In the last decades, the explosion of data from quantitative techniques has revolutionised our understanding of biological processes. In this scenario, advanced statistical methods and algorithms are becoming fundamental to decipher the dynamics of biochemical mechanisms such those involved in the regulation of gene expression. Here we develop mechanistic models and approximate inference techniques to reverse engineer the dynamics of gene regulation, from mRNA and/or protein time series data. We start from an existent variational framework for statistical inference in transcriptional networks. The framework is based on a continuous-time description of the mRNA dynamics in terms of stochastic differential equations, which are governed by latent switching variables representing the on/off activity of regulating transcription factors. The main contributions of this work are the following. We speeded-up the variational inference algorithm by developing a method to compute a posterior approximate distribution over the latent variables using a constrained optimisation algorithm. In addition to computational benefits, this method enabled the extension to statistical inference in networks with a combinatorial model of regulation. A limitation of this framework is the fact that inference is possible only in transcriptional networks with a single-layer architecture (where a single or couples of transcription factors regulate directly an arbitrary number of target genes). The second main contribution in this work is the extension of the inference framework to hierarchical structures, such as feed-forward loop. In the last contribution we define a general structure for transcription-translation networks. This work is important since it provides a general statistical framework to model complex dynamics in gene regulatory networks. The framework is modular and scalable to realistically large systems with general architecture, thus representing a valuable alternative to traditional differential equation models. All models are embedded in a Bayesian framework; inference is performed using a variational approach and compared to exact inference where possible. We apply the models to the study of different biological systems, from the metabolism in E. coli to the circadian clock in the picoalga O. tauri.

We develop and study the high-density limit of various new models in mathematical pop- ulation genetics. These models extend the Λ-Fleming–Viot process when there are two genetic types at the locus of study. Given a finite sample from a population undergoing these dynamics, a key tool for understanding the corresponding genealogy is the method of duality. We introduce the reproduction-linked mutation mechanism and consider how this affects the process of relative allelic frequencies and the genealogy. The second generalization incorporates two forms of natural selection – differential killing and differential birth. We contrast the structure of their genealogies. Several properties of the block size spectra of the Kingman and Beta coalescents are also investigated, including their behaviour as they come down from infinity.

This thesis consists of two parts. The first part (Chapters 2-4) focuses on the problem of inverting the signature of a path of bounded variation, and we present three results here. First, we give an explicit inversion formula for any axis path in terms of its signature. Second, we show that for relatively smooth paths, the derivative at the end point can be approximated arbitrarily closely by its signature sequence, and we provide explicit error estimates. As an application, we give an effective inversion procedure for piecewise linear paths. Finally, we prove a uniform estimate for the signatures of paths of bounded variations, and obtain a reconstruction theorem via that uniform estimate. Although this general reconstruction theorem is not computationally efficient, the techniques involved in deriving the uniform estimate are useful in other situations, and we also give an application in the case of expected signatures for Brownian motion. The second part (Chapter 5) deals with rough paths. After introducing proper backgrounds, we extend the uniform estimate above to the context of rough paths, and show how it can lead to simple proofs of distance bounds for Gaussian iterated integrals.

We study a natural generalization of the Λ-coalescent to a spatial continuum. We introduce the process, which is known as the Segregated Λ-coalescent, via its connections to the (non-spatial) Λ-coalescent and the Spatial Λ-Fleming-Viot process. The main new results contained in this thesis are as follows. The Segregated Λ-coalescent has a non-trivial construction which we present here in terms of stochastic flows. We describe the qualitative behaviour of the Segregated Λ-coalescent and compare it to the behaviour of the Λ-coalescent, showing in particular that the Segregated Λ-coalescent has an extra phase transition which is directly related to the introduction of space. We finish with some results concerning the rate at which the Segregated Λ-coalescent comes down from infinity.

Despite its recent advent, value at risk (VaR) became the most widely used technique for measuring future expected risk for both financial and non-financial institutions. VaR, the measure of the worst expected loss over a given horizon at a given confidence level, depends crucially on the distributional aspects of trading revenues. Existing VaR models do not capture adequately some empirical aspects of financial data such as the tail thickness, which is vital in VaR calculations. Tail thickness in financial variables results basically from stochastic volatility and event risk (jumps). Those two sources are not totally separated; under event risk, volatility updates faster than under normal market conditions. Generally, tail thickness is associated with hyper volatility updating. Existing VaR literature accounts partially for tail thickness either by including stochastic volatility or by including jump diffusion, but not both. Additionally, this literature does not account for fast updating of volatility associated with tail thickness. This dissertation fills the gap by developing analytical VaR models account for the total (maximum) tail thickness and the associated fast volatility updating. Those aspects are achieved by assuming that trading revenues are evolving according to a mixed non-affine stochastic volatility-jump diffusion process. The mixture of stochastic volatility and jumps diffusion accounts for the maximum tail thickness, whereas the nonaffine structure of stochastic volatility captures the fast volatility updating. The non-affine structure assumes that volatility dynamics are non-linearly related to the square root of current volatility rather than the traditional linear (affine) relationship. VaR estimates are obtained by deriving the conditional characteristic function, and then inverting it numerically via the Fourier Inversion technique to infer the cumulative distribution function. The application of the developed VaR models on a sample that contains six U.S banks during the period 1995-2002 shows that VaR models based on the non-affine stochastic volatility and jump diffusion process produce more reliable VaR estimates compared with the banks' own VaR models. The developed VaR models could significantly predict the losses that those banks incurred during the Russian crisis and the near collapse of the LTCM in 1998 when the banks' VaR models fail.

