Data augmentation for models based on rejection sampling

We present a data augmentation scheme to perform Markov chain Monte Carlo inference for models where data generation involves a rejection sampling algorithm. Our idea, which seems to be missing in the literature, is a simple scheme to instantiate the rejected proposals preceding each data point. The resulting joint probability over observed and rejected variables can be much simpler than the marginal distribution over the observed variables, which often involves intractable integrals. We consider three problems, the first being the modeling of flow-cytometry measurements subject to truncation. The second is a Bayesian analysis of the matrix Langevin distribution on the Stiefel manifold, and the third, Bayesian inference for a nonparametric Gaussian process density model. The latter two are instances of problems where Markov chain Monte Carlo inference is doubly-intractable. Our experiments demonstrate superior performance over state-of-the-art sampling algorithms for such problems.Comment: 6 figures. arXiv admin note: text overlap with arXiv:1311.090

Directional Estimation for Robotic Beating Heart Surgery

In robotic beating heart surgery, a remote-controlled robot can be used to carry out the operation while automatically canceling out the heart motion. The surgeon controlling the robot is shown a stabilized view of the heart. First, we consider the use of directional statistics for estimation of the phase of the heartbeat. Second, we deal with reconstruction of a moving and deformable surface. Third, we address the question of obtaining a stabilized image of the heart

Directional Estimation for Robotic Beating Heart Surgery

In robotic beating heart surgery, a remote-controlled robot can be used to carry out the operation while automatically canceling out the heart motion. The surgeon controlling the robot is shown a stabilized view of the heart. First, we consider the use of directional statistics for estimation of the phase of the heartbeat. Second, we deal with reconstruction of a moving and deformable surface. Third, we address the question of obtaining a stabilized image of the heart

Saddlepoint approximation to the distribution of the total distance of the von Mises-Fisher continuous time random walk

This article considers the random walk over Rp with any p ≥ 2, where a particle starts at the origin and progresses stepwise with fixed step lengths and von Mises–Fisher distributed step directions. The total number of steps follows a continuous time counting process. The saddlepoint approximation to the distribution of the distance between the origin and the position of the particle at any time is derived. Despite the p-dimensionality of the random walk, the computation of the proposed saddlepoint approximation is one-dimensional and thus simple. The high accuracy of the saddlepoint approximation is illustrated by a numerical comparison with Monte Carlo simulation

Small sphere distributions and related topics in directional statistics

This dissertation consists of two related topics in the statistical analysis of directional data. The research conducted for the dissertation is motivated by advancing the statistical shape analysis to understand the variation of shape changes in 3D objects. The first part of the dissertation studies a parametric approach for multivariate directional data lying on a product of spheres. Two kinds of concentric unimodal-small subsphere distributions are introduced. The first kind coincides with a special case of the Fisher-Bingham distribution; the second is a novel adaption that independently models horizontal and vertical variations. In its multi-subsphere version, the second kind allows for correlation of horizontal variations over different subspheres. For both kinds, we provide new computationally feasible algorithms for simulation and estimation, and propose a large-sample test procedure for several sets of hypotheses. Working as models to fit the major modes of variation, the proposed distributions properly describe shape changes of skeletally-represented 3D objects due to rotation, twisting and bending. In particular, the multi-subsphere version of the second kind accounts for the underlying horizontal dependence appropriately. The second part is a proposal of hypothesis test that is applicable to the analysis of principal nested spheres (PNS). In PNS, determining which subsphere to fit, among the geodesic (great) subsphere and non-geodesic (small) subsphere, is an important issue and it is preferred to fit a great subsphere when there is no major direction of variation in the directional data. The proposed test utilizes the measure of multivariate kurtosis. The change of the multivariate kurtosis for rotationally symmetric distributions is investigated based on modality. The test statistic is developed by modifying the sample kurtosis. The asymptotic sampling distribution of the test statistic is also investigated. The proposed test is seen to work well in numerical studies with various data situations

STK /WST 795 Research Reports

These documents contain the honours research reports for each year for the Department of Statistics.Honours Research Reports - University of Pretoria 20XXStatisticsBSs (Hons) Mathematical Statistics, BCom (Hons) Statistics, BCom (Hons) Mathematical StatisticsUnrestricte

Computer-intensive statistical methods:saddlepoint approximations with applications in bootstrap and robust inference

