Methods for generating variates from probability distributions

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Diverse probabilistic results are used in the design of random univariate generators. General methods based on these are classified and relevant theoretical properties derived. This is followed by a comparative review of specific algorithms currently available for continuous and discrete univariate distributions. A need for a Zeta generator is established, and two new methods, based on inversion and rejection with a truncated Pareto envelope respectively are developed and compared. The paucity of algorithms for multivariate generation motivates a classification of general methods, and in particular, a new method involving envelope rejection with a novel target distribution is proposed. A new method for generating first passage times in a Wiener Process is constructed. This is based on the ratio of two random numbers, and its performance is compared to an existing method for generating inverse Gaussian variates. New "hybrid" algorithms for Poisson and Negative Binomial distributions are constructed, using an Alias implementation, together with a Geometric tail procedure. These are shown to be robust, exact and fast for a wide range of parameter values. Significant modifications are made to Atkinson's Poisson generator (PA), and the resulting algorithm shown to be complementary to the hybrid method. A new method for Von Mises generation via a comparison of random numbers follows, and its performance compared to that of Best and Fisher's Wrapped Cauchy rejection method. Finally new methods are proposed for sampling from distribution tails, using optimally designed Exponential envelopes. Timings are given for Gamma and Normal tails, and in the latter case the performance is shown to be significantly better than Marsaglia's tail generation procedure.Governors of Dundee College of Technolog

Rejection Sampling with Vertical Weighted Strips

A number of distributions that arise in statistical applications can be expressed in the form of a weighted density: the product of a base density and a nonnegative weight function. Generating variates from such a distribution may be nontrivial and can involve an intractable normalizing constant. Rejection sampling may be used to generate exact draws, but requires formulation of a suitable proposal distribution. To be practically useful, the proposal must both be convenient to sample from and not reject candidate draws too frequently. A well-known approach to design a proposal involves decomposing the target density into a finite mixture, whose components may correspond to a partition of the support. This work considers such a construction that focuses on majorization of the weight function. This approach may be applicable when assumptions for adaptive rejection sampling and related algorithms are not met. An upper bound for the rejection probability based on this construction can be expressed to evaluate the efficiency of the proposal before sampling. A method to partition the support is considered where regions are bifurcated based on their contribution to the bound. Examples based on the von Mises Fisher distribution and Gaussian Process regression are provided to illustrate the method

A Simple Gamma Random Number Generator for Arbitrary Shape Parameters

This paper proposes an improved gamma random generator. In the past, a lot of gamma random number generators have been proposed, and depending on a shape parameter (say, alpha) they are roughly classified into two cases: (i) alpha lies on the interval (0,1) and (ii) alpha is greater than 1, where alpha=1 can be included in either case. In addition, Cheng and Feast (1980) extended the gamma random number generator in the case where alpha is greater than 1/n, where n denotes an arbitrary positive number. Taking n as a decreasing function of alpha, in this paper we propose a simple gamma random number generator with shape parameter alpha greater than zero. The proposed algorithm is very simple and shows quite good performance.Gamma Random Variable

Detecting recombination and its mechanistic association with genomic features via statistical models