Nesta tese abordamos o problema do hedging de mínima variância de derivativos em mercados incompletos usando a teoria de controle ótimo estocástico com critério quadrático de otimização. Desenvolvemos um modelo geral de apreçamento e hedging de derivativos em mercados incompletos, a tempo discreto, que é capaz de acomodar qualquer tipo de payoff com característica européia que dependa de n ativos de risco. Nesse modelo, o mercado pode apresentar diferentes modos de operação, o que foi formalizado matematicamente por meio de uma cadeia de Markov. Também desenvolvemos um modelo geral de apreçamento e hedging de derivativos em mercados incompletos, a tempo discreto e espaço de estados contínuo, que é capaz de acomodar qualquer tipo de payoff com característica européia que dependa de um ativo de risco cujos retornos sejam representados por um processo de difusão com saltos. Desenvolvemos, ainda, expressões analíticas fechadas para o apreçamento e hedging de uma opção de compra européia vanilla em duas situações: (1) quando os retornos do ativo de risco são representados por um processo de difusão com saltos, e (2) quando os retornos do ativo de risco são representados por um processo de Wiener. Por fim, realizamos simulações numéricas para o controle (hedging) de uma opção de compra européia vanilla quando os retornos do ativo de risco são representados por um processo de Wiener, e comparamos os resultados obtidos com a estratégia de controle derivada do modelo de Black & Scholes. / In this thesis we approach the mean-variance hedging problem of derivatives in incomplete markets employing the theory of stochastic optimal control with quadratic optimization criteria. We developed a general derivatives pricing and hedging model in incomplete markets, in discrete time, capable of accommodating any type of European payoff contingent on n risky assets. In this model, the market may exhibit different operating modes, which were mathematically formalized by means of a Markov chain. We also developed a general derivatives pricing and hedging model in incomplete markets, in discrete time and continuous state space, capable of accommodating any type of European payoff contingent on one risky asset whose returns are described by a jump diffusion process. Even further, we developed closed-form analytical expressions for the pricing and hedging of a European vanilla call option in two situations: (1) when the risky asset returns are described by a jump diffusion process, and (2) when the risky asset returns are described by a Wiener process. Finally, we simulated the control (hedging) of a European vanilla call option when the risky asset returns are described by a Wiener process, and compared the results to those obtained with the control strategy derived from the Black & Scholes model.

Survey is time-consuming and expensive. Therefore, it needs to be both effective and efficient. Some archaeologists have argued that current survey techniques are not effective (Shott 1985, 1989), but most archaeologists continue to employ these methods and therefore must believe they are effective. If our survey techniques are effective, why do simulations suggest otherwise? If they are ineffective, can we improve them? The answers to these practical questions depend on the topological characteristics of archaeological site distributions. In this study I analyze archaeological site distributions in the Valley of Oaxaca, Mexico, using lacunarity and fractal dimension. Fractal dimension is a parameter of fractal patterns, which are complex, space-filling designs exhibiting self-similarity and power-law scaling. Lacunarity is a statistical measure that describes the texture of a spatial dispersion. It is useful in understanding how archaeological tests should be spaced during surveys. Between these two measures, I accurately describe the regional topology and suggest new considerations for archaeological survey design. / Includes bibliography. / Thesis (M.A.)--Florida Atlantic University, 2014. / FAU Electronic Theses and Dissertations Collection

Thesis (M.Phil.)--Chinese University of Hong Kong. / Bibliography: leaf 34.

