by Lai Siu Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1989. / Bibliography: leaves 50-52.

We consider several problems motivated by issues faced in the mining industry. In recent years, it has become clear that mines have substantial tail risk in the form of environmental disasters, and this tail risk is not incorporated into common pricing and risk models. However, data sets of the extremal climate behavior that drive this risk are very small, and generally inadequate for properly estimating the tail behavior. We propose a data-driven methodology that comes up with reasonable worst-case scenarios, given the data size constraints, and we incorporate this into a real options based model for the valuation of mines. We propose several different iterations of the model, to allow the end-user to choose the degree to which they wish to specify the financial consequences of the disaster scenario. Next, in order to perform a risk analysis on a portfolio of mines, we propose a method of estimating the correlation structure of high-dimensional max-stable processes. Using the techniques of (Liu Et al, 2017) to map the relationship between normal correlations and max-stable correlations, we can then use techniques inspired by (Bickel et al, 2008, Liu et al, 2014, Rothman et al, 2009) to estimate the underlying correlation matrix, while preserving a sparse, positive-definite structure. The correlation matrices are then used in the calculation of model-robust risk metrics (VaR, CVAR) using the the Sample-Out-of-Sample methodology (Blanchet and Kang, 2017). We conclude with several new techniques that were developed in the field of robust performance analysis, that while not directly applied to mining, were motivated by our studies into distributionally robust optimization in order to address these problems.

博士論文著重研究關於多項式優化的理論，並討論其在風險管理及統計分析中的應用。我們主要研究的對象乃為在控制理論和穩健優化中常見的所謂S 引理。原始的S 引理最早由Yakubovich 所引入。它給出一個二吹多項式在另一個二吹多項式的非負域上為非負的等價條件。在本論文中，我們把S 引理推廣到一元高吹多項式。由於S 引理與穩健優化密切相關，所以我們的結果可廣泛應用於風險管理及統計分析，包括估算在高階矩約束下的非線性風險量度問題，以及利用半正定規劃來計算同時置信區域帶等重要課題。同時，在相關章節的末段，我們以數值實驗結果來引證有關的新理論的有效性和應用前景。 / In this thesis we study some structural results in polynomial optimization, with an emphasis paid to the applications from risk management problems and estimations in statistical analysis. The key underlying method being studied is related to the so-called S-lemma in control theory and robust optimization. The original S-lemma was developed by Yakubovich, which states an equivalent condition for a quadratic polynomial to be non-negative over the non-negative domain of other quadratic polynomial(s). In this thesis, we extend the S-Lemma to univariate polynomials of any degree. Since robust optimization has a strong connection to the S-Lemma, our results lead to many applications in risk management and statistical analysis, including estimating certain nonlinear risk measures under moment bound constraints, and an SDP formulation for simultaneous confidence bands. Numerical experiments are conducted and presented to illustrate the effectiveness of the methods. / Detailed summary in vernacular field only. / Wong, Man Hong. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 134-147). / Abstract also in Chinese. / Abstract --- p.i / 摘要 --- p.ii / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Meeting the S-Lemma --- p.5 / Chapter 3 --- A strongly robust formulation --- p.13 / Chapter 3.1 --- A more practical extension for robust optimization --- p.13 / Chapter 3.1.1 --- Motivation from modeling aspect --- p.13 / Chapter 3.1.2 --- Discussion of a more robust condition --- p.15 / Chapter 4 --- Theoretical developments --- p.19 / Chapter 4.1 --- Definition of several order relations --- p.19 / Chapter 4.2 --- S-Lemma with a single condition g(x)≥0 --- p.20 / Chapter 5 --- Confidence bands in polynomial regression --- p.47 / Chapter 5.1 --- An introduction --- p.47 / Chapter 5.1.1 --- A review on robust optimization, nonnegative polynomials and SDP --- p.49 / Chapter 5.1.2 --- A review on the confidence bands --- p.50 / Chapter 5.1.3 --- Our contribution --- p.51 / Chapter 5.2 --- Some preliminaries on optimization --- p.52 / Chapter 5.2.1 --- Robust optimization --- p.52 / Chapter 5.2.2 --- Semidefinite programming and LMIs --- p.53 / Chapter 5.2.3 --- Nonnegative polynomials with SDP --- p.55 / Chapter 5.3 --- Some preliminaries on linear regression and confidence region --- p.59 / Chapter 5.4 --- Optimization approach to the confidence bands construction --- p.63 / Chapter 5.5 --- Numerical experiments --- p.66 / Chapter 5.5.1 --- Linear regression example --- p.66 / Chapter 5.5.2 --- Polynomial regression example --- p.67 / Chapter 5.6 --- Conclusion --- p.70 / Chapter 6 --- Moment bound of nonlinear risk measures --- p.72 / Chapter 6.1 --- Introduction --- p.72 / Chapter 6.1.1 --- Motivation --- p.72 / Chapter 6.1.2 --- Robustness and moment bounds --- p.74 / Chapter 6.1.3 --- Literature review in general --- p.76 / Chapter 6.1.4 --- More literature review in actuarial science --- p.78 / Chapter 6.1.5 --- Our contribution --- p.79 / Chapter 6.2 --- Methodological fundamentals behind the moment bounds --- p.81 / Chapter 6.2.1 --- Dual formulations, duality and tight bounds --- p.82 / Chapter 6.2.2 --- SDP and LMIs for some dual problems --- p.84 / Chapter 6.3 --- Worst expectation and worst risk measures on annuity payments --- p.87 / Chapter 6.3.1 --- The worst mortgage payments --- p.88 / Chapter 6.3.2 --- The worst probability of repayment failure --- p.89 / Chapter 6.3.3 --- The worst expected downside risk of exceeding the threshold --- p.90 / Chapter 6.4 --- Numerical examples for risk management --- p.94 / Chapter 6.4.1 --- A mortgage example --- p.94 / Chapter 6.4.2 --- An annuity example --- p.97 / Chapter 6.5 --- Conclusion --- p.100 / Chapter 7 --- Computing distributional robust probability functions --- p.101 / Chapter 7.1 --- Distributional robust function with a single random variable --- p.105 / Chapter 7.2 --- Moment bound of joint probability --- p.108 / Chapter 7.2.1 --- Constraint (7.5) in LMIs --- p.112 / Chapter 7.2.2 --- Constraint (7.6) in LMIs --- p.112 / Chapter 7.2.3 --- Constraint (7.7) in LMIs --- p.116 / Chapter 7.3 --- Several model extensions --- p.119 / Chapter 7.3.1 --- Moment bound of probability of union events --- p.119 / Chapter 7.3.2 --- The variety of domain of x --- p.120 / Chapter 7.3.3 --- Higher moments incorporated --- p.123 / Chapter 7.4 --- Applications of the moment bound --- p.124 / Chapter 7.4.1 --- The Riemann integrable set approximation --- p.124 / Chapter 7.4.2 --- Worst-case simultaneous VaR --- p.124 / Chapter 7.5 --- Conclusion --- p.126 / Chapter 8 --- Concluding Remarks and Future Directions --- p.127 / Chapter A --- Nonnegative univariate polynomials --- p.129 / Chapter B --- First and second moment of (7.2) --- p.131 / Bibliography --- p.134

