Quaternion Structured Paranormal Distributions

Quaternions, a four dimensional extension of complex numbers, are increasingly being used in various signal processing applications. At the same time, there are few probabilistic representations for capturing such random variables. In this work we present a new copula-based distribution over quaternions. The distribution extends the classical Gaussian and independent distributions, while maintaining the circular symmetry conditions required by such applications, namely properness. Importantly, the new distribution allows for flexible control of both the marginals and the correlations. We develop methods for estimating the parameters of the new distribution, and demonstrate its potential advantages via numerical experiments.