近年来，随着机会约束规划被广泛应用以及凸分析和概率论的新进展，如何有效的处理机会约束成为一个炙手可热的研究方向。其中，一个成功的解决方法就是考虑其安全可解近似，也就是说将机会约束转化成一组方便处理的确定性约束，并且保持原机会约束在新的约束下成立。目前这样的方法主要应用于带有独立分布的数据扰动的机会约束规划，或者已知扰动的协方差矩阵的情况。同时，带有相关数据扰动的机会约束下的锥不等式广泛应用于供应链管理、金融、控制以及信号处理等学科，而现有的优化理论却极少涵盖。 / 在这篇论文中我们主要研究机会约束下的线性矩阵不等式，并假设扰动分布不必相互独立，其仅有的相关性信息只由一系列子扰动的独立关系结构提供。通过推导矩阵值随机变量的大偏差上界，我们得出这一类条件约束的安全可解近似。我们随后考虑了基于条件风险价值度量的机会约束规划问题, 以及带多项式扰动的机会约束优化问题。另外，通过构造相应的鲁棒对等式的不确定集合，我们把机会约束规划转换成鲁棒优化问题。由于这种近似可以表示为一组线性矩阵不等式，因而可以使用现成的优化软件方便地求解。最后，我们把该安全可解近似方法运用到一个控制理论问题，以及一个带风险价值约束的投资组合优化问题中。 / The wide applicability of chanceconstrained programming, together with advances in convex optimization and probability theory, has created a surge of interest in finding efficient methods for processing chance constraints in recent years. One of the successes is the development of so-called safe tractable approximations of chance-constrained programs, where a chance constraint is replaced by a deterministic and efficiently computable inner approximation. Currently, such an approach applies mainly to chance-constrained linear inequalities, in which the data perturbations are either independent or define a known covariance matrix. However, its applicability to the case of chanceconstrained conic inequalities with dependent perturbations--which arises in supply chain management, finance, control and signal processing applications--remains largely unexplored. / In this thesis, we consider the problem of processing chance-constrained affinely perturbed linear matrix inequalities, in which the perturbations are not necessarily independent, and the only information available about the dependence structure is a list of independence relations. Using large deviation bounds for matrix-valued random variables, we develop safe tractable approximations of those chance constraints. Extensions to the Matrix CVaR (Conditional Value-at-Risk) risk measure and general polynomials perturbations are also provided separately. Further more, we show that the chanceconstrained linear matrix inequalities optimization problem can be converted to a robust optimization problem by constructing the uncertainty set of the corresponding robust counterpart. A nice feature of our approximations is that they can be expressed as systems of linear matrix inequalities, thus allowing them to be solved easily and efficiently by off-the-shelf optimization solvers. We also provide a numerical illustration of our constructions through a problem in control theory and a portfolio VaR (Value-at-Risk) optimization problem. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Wang, Kuncheng. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 94-101). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Abstract --- p.i / Acknowledgement --- p.vi / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Motivations and Philosophy --- p.1 / Chapter 1.2 --- Background --- p.2 / Chapter 1.3 --- Literature Review --- p.4 / Chapter 1.4 --- Contribution --- p.7 / Chapter 2 --- Preliminaries --- p.10 / Chapter 2.1 --- Probabilistic Inequalities --- p.10 / Chapter 2.2 --- Exact Proper Fractional Covers --- p.12 / Chapter 2.2.1 --- Exact Proper Fractional Cover of Quadratic Perturbations --- p.15 / Chapter 3 --- Large Deviations of Sums of Dependent Random Matrices --- p.18 / Chapter 3.1 --- The Matrix Exponential Function and Its Properties --- p.18 / Chapter 3.2 --- Main Theorem --- p.19 / Chapter 4 --- From Large Deviations to ChanceConstrained LMIs --- p.26 / Chapter 4.1 --- General Results --- p.26 / Chapter 4.2 --- Application to ChanceConstrained Quadratically Perturbed Linear Matrix Inequalities --- p.30 / Chapter 4.3 --- Bounding the Matrix Moment Generating Functions --- p.31 / Chapter 4.4 --- Iterative Improvement of the Proposed Approximations --- p.42 / Chapter 5 --- Computational Studies --- p.49 / Chapter 5.1 --- Application to Control Problems --- p.49 / Chapter 5.2 --- Application to Value-at-Risk Portfolio Optimization --- p.57 / Chapter 6 --- ChanceConstrained LMIs with CVaR Risk Measure --- p.64 / Chapter 6.1 --- Matrix CVaR Risk Measure --- p.65 / Chapter 6.2 --- Some Useful Inequalities --- p.68 / Chapter 6.3 --- From Matrix CVaR to ChanceConstrained LMIs --- p.69 / Chapter 6.3.1 --- Bound π¹(A₀, · · · ,A[subscript m]) --- p.70 / Chapter 6.3.2 --- Bound π²(A₀, · · · ,A[subscript m]) --- p.71 / Chapter 6.3.3 --- Bound π³(A₀, · · · ,A[subscript m]) --- p.72 / Chapter 6.3.4 --- Convex Approximation of π[superscript i](A0, · · · ,Am) --- p.73 / Chapter 7 --- Extension to Polynomials Perturbations --- p.75 / Chapter 7.1 --- Decoupling Theory --- p.75 / Chapter 7.2 --- Safe Tractable Approximation by SecondOrder Cone Programming --- p.77 / Chapter 8 --- Construct Uncertainty Set for Chance Constraints --- p.81 / Chapter 8.1 --- Problem Statement --- p.82 / Chapter 8.2 --- Fractional Cover for Quartic Perturbations --- p.83 / Chapter 8.3 --- Probabilistic Guarantees --- p.85 / Chapter 8.3.1 --- Probabilistic Bound Based on Large Deviations --- p.85 / Chapter 8.4 --- The Value of Ω for Bounds --- p.88 / Chapter 8.5 --- Computational Study --- p.89 / Chapter 8.5.1 --- Independent Standard Normal Perturbations --- p.89 / Chapter 8.5.2 --- Independent Bounded Quadratic Perturbations --- p.91 / Chapter 9 --- Conclusion --- p.93 / Bibliography --- p.94

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