The saddlepoint approximation was introduced into statistics in 1954 by Henry E. Daniels. This basic result on approximating the density function of the sample mean has been generalized to many situations. The accuracy of this approximation is very good, particularly in the tails of the distribution and for small sample sizes, compared with normal or Edgeworth approximation methods. Before applying saddlepoint approximations to the bootstrap, this thesis will focus on saddlepoint approximations for the distribution of quadratic forms in normal variables and for the distribution of the waiting time in the coupon collector's problem. Both developments illustrate the modern art of statistics relying on the computer and embodying both numeric and analytic approximations. Saddlepoint approximations are extremely accurate in both cases. This is underlined in the first development by means of an extensive study and several applications to nonparametric regression, and in the second by several examples, including the exhaustive bootstrap seen from a collector's point of view. The remaining part of this thesis is devoted to the use of saddlepoint approximations in order to replace the computer-intensive bootstrap. The recent massive increases in computer power have led to an upsurge in interest in computer-intensive statistical methods. The bootstrap is the first computer-intensive method to become widely known. It found an immediate place in statistical theory and, more slowly, in practice. The bootstrap seems to be gaining ground as the method of choice in a number of applied fields, where classical approaches are known to be unreliable, and there is sustained interest from theoreticians in its development. But it is known that, for accurate approximations in the tails, the nonparametric bootstrap requires a large number of replicates of the statistic. As this is time-intensive other methods should be considered. Saddlepoint methods can provide extremely accurate approximations to resampling distributions. As a first step I develop fast saddlepoint approximations to bootstrap distributions that work in the presence of an outlier, using a saddlepoint mixture approximation. Then I look at robust M-estimates of location like Huber's M-estimate of location and its initially MAD scaled version. One peculiarity of the current literature is that saddlepoint methods are often used to approximate the density or distribution functions of bootstrap estimators, rather than related pivots, whereas it is the latter which are more relevant for inference. Hence the aim of the final part of this thesis is to apply saddlepoint approximations to the construction of studentized confidence intervals based on robust M-estimates. As examples I consider the studentized versions of Huber's M-estimate of location, of its initially MAD scaled version and of Huber's proposal 2. In order to make robust inference about a location parameter there are three types of robustness one would like to achieve: robustness of performance for the estimator of location, robustness of validity and robustness of efficiency for the resulting confidence interval method. Hence in the context of studentized bootstrap confidence intervals I investigate these in more detail in order to give recommendations for practical use, underlined by an extensive simulation study

Bayesian inference for dynamic pose estimation using directional statistics

The dynamic pose of an object, where the object can represent a spacecraft, aircraft, or mobile robot, among other possibilities, is defined to be the position, velocity, attitude, and angular velocity of the object. A new method to perform dynamic pose estimation is developed that leverages directional statistics and operates under the Bayesian estimation framework, as opposed to the minimum mean square error (MMSE) framework that conventional methods employ. No small attitude uncertainty assumption is necessary using this method, and, therefore, a more accurate estimate of the state can be obtained when the attitude uncertainty is large. Two new state densities, termed the Gauss-Bingham and Bingham-Gauss mixture (BGM) densities, are developed that probabilistically represent a state vector comprised of an attitude quaternion and other Euclidean states on their natural manifold, the unit hypercylinder. When the Euclidean states consist of position, velocity, and angular velocity, the state vector represents the dynamic pose. An uncertainty propagation scheme is developed for a Gauss-Bingham-distributed state vector, and two demonstrations of this uncertainty propagation scheme are presented that show its applicability to quantify the uncertainty in dynamic pose, especially when the attitude uncertainty becomes large. The BGM filter is developed, which is an approximate Bayesian filter in which the true temporal and measurement evolution of the BGM density, as quantified by the Chapman-Kolmogorov equation and Bayes\u27 rule, are approximated by a BGM density. The parameters of the approximating BGM density are found via integral approximation on a component-wise basis, which is shown to be the Kullback-Leibler divergence optimal parameters of each component. The BGM filter is then applied to three simulations in order to compare its performance to a multiplicative Kalman filter and demonstrate its efficacy in estimating dynamic pose. The BGM filter is shown to be more statistically consistent than the multiplicative Kalman filter when the attitude uncertainty is large --Abstract, page iii