Classifying individuals into binary disease categories can be challenging due to complex relationships across different exposures of interest. In this thesis, we investigate three different approaches for disease classification using multiple biomarkers. First, we consider combining information from literature reviews and INTERHEART data set to identify the threshold of ApoB, ApoA1 and the ratio of these two biomarkers to classify individuals at risk of developing myocardial infarction. We develop a Bayesian estimation procedure for this purpose that utilizes the conditional probability distribution of these biomarkers. This method is flexible compared to standard logistic regression approach and allows us to identify a precise threshold of these biomarkers. Second, we consider the problem of disease classification using two dependent biomarkers. An independently identified threshold for this purpose usually leads to a conflicting classification for some individuals. We develop and describe a method of determining the joint threshold of two dependent biomarkers for a disease classification, based on the joint probability distribution function constructed through copulas. This method will allow researchers uniquely classify individuals at risk of developing the disease. Third, we consider the problem of classifying an outcome using a gene and miRNA expression data sets. Linear principal component analysis (PCA) is a widely used approach to reduce the dimension of such data sets and subsequently use it for classification, but many authors suggest using kernel PCA for this purpose. Using real and simulated data sets, we compare these two approaches and assess the performance of components towards genetic data integration for an outcome classification. We conclude that reducing dimensions using linear PCA followed by a logistic regression model for classification seems to be acceptable for this purpose. We also observe that integrating information from multiple data sets using either of these approaches leads to a better performance of an outcome classification. / Thesis / Doctor of Philosophy (PhD)

Flooding is of particular concern in low-lying coastal zones that are prone to impacts from multiple flooding drivers, such as coastal (storm surge and waves), fluvial (excessive river discharge), and pluvial (excessive surface runoff). Failure to account for dependence (and its changes over time) between flood drivers, when dependence exists between them, may lead to underestimation of flood risk and under-design of flood defense measures. Characterizing the dependence between compound flooding drivers in space and across seasons (tropical and extra-tropical), and how this dependence changed over time is essential in this context. In this dissertation, compound flooding potential from all relevant flooding drivers is assessed at 35 locations along the contiguous United States (CONUS) coastline. Different dependence measures are derived and analyzed using observations and state-of-the-art re-analysis data sets. In addition, temporal changes in the extremal dependence are assessed, using a sliding time window approach and possible associations with large-scale climate indices are explored. The effects of changes in dependence and marginal distributions over time between coastal and fluvial flooding drivers are investigated in more detail for a selected case study location. To overcome the computational expense of numerical modeling for flood mapping of large sets of events, a framework is introduced based on hybrid statistical modeling and one-dimensional hydraulic modeling combined with a flood inundation tool capable of propagating spatially variable along-river water surface elevations inland. The framework to delineate the flood transition zone is implemented for the Potomac River and different flood scenarios are analyzed to assess how different combinations of coastal water levels and river discharge modulate the flood hazard (specifically flood depth and extent). Overall, the findings provide new insights into characterizing compound flooding potential, its changes in space and time and how incorporating flood driver dependencies affects flood hazard.

Les copulas archimédiennes hiérarchiques ont récemment gagné en intérêt puisqu’elles généralisent la famille de copules archimédiennes, car elles introduisent une asymétrie partielle. Des algorithmes d’échantillonnages et des méthodes ont largement été développés pour de telles copules. Néanmoins, concernant l’estimation par maximum de vraisemblance et les tests d’adéquations, il est important d’avoir à disposition la densité de ces variables aléatoires. Ce travail remplie ce manque. Après une courte introduction aux copules et aux copules archimédiennes hiérarchiques, une équation générale sur les dérivées des noeuds et générateurs internes apparaissant dans la densité des copules archimédiennes hiérarchique. sera dérivée. Il en suit une formule tractable pour la densité des copules archimédiennes hiérarchiques. Des exemples incluant les familles archimédiennes usuelles ainsi que leur transformations sont présentés. De plus, une méthode numérique efficiente pour évaluer le logarithme des densités est présentée. / Nested Archimedean copulas recently gained interest since they generalize the well-known class of Archimedean copulas to allow for partial asymmetry. Sampling algorithms and strategies have been well investigated for nested Archimedean copulas. However, for likelihood based inference such as estimation or goodness-of-fit testing it is important to have the density. The present work fills this gap. After a short introduction on copula and nested Archimedean copulas, a general formula for the derivatives of the nodes and inner generators appearing in nested Archimedean copulas is developed. This leads to a tractable formula for the density of nested Archimedean copulas. Various examples including famous Archimedean families and transformations of such are given. Furthermore, a numerically efficient way to evaluate the log-density is presented.

