The idea of `probabilistic distances' (also called divergences), which in some sense assess how `close' two probability distributions are from one another, has been widely employed in probability, statistics, information theory, and related fields. Of particular importance due to their generality and applicability are the Renyi divergence measures. While the closely related concept of Renyi entropy of a probability distribution has been studied extensively, and closed-form expressions for the most common univariate and multivariate continuous distributions have been obtained and compiled, the literature currently lacks the corresponding compilation for continuous Renyi divergences. The present thesis addresses this issue for analytically tractable cases. Closed-form expressions for Kullback-Leibler divergences are also derived and compiled, as they can be seen as an extension by continuity of the Renyi divergences. Additionally, we establish a connection between Renyi divergence and the variance of the log-likelihood ratio of two distributions, which extends the work of Song (2001) on the relation between Renyi entropy and the log-likelihood function, and which becomes practically useful in light of the Renyi divergence expressions we have derived. Lastly, we consider the Renyi divergence rate between two zero-mean stationary Gaussian processes. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2011-08-30 13:37:41.792

The classical parametric estimation approach, maximum likelihood, while providing maximally efficient estimators at the correct model, lacks robustness. As a modification of maximum likelihood, Huber (1964) introduced M-estimators, which are very general but often ad hoc. Basu et al. (1998) developed a family of density-based divergences, many of which exhibit robustness. It turns out that maximum likelihood is a special case of this general class of divergence functions, which are indexed by a parameter alpha. Basu noted that only values of alpha in the [0,1] range were of interest -- with alpha = 0 giving the maximum likelihood solution and alpha = 1 the L2E solution (Scott, 2001). As alpha increases, there is a clear tradeoff between increasing robustness and decreasing efficiency. This thesis develops a family of robust location and scale estimators by applying Basu's alpha-divergence function to a multivariate partial density component model (Scott, 2004). The usefulness of alpha values greater than 1 will be explored, and the new estimator will be applied to simulated cases and applications in parametric density estimation and regression.

The energy costs of data movement are limiting the performance scaling of future generations of high performance computing architectures targeted to data intensive applications. The result has been a resurgence in the interest in processing-in-memory (PIM) architectures. This challenge has spawned the development of a scalable, parametric data parallel architecture referred at the Heterogeneous Architecture Research Prototype (HARP) - a single instruction multiple thread (SIMT) architecture for integration into DRAM systems, particularly 3D memory stacks as a distinct processing layer to exploit the enormous internal memory bandwidth. However, this potential can only be realized with an optimizing compilation environment. This thesis addresses this challenge by i) the construction of an open source compiler for HARP, and ii) integrating optimizations for handling control flow divergence for HARP instances. The HARP compiler is built using the LLVM open source compiler infrastructure. Apart from traditional code generation, the HARP compiler backend handles unique challenges associated with the HARP instruction set. Chief among them is code generation for control divergence management techniques. The HARP architecture and compiler supports i) a hardware reconvergence stack and ii) predication to handle divergent branches. The HARP compiler addresses several challenges associated with generating code for these two control divergence management techniques and implements multiple analyses and transformations for code generation. Both of these techniques have unique advantages and disadvantages depending upon whether the conditional branch is likely to be unanimous or not. Two decision frameworks, guided by static analysis and dynamic profile information are implemented to choose between the control divergence management techniques by analyzing the nature of the conditional branches and utilizing this information during compilation.

