Aspects of categorical data analysis.

Thesis (M.Sc.)-University of Natal, Durban, 1998.The purpose of this study is to investigate and understand data which are grouped into categories. At the onset, the study presents a review of early research contributions and controversies surrounding categorical data analysis. The concept of sparseness in a contingency table refers to a table where many cells have small frequencies. Previous research findings showed that incorrect results were obtained in the analysis of sparse tables. Hence, attention is focussed on the effect of sparseness on modelling and analysis of categorical data in this dissertation. Cressie and Read (1984) suggested a versatile alternative, the power divergence statistic, to statistics proposed in the past. This study includes a detailed discussion of the power-divergence goodness-of-fit statistic with areas of interest covering a review on the minimum power divergence estimation method and evaluation of model fit. The effects of sparseness are also investigated for the power-divergence statistic. Comparative reviews on the accuracy, efficiency and performance of the power-divergence family of statistics under large and small sample cases are presented. Statistical applications on the power-divergence statistic have been conducted in SAS (Statistical Analysis Software). Further findings on the effect of small expected frequencies on accuracy of the X2 test are presented from the studies of Tate and Hyer (1973) and Lawal and Upton (1976). Other goodness-of-fit statistics which bear relevance to the sparse multino-mial case are discussed. They include Zelterman's (1987) D2 goodness-of-fit statistic, Simonoff's (1982, 1983) goodness-of-fit statistics as well as Koehler and Larntz's tests for log-linear models. On addressing contradictions for the sparse sample case under asymptotic conditions and an increase in sample size, discussions are provided on Simonoff's use of nonparametric techniques to find the variances as well as his adoption of the jackknife and bootstrap technique

FPGAs in Bioinformatics: Implementation and Evaluation of Common Bioinformatics Algorithms in Reconfigurable Logic

Life. Much effort is taken to grant humanity a little insight in this fascinating and complex but fundamental topic. In order to understand the relations and to derive consequences humans have begun to sequence their genomes, i.e. to determine their DNA sequences to infer information, e.g. related to genetic diseases. The process of DNA sequencing as well as subsequent analysis presents a computational challenge for recent computing systems due to the large amounts of data alone. Runtimes of more than one day for analysis of simple datasets are common, even if the process is already run on a CPU cluster. This thesis shows how this general problem in the area of bioinformatics can be tackled with reconfigurable hardware, especially FPGAs. Three compute intensive problems are highlighted: sequence alignment, SNP interaction analysis and genotype imputation. In the area of sequence alignment the software BLASTp for protein database searches is exemplarily presented, implemented and evaluated.SNP interaction analysis is presented with three applications performing an exhaustive search for interactions including the corresponding statistical tests: BOOST, iLOCi and the mutual information measurement. All applications are implemented in FPGA-hardware and evaluated, resulting in an impressive speedup of more than in three orders of magnitude when compared to standard computers. The last topic of genotype imputation presents a two-step process composed of the phasing step and the actual imputation step. The focus lies on the phasing step which is targeted by the SHAPEIT2 application. SHAPEIT2 is discussed with its underlying mathematical methods in detail, and finally implemented and evaluated. A remarkable speedup of 46 is reached here as well

Evaluation of lntelligent Medical Systems

This thesis presents novel, robust, analytic and algorithmic methods for calculating Bayesian posterior intervals of receiver operating characteristic (ROC) curves and confusion matrices used for the evaluation of intelligent medical systems tested with small amounts of data. Intelligent medical systems are potentially important in encapsulating rare and valuable medical expertise and making it more widely available. The evaluation of intelligent medical systems must make sure that such systems are safe and cost effective. To ensure systems are safe and perform at expert level they must be tested against human experts. Human experts are rare and busy which often severely restricts the number of test cases that may be used for comparison. The performance of expert human or machine can be represented objectively by ROC curves or confusion matrices. ROC curves and confusion matrices are complex representations and it is sometimes convenient to summarise them as a single value. In the case of ROC curves, this is given as the Area Under the Curve (AUC), and for confusion matrices by kappa, or weighted kappa statistics. While there is extensive literature on the statistics of ROC curves and confusion matrices they are not applicable to the measurement of intelligent systems when tested with small data samples, particularly when the AUC or kappa statistic is high. A fundamental Bayesian study has been carried out, and new methods devised, to provide better statistical measures for ROC curves and confusion matrices at low sample sizes. They enable exact Bayesian posterior intervals to be produced for: (1) the individual points on a ROC curve; (2) comparison between matching points on two uncorrelated curves; . (3) the AUC of a ROC curve, using both parametric and nonparametric assumptions; (4) the parameters of a parametric ROC curve; and (5) the weight of a weighted confusion matrix. These new methods have been implemented in software to provide a powerful and accurate tool for developers and evaluators of intelligent medical systems in particular, and to a much wider audience using ROC curves and confusion matrices in general. This should enhance the ability to prove intelligent medical systems safe and effective and should lead to their widespread deployment. The mathematical and computational methods developed in this thesis should also provide the basis for future research into determination of posterior intervals for other statistics at small sample sizes

