This thesis focuses on assessing and improving statistical methods implemented in two areas of public health research. The first topic involves estimation of national influenza-associated mortality rates via mathematical modeling. The second topic involves the timely detection of infectious disease outbreaks using statistical process control monitoring. For over fifty years, the Centers for Disease Control and Prevention has been estimating annual rates of U.S. deaths attributable to influenza. These estimates have been used to determine costs and benefits associated with influenza prevention and control strategies. Quantifying the effect of influenza on mortality, however, can be challenging since influenza infections typically are not confirmed virologically nor specified on death certificates. Consequently, a wide range of ecologically based, mathematical modeling approaches have been applied to specify the association between influenza and mortality. To date, all influenza-associated death estimates have been based on mortality data first aggregated at the national level and then modeled. Unfortunately, there are a number of local-level seasonal factors that may confound the association between influenza and mortality - thus suggesting that data be modeled at the local level and then pooled to make national estimates of death. The first component of the thesis topic involving mortality estimation addresses this issue by introducing and implementing a two-stage hierarchical Bayesian modeling approach. In the first stage, city-level data with varying trends in mortality and weather were modeled using semi-parametric, generalized additive models. In the second stage, the log-relative risk estimates calculated for each city in stage 1 represented the “outcome” variable, and were modeled two ways: (1) assuming spatial independence across cities using a Bayesian generalized linear model, and (2) assuming correlation among cities using a Bayesian spatial correlation model. Results from these models were compared to those from a more-conventional approach. The second component of this topic examines the extent to which seasonal confounding and collinearity affect the relationship between influenza and mortality at the local (city) level. Disentangling the effects of temperature, humidity, and other seasonal confounders on the association between influenza and mortality is challenging since these covariates are often temporally collinear with influenza activity. Three modeling strategies with varying representations of background seasonality were compared. Seasonal covariates entered into the model may have been measured (e.g., ambient temperature) or unmeasured (e.g., time-based smoothing splines or Fourier terms). An advantage of modeling background seasonality via time splines is that the amount of seasonal curvature can be controlled by the number of degrees of freedom specified for the spline. A comparison of the effects of influenza activity on mortality based on these varying representations of seasonal confounding is assessed. The third component of this topic explores the relationship between mortality rates and influenza activity using a flexible, natural cubic spline function to model the influenza term. The conventional approach of fitting influenza-activity terms linearly in regression was found to be too constraining. Results show that the association is best represented nonlinearly. The second area of focus in this thesis involves infectious disease outbreak detection. A fundamental goal of public health surveillance, particularly syndromic surveillance, is the timely detection of increases in the rate of unusual events. In syndromic surveillance, a significant increase in the incidence of monitored disease outcomes would trigger an alert, possibly prompting the implementation of an intervention strategy. Public health surveillance generally monitors count data (e.g., counts of influenza-like illness, sales of over-the-counter remedies, and number of visits to outpatient clinics). Statistical process control charts, designed for quality control monitoring in industry, have been widely adapted for use in disease and syndromic surveillance. The behavior of these detection methods on discrete distributions, however, has not been explored in detail. For this component of the thesis, a simulation study was conducted to compare the CuSum and EWMA methods for detection of increases in negative binomial rates with varying amounts of dispersion. The goal of each method is to detect an increase in the mean number of cases as soon as possible after an upward rate shift has occurred. The performance of the CuSum and EWMA detection methods is evaluated using the conditional expected delay criterion, which is a measure of the detection delay, i.e., the time between the occurrence of a shift and when that shift is detected. Detection capabilities were explored under varying shift sizes and times at which the shifts occurred.

This thesis provides a study on stochastic models of financial markets related to problems of asset pricing and hedging, optimal portfolio managing and statistical changepoint detection in trends of asset prices. Chapter 1 develops a general model of a system of interconnected stochastic markets associated with a directed acyclic graph. The main result of the chapter provides sufficient conditions of hedgeability of contracts in the model. These conditions are expressed in terms of consistent price systems, which generalise the notion of equivalent martingale measures. Using the general results obtained, a particular model of an asset market with transaction costs and portfolio constraints is studied. In the second chapter the problem of multi-period utility maximisation in the general market model is considered. The aim of the chapter is to establish the existence of systems of supporting prices, which play the role of Lagrange multipliers and allow to decompose a multi-period constrained utility maximisation problem into a family of single-period and unconstrained problems. Their existence is proved under conditions similar to those of Chapter 1.The last chapter is devoted to applications of statistical sequential methods for detecting trend changes in asset prices. A model where prices are driven by a geometric Gaussian random walk with changing mean and variance is proposed, and the problem of choosing the optimal moment of time to sell an asset is studied. The main theorem of the chapter describes the structure of the optimal selling moments in terms of the Shiryaev–Roberts statistic and the posterior probability process.

