Interval-valued circular time series: A toroidal score-driven approach with an application to wind direction

Abstract

Circular data recorded over time, such as wind direction, are increasingly observed in intervals rather than as single point-valued observations, yet no existing framework accommodates data that are simultaneously circular, interval-valued, and temporally dependent. Time series models for point-valued circular data, including recently proposed score-driven approaches, assume that each observation is a single angle. This paper addresses this gap by proposing a dynamic conditional score framework for interval-valued circular time series. The lower and upper boundary sequences of the observed intervals are treated as circular time series governed by score-driven dynamics, which preserves the periodicity of the circular domain and the dependency of boundaries, in which the boundaries are coupled through a toroidal distribution and a jointly derived score. Maximum likelihood estimation is discussed, and their finite-sample performance is evaluated through simulation. An application to interval-valued wind direction data illustrates the practical value of retaining the interval structure within a fully rotationally invariant circular framework.