Les copulas archimédiennes hiérarchiques ont récemment gagné en intérêt puisqu’elles généralisent la famille de copules archimédiennes, car elles introduisent une asymétrie partielle. Des algorithmes d’échantillonnages et des méthodes ont largement été développés pour de telles copules. Néanmoins, concernant l’estimation par maximum de vraisemblance et les tests d’adéquations, il est important d’avoir à disposition la densité de ces variables aléatoires. Ce travail remplie ce manque. Après une courte introduction aux copules et aux copules archimédiennes hiérarchiques, une équation générale sur les dérivées des noeuds et générateurs internes apparaissant dans la densité des copules archimédiennes hiérarchique. sera dérivée. Il en suit une formule tractable pour la densité des copules archimédiennes hiérarchiques. Des exemples incluant les familles archimédiennes usuelles ainsi que leur transformations sont présentés. De plus, une méthode numérique efficiente pour évaluer le logarithme des densités est présentée. / Nested Archimedean copulas recently gained interest since they generalize the well-known class of Archimedean copulas to allow for partial asymmetry. Sampling algorithms and strategies have been well investigated for nested Archimedean copulas. However, for likelihood based inference such as estimation or goodness-of-fit testing it is important to have the density. The present work fills this gap. After a short introduction on copula and nested Archimedean copulas, a general formula for the derivatives of the nodes and inner generators appearing in nested Archimedean copulas is developed. This leads to a tractable formula for the density of nested Archimedean copulas. Various examples including famous Archimedean families and transformations of such are given. Furthermore, a numerically efficient way to evaluate the log-density is presented.

In 2005 Attilio Meucci presented his article Beyond Black-Litterman: Views on Non-Normal Markets which introduces the copula opinion pooling approach using generic non-normal market assumptions. Copulas and opinion pooling are used to express views on the market which provides a posterior market distribution that smoothly blends an arbitrarily distributed market prior distribution with arbitrarily chosen views. This thesis explains how to use this method in practice and investigates its performance in different investment situations. The method is tested on three portfolios, each showing some different feature. The conclusions that can be drawn are e.g. that the method can be used in many different investment situations in many different ways, implementation and calculations can be made within a few seconds for a large data set and the method could be useful for portfolio managers using mathematical methods. The presented examples together with the method generate reasonable results.

In my thesis I~ deal with the design, implementation and testing of the advanced parallel Estimation of Distribution Algorithm (EDA) utilizing copula theory to create a~ probabilistic model. A~new population is created by the process of sampling the joint distribution function, which models the current distribution of the subpopulation of promising individuals . The usage of copulas increases the efficiency of the learning process and sampling the probabilistic model. It can be separated into mutually independent marginal distributions and the copula , which represents the correlations between the variables of the solved problem. This concept initiated the usage of the parallel island architecture , in which the migration of probabilistic models belonging to individual islands ' subpopulations was used instead of the migration of individuals . The statistical tests used in the comparison of the proposed algorithm ( mCEDA = migrating Copula - based Estimation of Distribution Algorithm ) and the algorithms of other authors confirmed the effectiveness of the proposed concept .

In order to ensure solvency, financial institutions must evaluate their credit risk exposure and determine how much economic capital is required to hold as a cushion. This thesis compares three factor models, namely Asymptotic Single Risk Factor (“ASRF”), Inter-sector and Intra-sector factor models and evaluates how their different characteristics affect the economic capital outcomes. The thesis also investigates how these outcomes are affected when assuming asset dependency through a Student's-$t$ copula. Focus will also be put on how different types and levels of asset correlation affect the models' credit risk results. We use a fictitious loan portfolio consisting of 138 Swedish firms with equity data from between 2007 and 2019 in order to calculate asset correlations and economic capital. Our main findings are that the asset correlations severely impact the outcomes of the credit risk models and that practitioners must calibrate and stress test their models regularly with respect to how correlations vary between different firms. The thesis also finds that using copulas for credit portfolios provides more conservative risk outcomes but makes the models less sensitive to correlation level input. / För att finansiella institutioner ska försäkra sig om att vara solventa måste de utvärdera sin exponering mot kreditrisk och därmed avgöra hur mycket ekonomiskt kapital de behöver hålla som buffert. Denna uppsats jämför tre faktormodeller vid namn Asymptotic Systematic Risk Factor (“ASRF”), Inter-sektor, och Intra-sektor med syfte att undersöka hur deras olika karaktärsdrag påverkar estimaten för ekonomiskt kapital. Vi utvärderar även hur utfallen påverkas av införandet av copula-beroende mellan portföljtillgångarna. Fokus kommer även att läggas på hur olika typer och nivåer av korrelation mellan bolag påverkar de olika modellernas kreditriskutfall. Vi använder oss av en fiktiv låneportfölj bestående av 138 svenska bolag med aktieprisdata mellan 2007 och 2019 för att beräkna korrelationer och ekonomiskt kapital. Uppsatsens främsta resultat pekar på att korrelationerna har en väldigt stor påverkan på det ekonomiska kapitalet och att analytiker rekommenderas att kontinuerligt kalibrera och stresstesta sina modeller med avseende på hur korrelationerna kan skilja sig mellan olika bolag. Vi fann även att copula-beroende gav mycket mer konservativa utfall, det vill säga ett högre ekonomiskt kapital, men var mindre känslig för korrelationsnivåerna mellan bolagen i portföljen.

