Random Sampling from the Watson Distribution

In this paper we present and discuss two methods for efficient sampling from the Watson distribution. The first approach adapts the rejection sampling algorithm from Kent et al. (2018), which originally samples Bingham distribution using angular central Gaussian envelopes. We show that for the case of the Watson distribution, this allows for a closed form expression for the parameters that maximize the efficiency of the sampling procedure, which is then further investigated and bounded by derived asymptotic results. What is more, we present a sampling algorithm that removes the curse of dimensionality by a smart matrix inversion, which allows for fast runtimes even for complex problems with high dimension. The second method relates to Saw (1978), and simulates from a projected distribution using adaptive rejection sampling. Also for this sampling procedure, the derived algorithm offers fast sampling for large dimension. This is not the case for similar algorithms in the field, which usually require an expensive rotation of the sampled results using a QR-decomposition. Finally, we propose some simple generators for the trivial cases and compare the two main methods in a simulation study.