series are explicitly presented. Furthermore, the interval-valued time series is contextualized in the Box–Jenkins framework: an interval-valued autoregression model, along with its parameter estimation method, is introduced. Finally, experiments on both simulated and real data are presented to justify the efficiency of the parameters estimation method and the availability of the proposed model.