This thesis examines the evolutionary divergence between northern European and Italian populations of Chorthippus parallelus. Several differing approaches were taken, which identified the inception of various components of the speciation process between these parapatric populations which meet in the Alps. Firstly, partial post-zygotic reproductive isolation was demonstrated using hybrid crosses. The male hybrid offspring of both reciprocal crosses were sterile, displaying severe testicular dysfunction, while the female hybrids showed no deleterious effects of hybridisation. In this grasshopper the males are the heterogametic sex, possessing a single X chromosome, and so this pattern of hybrid sterility conforms to Haldane's rule. Secondly, investigation of the calling song of the male grasshopper, a component of the mate recognition system, suggested the presence of pre-mating reproductive isolation. Males from the different races were found to sing calling song of a significantly different structure. Finally, examination of DNA sequence divergence in a mitochondrial DNA marker demonstrated Significant levels of genetic differentiation between the races. This population divergence and incipient reproductive isolation parallels that found between the north European and Iberian populations of this grasshopper, and provides further evidence that the divergent geographical races have resulted from allopatric divergence while in isolated refugial populations during the glacial periods of the Pleistocene Epoch. These approaches were repeated to investigate genetic divergence between localised populations within the Italian peninsula. No hybrid dysfunction was observed between these populations, suggesting that they are recently derived from one continuous population. This was probably the refugial population of the last ice-age. Additionally, investigation with the mtDNA marker gave preliminary evidence for population expansion from the south to the north of Italy. Interestingly, the male calling song was Significantly different in populations from the north and the south of Italy, suggesting that a component of pre-mating reproductive isolation may have evolved prior to post-mating isolation in allopatric populations of C. parallelus.

Thesis (B.A.(Hons.))--University of Adelaide, 1977. / Includes bibliographical references.

Radial basis functions (RBFs) are probably best known for their applications to scattered data problems. Until the 1990s, RBF theory only involved functions that were scalar-valued. Matrix-valued RBFs were subsequently introduced by Narcowich and Ward in 1994, when they constructed divergence-free vector-valued functions that interpolate data at scattered points. In 2002, Lowitzsch gave the first error estimates for divergence-free interpolants. However, these estimates are only valid when the target function resides in the native space of the RBF. In this paper we develop Sobolev-type error estimates for cases where the target function is less smooth than functions in the native space. In the process of doing this, we give an alternate characterization of the native space, derive improved stability estimates for the interpolation matrix, and give divergence-free interpolation and approximation results for band-limited functions. Furthermore, we introduce a new class of matrix-valued RBFs that can be used to produce curl-free interpolants.

This study investigates the use of the divergence speed as an additional constraint to a multi-disciplinary optimization (MDO) problem. The goal of the project is to expand the MDO toolbox by adding an aeroelastic module used where the aeroelastic characteristics present a possible safety hazard. This paper examines aeroelastic theory and MDO disciplines. The divergence constraint function is developed on a BAH wing. The optimization problem is executed on the HANSA HFB 320 transport jet using the FEAP structural solver and a Vortex Lattice Method as the aerodynamic solver. The study shows that divergence speed can function as a safety constraint but the stress constraints determine the optimum design. Furthermore, obtaining a true divergence constraint will require a finer mesh, a more efficient aerodynamic solver and non-finite difference approach to gradient determination. Thus, the addition of the divergence constraint does not yet directly benefit this MDO framework.

This study investigates the use of the divergence speed as an additional constraint to a multi-disciplinary optimization (MDO) problem. The goal of the project is to expand the MDO toolbox by adding an aeroelastic module used where the aeroelastic characteristics present a possible safety hazard. This paper examines aeroelastic theory and MDO disciplines. The divergence constraint function is developed on a BAH wing. The optimization problem is executed on the HANSA HFB 320 transport jet using the FEAP structural solver and a Vortex Lattice Method as the aerodynamic solver. The study shows that divergence speed can function as a safety constraint but the stress constraints determine the optimum design. Furthermore, obtaining a true divergence constraint will require a finer mesh, a more efficient aerodynamic solver and non-finite difference approach to gradient determination. Thus, the addition of the divergence constraint does not yet directly benefit this MDO framework.

Thesis (Ph.D.)--Michigan State University. Dept. of Biochemistry and Molecular Biology, 2008. / Title from PDF t.p. (viewed on Mar. 30, 2009) Includes bibliographical references (p. 127-139). Also issued in print.

No / This paper presents a novel approach for probabilistic clustering, motivated by a real-world problem of modelling driving behaviour. The main aim is to establish clusters of drivers with similar journey behaviour, based on a large sample of historic journeys data. The proposed approach is to establish similarity between driving behaviours by using the Kullback-Leibler and Jensen-Shannon divergence metrics based on empirical multi-dimensional probability density functions. A graph-clustering algorithm is proposed based on modifications of the Markov Cluster algorithm. The paper provides a complete mathematical formulation, details of the algorithms and their implementation in Python, and case study validation based on real-world data.