Probabilistic machine learning and circular statistics—the branch of statistics concerned with data as angles and directions—are two research communities that have grown mostly in isolation from one another. On the one hand, probabilistic machine learning community has developed powerful frameworks for problems whose data lives on Euclidean spaces, such as Gaussian Processes, but have generally neglected other topologies studied by circular statistics. On the other hand, the approximate inference frameworks from probabilistic machine learning have only recently started to the circular statistics landscape. This thesis intends to redress the gap between these two fields by contributing to both fields with models and approximate inference algorithms. In particular, we introduce the multivariate Generalised von Mises distribution (mGvM), which allows the use of kernels in circular statistics akin to Gaussian Processes, and an augmented representation. These models account for a vast number of applications comprising both latent variable modelling and regression of circular data. Then, we propose methods to conduct approximate inference on these models. In particular, we investigate the use of Variational Inference, Expectation Propagation and Markov chain Monte Carlo methods. The variational inference route taken was a mean field approach to efficiently leverage the mGvM tractable conditionals and create a baseline for comparison with other methods. Then, an Expectation Propagation approach is presented drawing on the Expectation Consistent Framework for Ising models and connecting the approximations used to the augmented model presented. In the final MCMC chapter, efficient Gibbs and Hamiltonian Monte Carlo samplers are derived for the mGvM and the augmented model.

The main objective of this thesis is the development of a nonparametric sequential estimation technique for the off-pulse interval(s) of a source function originating from a pulsar. It is important to identify the off-pulse interval of each pulsar accurately, since the properties of the off-pulse emissions are further researched by astrophysicists in an attempt to detect potential emissions from the associated pulsar wind nebula (PWN). The identification technique currently used in the literature is subjective in nature, since it is based on the visual inspection of the histogram estimate of the pulsar light curve. The developed nonparametric estimation technique is not only objective in nature, but also accurate in the estimation of the off-pulse interval of a pulsar, as evident from the simulation study and the application of the developed technique to observed pulsar data. The first two chapters of this thesis are devoted to a literature study that provides background information on the pulsar environment and -ray astronomy, together with an explanation of the on-pulse and off-pulse interval of a pulsar and the importance thereof for the present study. This is followed by a discussion on some fundamental circular statistical ideas, as well as an overview of kernel density estimation techniques. These two statistical topics are then united in order to illustrate kernel density estimation techniques applied to circular data, since this concept is the starting point of the developed nonparametric sequential estimation technique. Once the basic theoretical background of the pulsar environment and circular kernel density estimation has been established, the new sequential off-pulse interval estimator is formulated. The estimation technique will be referred to as `SOPIE'. A number of tuning parameters form part of SOPIE, and therefore the performed simulation study not only serves as an evaluation of the performance of SOPIE, but also as a mechanism to establish which tuning parameter configurations consistently perform better than some other configurations. In conclusion, the optimal parameter configurations are utilised in the application of SOPIE to pulsar data. For several pulsars, the sequential off-pulse interval estimators are compared to the off-pulse intervals published in research papers, which were identified with the subjective \eye-ball" technique. It is found that the sequential off-pulse interval estimators are closely related to the off-pulse intervals identified with subjective visual inspection, with the benefit that the estimated intervals are objectively obtained with a nonparametric estimation technique. / PhD (Statistics), North-West University, Potchefstroom Campus, 2014