In den letzten Jahren gab es ein drastisches Wachstum in verfügbaren Finanzdaten. Finanzmärkte haben starke und oft nicht ganz vorhersagbare Änderungen ihrer Dynamik erlebt. Diese Tendenz hat dazu geführt, dass die Methoden der Risikomodellierung sowohl das Problem der hohen Dimensionalität als auch dynamische nicht Gaußsche Strukturen behandeln müssen. Das Ziel dieser Dissertation ist es, Methoden der Risikomodellierung vorzuschlagen, die gleichzeitig Reduzierung der Dimensionalität und dynamische Struktur in drei Anwendungen erlauben: 1) Asset Allocation und Hedging, 2) stochastische Modellierung von multivariaten Prozessen, 2) Messung der systemischen Risiken. Die vorgeschlagenen Methoden demonstrieren gute Ergebnisse im Vergleich mit den existierenden Methoden der Risikomodellierung und führen neue Verfahren zur Erkennung der extremen Risiken und Anomalien auf Finanzmärkten sowie zur deren Management. / Over the recent years, there have been a significant increase in financial data availability. On the other hand, financial markets have experienced sharp and often unforeseen changes in their dynamics. This tendency has caused the need for risk modeling approaches addressing both high dimensionality problem and accustoming for dynamic non Gaussian structure. The primary aim of this dissertation is to propose several risk modeling approaches which allow for simultaneous dimension reduction and dynamic structures in three setups: 1) asset allocation and hedging, 2) stochastic surface modeling and 3) systemic risk determination. Proposed models demonstrate good performance when compared to existing approaches for risk modeling and introduce new flexible ways to detect extreme risks and anomalies on financial markets as well as methods for their modeling and management.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Chih-Hang Wu / In recent years the stock markets have shown tremendous volatility with significant spikes and drops in the stock prices. Within the past decade, there have been numerous jumps in the market; one key example was on September 17, 2001 when the Dow industrial average dropped 684 points following the 9-11 attacks on the United States. These evident jumps in the markets show the inaccuracy of the Black-Scholes model for pricing options. Merton provided the first research to appease this problem in 1976 when he extended the Black-Scholes model to include jumps in the market. In recent years, Kou has shown that the distribution of the jump sizes used in Merton’s model does not efficiently model the actual movements of the markets. Consequently, Kou modified Merton’s model changing the jump size distribution from a normal distribution to the double exponential distribution. Kou’s research utilizes mathematical equations to estimate the value of an American put option where the underlying stocks follow a jump-diffusion process. The research contained within this thesis extends on Kou’s research using Monte Carlo simulation (MCS) coupled with least-squares regression to price this type of American option. Utilizing MCS provides a continuous exercise and pricing region which is a distinct difference, and advantage, between MCS and other analytical techniques. The aim of this research is to investigate whether or not MCS is an efficient means to pricing American put options where the underlying stock undergoes a jump-diffusion process. This thesis also extends the simulation to utilize copulas in the pricing of baskets, which contains several of the aforementioned type of American options. The use of copulas creates a joint distribution from two independent distributions and provides an efficient means of modeling multiple options and the correlation between them. The research contained within this thesis shows that MCS provides a means of accurately pricing American put options where the underlying stock follows a jump-diffusion. It also shows that it can be extended to use copulas to price baskets of options with jump-diffusion. Numerical examples are presented for both portions to exemplify the excellent results obtained by using MCS for pricing options in both single dimension problems as well as multidimensional problems.

Understanding financial asset return correlation is a key facet in asset allocation and investor’s portfolio optimization strategy. For the last decades, several studies have investigated this relationship between stock and bond returns. But, fewer studies have dealt with multi-asset return dynamics. While initial literature attempted to understand the fundamental pattern of comovements, later studies model the economic state variables influencing such time-varying comovements of primarily stock and bond returns. Research widely acknowledges that return distributions of financial assets are non-normal. When the joint distributions of the asset returns follow a non-elliptical structure, linear correlation fails to provide sufficient information of their dependence structure. In particular two issues arise from this existing empirical evidence. The first is to propose a more reliable alternative density specification for a higher-dimensional case. The second is to formulate a measure of the variables’ dependence structure which is more instructive than linear correlation. In this work I use a time-varying conditional multivariate elliptical and non-elliptical copula to examine the return comovements of three different asset classes: financial assets, commodities and real estate in the US market. I establish the following stylized facts about asset return comovements. First, the static measures of asset return comovements overestimate the asset return comovements in the economic expansion phase, while underestimating it in the periods of economic contraction. Second, Student t-copulas outperform both elliptical and non-elliptical copula models, thus confirming the ii dominance of Student t-distribution. Third, findings show a significant increase in asset return comovements post August 2007 subprime crisis ... [cont.].