Given a planar curve, imagine rolling a sphere along that curve without slipping or twisting, and by this means tracing out a curve on the sphere. It is well known that such a rolling operation induces a local isometry between the sphere and the plane so that the two curves uniquely determine each other, and moreover, the operation extends to a general class of manifolds in any dimension. We use rolling to construct an analogue of a Gaussian process on a manifold starting from a Euclidean Gaussian process. The resulting model is generative, and is amenable to statistical inference given data as curves on a manifold. We illustrate with examples on the unit sphere, symmetric positive-definite matrices, and with a robotics application involving 3D orientations.

Data of the form of event times arise in various applications. A simple model for such data is a non-homogeneous Poisson process (NHPP) which is specified by a rate function that depends on time. We consider the problem of having access to multiple independent samples of event time data, observed on a common interval, from which we wish to classify or cluster the samples according to their rate functions. Each rate function is unknown but assumed to belong to a finite number of rate functions each defining a distinct class. We model the rate functions using a spline basis expansion, the coefficients of which need to be estimated from data. The classification approach consists of using training data for which the class membership is known, to calculate maximum likelihood estimates of the coefficients for each group, then assigning test samples to a class by a maximum likelihood criterion. For clustering, by analogy to the Gaussian mixture model approach for Euclidean data, we consider a mixture of NHPP models and use the expectation-maximisation algorithm to estimate the coefficients of the rate functions for the component models and cluster membership probabilities for each sample. The classification and clustering approaches perform well on both synthetic and real-world data sets. Code associated with this paper is available at https://github.com/duncan-barrack/NHPP .