With the increasing use of geometry scanners to create 3D models, there is a rising need for fast and robust mesh smoothing to remove inevitable noise in the measurements. While most previous work has favored diffusion-based iterative techniques for feature-preserving smoothing, we propose a radically different approach, based on robust statistics and local first-order predictors of the surface. The robustness of our local estimates allows us to derive a non-iterative feature-preserving filtering technique applicable to arbitrary "triangle soups". We demonstrate its simplicity of implementation and its efficiency, which make it an excellent solution for smoothing large, noisy, and non-manifold meshes. / Singapore-MIT Alliance (SMA)

Thesis (Ph. D.)--University of Hong Kong, 2001. / Includes bibliographical references.

published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

In longitudinal and spatio-temporal data analysis, repeated measurements from a subject can be either regional- or temporal-dependent. The correct specification of the within-subject covariance matrix cultivates an efficient estimation for mean regression coefficients. In this thesis, robust estimation for the mean and covariance jointly for the regression model of longitudinal data within the framework of generalized estimating equations (GEE) is developed. The proposed approach integrates the robust method and joint mean-covariance regression modeling. Robust generalized estimating equations using bounded scores and leverage-based weights are employed for the mean and covariance to achieve robustness against outliers. The resulting estimators are shown to be consistent and asymptotically normally distributed. Robust variable selection method in a joint mean and covariance model is considered, by proposing a set of penalized robust generalized estimating equations to estimate simultaneously the mean regression coefficients, the generalized autoregressive coefficients and innovation variances introduced by the modified Cholesky decomposition. The set of estimating equations select important covariate variables in both mean and covariance models together with the estimating procedure. Under some regularity conditions, the oracle property of the proposed robust variable selection method is developed. For these two robust joint mean and covariance models, simulation studies and a hormone data set analysis are carried out to assess and illustrate the small sample performance, which show that the proposed methods perform favorably by combining the robustifying and penalized estimating techniques together in the joint mean and covariance model. Capturing dynamic change of time-varying correlation structure is both interesting and scientifically important in spatio-temporal data analysis. The time-varying empirical estimator of the spatial correlation matrix is approximated by groups of selected basis matrices which represent substructures of the correlation matrix. After projecting the correlation structure matrix onto the space spanned by basis matrices, varying-coefficient model selection and estimation for signals associated with relevant basis matrices are incorporated. The unique feature of the proposed model and estimation is that time-dependent local region signals can be detected by the proposed penalized objective function. In theory, model selection consistency on detecting local signals is provided. The proposed method is illustrated through simulation studies and a functional magnetic resonance imaging (fMRI) data set from an attention deficit hyperactivity disorder (ADHD) study. / published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

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