The ability to make decisions is a crucial ability of many autonomous systems. In many scenarios the consequence of a decision is unknown and often stochastic. The same decision may lead to a different outcome every time it is taken. An agent that can learn to make decisions based purely on its past experience needs less tuning and is likely more robust. An agent must often balance between learning the payoff of actions by exploring, and exploiting the knowledge they currently have. The multi-armed bandit problem exhibits such an exploration-exploitation dilemma. Thompson Sampling is a strategy for the problem, first proposed in 1933. In the last several years there has been renewed interest in it, with the emergence of strong empirical and theoretical justification for its use. This thesis seeks to take advantage of the benefits of Thompson Sampling while applying it to other decision-making models. In doing so we propose different algorithms for these scenarios. Firstly we explore a switching multi-armed bandit problem. In real applications the most appropriate decision to take often changes over time. We show that an agent assuming switching is often robust to many types of changing environment. Secondly we consider the best arm identification problem. Unlike the multi-armed bandit problem, where an agent wants to increase reward over the entire period of decision making, the best arm identification is concerned in increasing the reward gained by a final decision. This thesis argues that both problems can be tackled effectively using Thompson Sampling based approaches and provides empirical evidence to support this claim.

Modeling the complex relationships between habitat characteristics and a species' habitat preferences pose many difficult problems for ecological researchers. These problems are complicated further when information is collected over a range of time or space. Additionally, the variety of factors affecting these choices is difficult to understand and even more difficult to accurately collect information about. In light of these concerns, we evaluate the performance of current standard habitat preference models that are based on Bayesian methods and then present some extensions and supplements to those methods that prove to be very useful. More specifically, we demonstrate the value of extending the standard Bayesian hierarchical model using finite mixture model methods. Additionally, we demonstrate that an extension of the Bayesian hierarchical changepoint model to allow for estimating multiple changepoints simultaneously can be very informative when applied to data about multiple habitat locations or species. These models allow the researcher to compare the sites or species with respect to a very specific ecological question and consequently provide definitive answers that are often not available with more commonly used models containing many explanatory factors. Throughout our work we use a complex data set containing information about horseshoe crab spawning habitat preferences in the Delaware Bay over a five-year period. These data epitomize some of the difficult issues inherent to studying habitat preferences. The data are collected over time at many sites, have missing observations, and include explanatory variables that, at best, only provide surrogate information for what researchers feel is important in explaining spawning preferences throughout the bay. We also looked at a smaller data set of freshwater mussel habitat selection preferences in relation to bridge construction on the Kennerdell River in Western Pennsylvania. Together, these two data sets provided us with insight in developing and refining the methods we present. They also help illustrate the strengths and weaknesses of the methods we discuss by assessing their performance in real situations where data are inevitably complex and relationships are difficult to discern. / Ph. D.

We develop a family of Bayesian algorithms built around Gaussian processes for various problems posed by sensor networks. We firstly introduce an iterative Gaussian process for multi-sensor inference problems, and show how our algorithm is able to cope with data that may be noisy, missing, delayed and/or correlated. Our algorithm can also effectively manage data that features changepoints, such as sensor faults. Extensions to our algorithm allow us to tackle some of the decision problems faced in sensor networks, including observation scheduling. Along these lines, we also propose a general method of global optimisation, Gaussian process global optimisation (GPGO), and demonstrate how it may be used for sensor placement. Our algorithms operate within a complete Bayesian probabilistic framework. As such, we show how the hyperparameters of our system can be marginalised by use of Bayesian quadrature, a principled method of approximate integration. Similar techniques also allow us to produce full posterior distributions for any hyperparameters of interest, such as the location of changepoints. We frame the selection of the positions of the hyperparameter samples required by Bayesian quadrature as a decision problem, with the aim of minimising the uncertainty we possess about the values of the integrals we are approximating. Taking this approach, we have developed sampling for Bayesian quadrature (SBQ), a principled competitor to Monte Carlo methods. We conclude by testing our proposals on real weather sensor networks. We further benchmark GPGO on a wide range of canonical test problems, over which it achieves a significant improvement on its competitors. Finally, the efficacy of SBQ is demonstrated in the context of both prediction and optimisation.

The need to quantify distance between two groups of objects is prevalent throughout the signal processing world. The difference of group means computed using the Euclidean, or L2 distance, is one of the predominant distance measures used to compare feature vectors and groups of vectors, but many problems arise with it when high data dimensionality is present. Maximum mean discrepancy (MMD) is a recent unsupervised kernel-based pattern recognition method which may improve differentiation between two distinct populations over many commonly used methods such as the difference of means, when paired with the proper feature representations and kernels. MMD-based distance computation combines many powerful concepts from the machine learning literature, such as data distribution-leveraging similarity measures and kernel methods for machine learning. Due to this heritage, we posit that dissimilarity-based classification and changepoint detection using MMD can lead to enhanced separation between different populations. To test this hypothesis, we conduct studies comparing MMD and the difference of means in two subareas of image analysis and understanding: first, to detect scene changes in video in an unsupervised manner, and secondly, in the biomedical imaging field, using clinical ultrasound to assess tumor response to treatment. We leverage effective computer vision data descriptors, such as the bag-of-visual-words and sparse combinations of SIFT descriptors, and choose from an assessment of several similarity kernels (e.g. Histogram Intersection, Radial Basis Function) in order to engineer useful systems using MMD. Promising improvements over the difference of means, measured primarily using precision/recall for scene change detection, and k-nearest neighbour classification accuracy for tumor response assessment, are obtained in both applications.

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