A Dirichlet process model for directional-linear data with application to bloodstain pattern analysis

Directional data require specialized models because of the non-Euclidean nature of their domain. When a directional variable is observed jointly with linear variables, modeling their dependence adds an additional layer of complexity. A Bayesian nonparametric approach is introduced to analyze directional-linear data. Firstly, the projected normal distribution is extended to model the joint distribution of linear variables and a directional variable with arbitrary dimension projected from a higher-dimensional augmented multivariate normal distribution. The new distribution is called the semi-projected normal distribution (SPN) and can be used as the mixture distribution in a Dirichlet process model to obtain a more flexible class of models for directional-linear data. Then, a conditional inverse-Wishart distribution is proposed as part of the prior distribution to address an identifiability issue inherited from the projected normal and preserve conjugacy with the SPN. The SPN mixture model shows superior performance in clustering on synthetic data compared to the semi-wrapped Gaussian model. The experiments show the ability of the SPN mixture model to characterize bloodstain patterns. A hierarchical Dirichlet process model with the SPN distribution is built to estimate the likelihood of bloodstain patterns under a posited causal mechanism for use in a likelihood ratio approach to the analysis of forensic bloodstain pattern evidence.