The Graphical Representation of Structured Multivariate Data

During the past two decades or so, graphical representations have been used increasingly for the examination, summarisation and communication of statistical data. Many graphical techniques exist for exploratory data analysis (ie. for deciding which model it is appropriate to fit to the data) and a number of graphical diagnostic techniques exist for checking the appropriateness of a fitted model. However, very few techniques exist for the representation of the fitted model itself. This thesis is concerned with the development of some new and existing graphical representation techniques for the communication and interpretation of fitted statistical models. The first part of this thesis takes the form of a general overview of the use in statistics of graphical representations for exploratory data analysis and diagnostic model checking. In relation to the concern of this thesis, particular consideration is given to the few graphical techniques which already exist for the representation of fitted models. A number of novel two-dimensional approaches are then proposed which go partway towards providing a graphical representation of the main effects and interaction terms for fitted models. This leads on to a description of conditional independence graphs, and consideration of the suitability of conditional independence graphs as a technique for the representation of fitted models. Conditional independence graphs are then developed further in accordance with the research aims. Since it becomes apparent that it is not possible to use any of the approaches taken m order to develop a simple two-dimensional pen-and-paper technique for the unambiguous graphical representation of all fitted statistical models, an interactive computer package based on the conditional independence graph approach is developed for the construction, communication and interpretation of graphical representations for fitted statistical models. This package, called the "Conditional Independence Graph Enhancer" (CIGE), does provide unambiguous graphical representations for all fitted statistical models considered

Stream sketches, sampling, and sabotage

Exact solutions are unattainable for important problems. The calculations are limited by the memory of our computers and the length of time that we can wait for a solution. The field of approximation algorithms has grown to address this problem; it is practically important and theoretically fascinating. We address three questions along these lines. What are the limits of streaming computation? Can we efficiently compute the likelihood of a given network of relationships? How robust are the solutions to combinatorial optimization problems? High speed network monitoring and rapid acquisition of scientific data require the development of space efficient algorithms. In these settings it is impractical or impossible to store all of the data, nonetheless the need for analyzing it persists. Typically, the goal is to compute some simple statistics on the input using sublinear, or even polylogarithmic, space. Our main contributions here are the complete classification of the space necessary for several types of statistics. Our sharpest results characterize the complexity in terms of the domain size and stream length. Furthermore, our algorithms are universal for their respective classes of statistics. A network of relationships, for example friendships or species-habitat pairings, can often be represented as a binary contingency table, which is {0,1}-matrix with given row and column sums. A natural null model for hypothesis testing here is the uniform distribution on the set of binary contingency tables with the same line sums as the observation. However, exact calculation, asymptotic approximation, and even Monte-Carlo approximation of p-values are so-far practically unattainable for many interesting examples. This thesis presents two new algorithms for sampling contingency tables. One is a hybrid algorithm that combines elements of two previously known algorithms. It is intended to exploit certain properties of the margins that are observed in some data sets. Our other algorithm samples from a larger set of tables, but it has the advantage of being fast. The robustness of a system can be assessed from optimal attack strategies. Interdiction problems ask about the worst-case impact of a limited change to an underlying optimization problem. Most interdiction problems are NP-hard, and furthermore, even designing efficient approximation algorithms that allow for estimating the order of magnitude of a worst-case impact has turned out to be very difficult. We suggest a general method to obtain pseudoapproximations for many interdiction problems