Recombination is a powerful weapon in the evolutionary arsenal of retroviruses such as HIV. It enables the production of chimeric variants or recombinants that may confer a selective advantage to the pathogen over the host immune response. Recombinants further accentuate differences in virulence, disease progression and drug resistance mutation patterns already observed in non-recombinant variants of HIV. This thesis describes the development of a rapid genotyper for HIV sequences employing supervised learning algorithms and its application to complex HIV recombinant data, the application of a hierarchical model for detection of recombination hotspots in the HIV-1 genome and the extension of this model enabling estimation of the association between recombination probabilities and covariates of interest. The rapid genotyper for HIV-1 explores a solution to the genotyping problem in the machine learning paradigm. Of the algorithms tested, the genotyper built using Bayesian additive regression trees (BART) was most successful in efficiently classifying complex recombinants that pose a challenge to other currently available genotyping methods. We also developed a novel method, bootSMOTE, for generating synthetic data in order to supplement insufficient training data. We found that supplementation with synthetic recombinants especially boosts identification of complex recombinants. We describe the genotyper software available for download as well as a web interface enabling rapid classiffication of HIV-1 sequences. Hotspots for recombination in the HIV-1 genome are modeled using spatially smoothed changepoint processes. This hierarchical model uses a phylogenetic recombination detection model of dual changepoint processes at the lower level. The upper level applies a Gaussian Markov random eld (GMRF) hyperprior to population-level recombination probabilities in order to efficiently combine the information from many individual recombination events as inferred at the lower level. Focusing on 544 unique recombinant sequences, we found a novel hotspot in the pol gene of HIV-1 while confirming the presence of a high recombination activity in the env gene. Valuable insights into the molecular mechanism of recombination may be gained by extending the GMRF model to include covariates of interest. We add a level to the hierarchical model and allow for the simultaneous inference of recombination probabilities as well their association with genomic covariates of interest. Using a set of 527 unique recombinants, we confirmed the presence of the pol hotspot. Interestingly, we found significant positive associations of spatial fluctuations in recombination probabilities with genomic regions prone to forming secondary structure as well as significant negative associations with regions that support tight RNA-DNA hybrid formation. Overall, our results support the theory that pause sites along the genome promote recombination

Hierarchical Variance Reduction Techniques for Monte Carlo Rendering

Ever since the first three-dimensional computer graphics appeared half a century ago, the goal has been to model and simulate how light interacts with materials and objects to form an image. The ultimate goal is photorealistic rendering, where the created images reach a level of accuracy that makes them indistinguishable from photographs of the real world. There are many applications ñ visualization of products and architectural designs yet to be built, special effects, computer-generated films, virtual reality, and video games, to name a few. However, the problem has proven tremendously complex; the illumination at any point is described by a recursive integral to which a closed-form solution seldom exists. Instead, computer simulation and Monte Carlo methods are commonly used to statistically estimate the result. This introduces undesirable noise, or variance, and a large body of research has been devoted to finding ways to reduce the variance. I continue along this line of research, and present several novel techniques for variance reduction in Monte Carlo rendering, as well as a few related tools. The research in this dissertation focuses on using importance sampling to pick a small set of well-distributed point samples. As the primary contribution, I have developed the first methods to explicitly draw samples from the product of distant high-frequency lighting and complex reflectance functions. By sampling the product, low noise results can be achieved using a very small number of samples, which is important to minimize the rendering times. Several different hierarchical representations are explored to allow efficient product sampling. In the first publication, the key idea is to work in a compressed wavelet basis, which allows fast evaluation of the product. Many of the initial restrictions of this technique were removed in follow-up work, allowing higher-resolution uncompressed lighting and avoiding precomputation of reflectance functions. My second main contribution is to present one of the first techniques to take the triple product of lighting, visibility and reflectance into account to further reduce the variance in Monte Carlo rendering. For this purpose, control variates are combined with importance sampling to solve the problem in a novel way. A large part of the technique also focuses on analysis and approximation of the visibility function. To further refine the above techniques, several useful tools are introduced. These include a fast, low-distortion map to represent (hemi)spherical functions, a method to create high-quality quasi-random points, and an optimizing compiler for analyzing shaders using interval arithmetic. The latter automatically extracts bounds for importance sampling of arbitrary shaders, as opposed to using a priori known reflectance functions. In summary, the work presented here takes the field of computer graphics one step further towards making photorealistic rendering practical for a wide range of uses. By introducing several novel Monte Carlo methods, more sophisticated lighting and materials can be used without increasing the computation times. The research is aimed at domain-specific solutions to the rendering problem, but I believe that much of the new theory is applicable in other parts of computer graphics, as well as in other fields

Glosarium Matematika

273 p.; 24 cm