The angular halfspace depth ( \(ahD\) ) was, already in 1987, the first depth function proposed for the nonparametric analysis of directional data. Mainly due to its presumed high computational cost and lack of efficient computational algorithms, it was never widely used in directional data analysis. We address the problem of the exact computation of \(ahD\) in any dimension d. We proceed in two steps: (i) We express \(ahD\) as a generalized (Euclidean) halfspace depth in dimension \(d-1\) , using a projection approach. That allows us to develop fast exact computational algorithms for \(ahD\) in dimensions \(d=1, 2, 3\) . (ii) In spaces of dimension 3]]d 3 we design an inductive procedure that reduces the dimensionality d in the computation of \(ahD\) , until the algorithms for \(d \le 3\) can be used. Using our advances we develop a family of powerful algorithms for the computation of \(ahD\) in any dimension d. Our procedures are implemented efficiently in C++ with an interface in R. A detailed analysis of the complexity of the novel algorithms is performed. Surprisingly, we show that computing \(ahD\) of multiple points with respect to the same dataset is substantially faster than the same task for the classical (Euclidean) halfspace depth.
