UC Irvine

Directional data are observations that can be denoted by unit vectors in Euclidean space and thus are usually used to denote directions. The sample space of a directional variable can be the surface of a circle, a sphere, or a hyper-sphere. Traditional statistical methods are not suitable for analyzing directional data due to the non-Euclidean structure of their sample spaces. In forensic science, the analysis of bloodstain patterns typically requires considering the directions of the constituent bloodstains for crime scene reconstruction. One of the major goals of bloodstain pattern analysis is to evaluate different hypotheses regarding the causal mechanism of a bloodstain pattern. Statistical summaries of the directional attributes of bloodstains can be useful indicators of the causal mechanism. In this dissertation, we first develop an image processing algorithm to approximate bloodstains with ellipses. Afterwards, we employ directional statistics on the orientation of the ellipses to extract useful features and incorporate the distributions of the features into a likelihood ratio framework that has been widely used in various forensic fields to evaluate different hypotheses regarding crime scene evidence. This work provides a conceptual demonstration of the likelihood ratio application to bloodstain pattern analysis. Finally, we extend the approach by proposing a nonparametric Bayesian model that can model directional and linear variables jointly. The resulting Dirichlet process semi-projected normal mixture model can evaluate the likelihood of a bloodstain pattern based on the ellipse representation without pre-designed features.