The main objective of this thesis is the development of a nonparametric sequential estimation technique for the off-pulse interval(s) of a source function originating from a pulsar. It is important to identify the off-pulse interval of each pulsar accurately, since the properties of the off-pulse emissions are further researched by astrophysicists in an attempt to detect potential emissions from the associated pulsar wind nebula (PWN). The identification technique currently used in the literature is subjective in nature, since it is based on the visual inspection of the histogram estimate of the pulsar light curve. The developed nonparametric estimation technique is not only objective in nature, but also accurate in the estimation of the off-pulse interval of a pulsar, as evident from the simulation study and the application of the developed technique to observed pulsar data. The first two chapters of this thesis are devoted to a literature study that provides background information on the pulsar environment and -ray astronomy, together with an explanation of the on-pulse and off-pulse interval of a pulsar and the importance thereof for the present study. This is followed by a discussion on some fundamental circular statistical ideas, as well as an overview of kernel density estimation techniques. These two statistical topics are then united in order to illustrate kernel density estimation techniques applied to circular data, since this concept is the starting point of the developed nonparametric sequential estimation technique. Once the basic theoretical background of the pulsar environment and circular kernel density estimation has been established, the new sequential off-pulse interval estimator is formulated. The estimation technique will be referred to as `SOPIE'. A number of tuning parameters form part of SOPIE, and therefore the performed simulation study not only serves as an evaluation of the performance of SOPIE, but also as a mechanism to establish which tuning parameter configurations consistently perform better than some other configurations. In conclusion, the optimal parameter configurations are utilised in the application of SOPIE to pulsar data. For several pulsars, the sequential off-pulse interval estimators are compared to the off-pulse intervals published in research papers, which were identified with the subjective \eye-ball" technique. It is found that the sequential off-pulse interval estimators are closely related to the off-pulse intervals identified with subjective visual inspection, with the benefit that the estimated intervals are objectively obtained with a nonparametric estimation technique. / PhD (Statistics), North-West University, Potchefstroom Campus, 2014

The automotive safety company Veoneer are producers of high end driver visual assistance systems, but the knowledge about the absolute accuracy of their dynamic calibration algorithms that estimate the vehicle’s orientation is limited. In this thesis, a novel measurement system is proposed to be used in gathering reference data of a vehicle’s orientation as it is in motion, more specifically the pitch and roll angle of the vehicle. Focus has been to estimate how the uncertainty of the measurement system is affected by errors introduced during its construction, and to evaluate its potential in being a viable tool in gathering reference data for algorithm performance evaluation. The system consisted of three laser distance sensors mounted on the body of the vehicle, and a range of data acquisition sequences with different perturbations were performed by driving along a stretch of road in Linköping with weights loaded in the vehicle. The reference data were compared to camera system data where the bias of the calculated angles were estimated, along with the dynamic behaviour of the camera system algorithms. The experimental results showed that the accuracy of the system exceeded 0.1 degrees for both pitch and roll, but no conclusions about the bias of the algorithms could be drawn as there were systematic errors present in the measurements. / Bilsäkerhetsföretaget Veoneer är utvecklare av avancerade kamerasystem inom förarassistans, men kunskapen om den absoluta noggrannheten i deras dynamiska kalibreringsalgoritmer som skattar fordonets orientering är begränsad. I denna avhandling utvecklas och testas ett nytt mätsystem för att samla in referensdata av ett fordons orientering när det är i rörelse, mer specifikt dess pitchvinkel och rollvinkel. Fokus har legat på att skatta hur osäkerheten i mätsystemet påverkas av fel som introducerats vid dess konstruktion, samt att utreda dess potential när det kommer till att vara ett gångbart alternativ för att samla in referensdata för evaluering av prestandan hos algoritmerna. Systemet bestod av tre laseravståndssensorer monterade på fordonets kaross. En rad mätförsök utfördes med olika störningar introducerade genom att köra längs en vägsträcka i Linköping med vikter lastade i fordonet. Det insamlade referensdatat jämfördes med data från kamerasystemet där bias hos de framräknade vinklarna skattades, samt att de dynamiska egenskaperna kamerasystemets algoritmer utvärderades. Resultaten från mätförsöken visade på att noggrannheten i mätsystemet översteg 0.1 grader för både pitchvinklarna och rollvinklarna, men några slutsatser kring eventuell bias hos algoritmerna kunde ej dras då systematiska fel uppstått i mätresultaten.