A Novel Directional Metrics and Wold Decomposition for Extended Interval Time Series

In this study, we propose a novel conceptualization and methodology for extended intervals, defining them as subsets of the Cartesian product , where R × Z2, where Z2 = {0,1} represents the direction of interval traversal. This new framework simplifies the representation and computation of extended intervals by introducing a unique metric dγ derived from a function γ(t), which we prove to be an adapted measure. This metric not only facilitates efficient calculation but also demonstrates excellent properties for practical implementation, as evidenced by its integration into the R software. Our approach significantly extends traditional interval analysis by incorporating directional information, which is crucial for accurately modeling and analyzing variability in interval data. Furthermore, we explore the application of this framework to extended interval-valued ARMA (AutoRegressive Moving Average) time series, providing a comprehensive theoretical foundation along with the proof of the Wold decomposition theorem for stationary extended interval-valued time series. The practical utility of our method is demonstrated through numerical simulations and empirical analysis of some financial markets time series. Our results show that extended intervals offer a robust and intuitive means to capture both the direction and magnitude of variations, making them highly valuable for time series forecasting and analysis in finance and other fields.