Magister Scientiae - MSc / The aim of this research project was to estimate parameters for the distribution of annual maximum flood levels for the Zambezi River at Katima Mulilo. The estimation of parameters was done by using the maximum likelihood method. The study aimed to explore data of the Zambezi's annual maximum flood heights at Katima Mulilo by means of fitting the Gumbel, Weibull and the generalized extreme value distributions and evaluated their goodness of fit. / South Africa

This thesis reviewed various heavy tailed distributions and Extreme Value Theory (EVT) to estimate the catastrophic losses simulated from Florida Public Hurricane Loss Projection Model (FPHLPM). We have compared risk measures such as Probable Maximum Loss (PML) and Tail Value at Risk (TVaR) of the selected distributions with empirical estimation to capture the characteristics of the loss data as well as its tail distribution. Generalized Pareto Distribution (GPD) is the main focus for modeling the tail losses in this application. We found that the hurricane loss data generated from FPHLPM were consistent with historical losses and were not as heavy as expected. The tail of the stochastic annual maximum losses can be explained by an exponential distribution. This thesis also touched on the philosophical implication of small probability, high impact events such as Black Swan and discussed the limitations of quantifying catastrophic losses for future inference using statistical methods.

This study attempts to quantify the shocks to a banking network and analyze the transfer of shocks through the network. We consider two sources of shocks: external shocks due to market and macroeconomic factors which impact the entire banking system, and idiosyncratic shocks due to failure of a single bank. The external shocks will be estimated by using two methods: (i) non-parametric simulation of the time series of shocks that occurred to the banking system in the past, and (ii) using the extreme value theory (EVT) to model the tail part of the shocks. The external shocks we considered in this study are due to exchange rate and treasury bill rate volatility. Also, an ARMA/GARCH model is used to extract iid residuals for this purpose. In the next step, the probability of the failure of banks in the system is studied by using Monte Carlo simulation. We calibrate the model such that the network resembles the Canadian banking system.

In traditional classification problems, all classes in the test set are assumed to also occur in the training set, also referred to as the closed-set assumption. However, in practice, new classes may occur in the test set, which reduces the performance of machine learning models trained under the closed-set assumption. Machine learning models should be able to accurately classify instances of classes known during training while concurrently recognizing instances of previously unseen classes (also called the open set assumption). This open set assumption is motivated by real world applications of classifiers wherein its improbable that sufficient data can be collected a priori on all possible classes to reliably train for them. For example, motivated by the DARPA WASH project at WPI, a disease classifier trained on data collected prior to the outbreak of COVID-19 might erroneously diagnose patients with the flu rather than the novel coronavirus. State-of-the-art open set methods based on the Extreme Value Theory (EVT) fail to adequately model class distributions with unequal variances. We propose the Variational Open-Set Recognition (VOSR) model that leverages all class-belongingness probabilities to reject unknown instances. To realize the VOSR model, we design a novel Multi-Modal Variational Autoencoder (MMVAE) that learns well-separated Gaussian Mixture distributions with equal variances in its latent representation. During training, VOSR maps instances of known classes to high-probability regions of class-specific components. By enforcing a large distance between these latent components during training, VOSR then assumes unknown data lies in the low-probability space between components and uses a multivariate form of Extreme Value Theory to reject unknown instances. Our VOSR framework outperforms state-of-the-art open set classification methods with a 15% F1 score increase on a variety of benchmark datasets.

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Dissertation submitted for Masters of Science degree in Mathematical Statistics in the FacultyofScience, SchoolofStatisticsandActuarialScience, University of the Witwatersrand Johannesburg, January 2018 / This dissertation focuses on demonstrating the use of extreme value theory in modelling temperature in South Africa. The purpose of modelling temperature is to investigate the frequency of occurrences of extremely low and extremely high temperatures and how they influence the demand of electricity over time. The data comprise a time series of average hourly temperatures that are collected by the South African Weather Service over the period 2000−2010 and supplied by Eskom. The generalized extreme value distribution (GEVD) for r largest order statistics is fitted to the average maximum daily temperature (non-winter season) using the maximum likelihood estimation method and used to estimate extreme high temperatures which result in high demand of electricity due to use of cooling systems. The estimation of the shape parameter reveals evidence that the Weibull family of distributions is an appropriate fit to the data. A frequency analysis of extreme temperatures is carried out and the results show that most of the extreme temperatures are experienced during the months January, February, November and December of each year. The generalized Pareto distribution (GPD) is firstly used for modelling the average minimum daily temperatures for the period January 2000 to August 2010. A penalized regression cubic smoothing spline is used as a time varying threshold. We then extract excessesabovethecubicregressionsmoothingsplineandfitanon-parametricmixturemodel to get a sufficiently high threshold. The data exhibit evidence of short-range dependence and high seasonality which lead to the declustering of the excesses above the threshold and fit the GPD to cluster maxima. The estimate of the shape parameter shows that the Weibullfamilyofdistributionsisappropriateinmodellingtheuppertailofthedistribution. The stationary GPD and the piecewise linear regression models are used in modelling the influence of temperature above the reference point of 22◦C on the demand of electricity. The stationary and non-stationary point process models are fitted and used in determining the frequency of occurrence of extremely high temperatures. The orthogonal and the reparameterizationapproachesofdeterminingthefrequencyandintensityofextremeshave i been used to establish that, extremely hot days occur in frequencies of 21 and 16 days per annum, respectively. For the fact that temperature is established as a major driver of electricity demand, this dissertation is relevant to the system operators, planners and decision makers in Eskom and most of the utility and engineering companies. Our results are furtherusefultoEskomsinceitisduringthenon-winterperiodthattheyplanformaintenance of their power plants. Modelling temperature is important for the South African economy since electricity sector is considered as one of the most weather sensitive sectors of the economy. Over and above, the modelling approaches that are presented in this dissertation are relevant for modelling heat waves which impose several impacts on energy, economy and health of our citizens. / XL2018

This dissertation implements an optimal trading strategy based on the machine learning method and extreme value theory (EVT) to obtain an excess return on investments in the capital market. The trading strategy outperforms the benchmark S&P 500 index with higher returns and lower volatility through effective market timing. In addition, this dissertation starts by modeling the market tail risk using the EVT and reinforcement learning methods, distinguishing from the traditional value at risk method. In this dissertation, I used EVT to extract the characteristics of the tail risk, which are inputs for reinforcement learning. This process is proved to be effective in market timing, and the trading strategy could avoid market crash and achieve a long-term excess return. In sum, this study has several contributions. First, this study takes a new method to analyze stock price (in this dissertation, I use the S&P 500 index as a stock). I combined the EVT and reinforcement learning to study the price tail risk and predict stock crash efficiently, which is a new method for tail risk research. Thus, I can predict the stock crash or provide the probability of risk, and then, the trading strategy can be built. The second contribution is that this dissertation provides a dynamic market timing trading strategy, which can significantly outperform the market index with a lower volatility and a higher Sharpe ratio. Moreover, the dynamic trading process can provide investors an intuitive sense on the stock market and help in decision-making. Third, the success of the strategy shows that the combination of EVT and reinforcement learning can predict the stock crash very well, which is a great improvement on the extreme event study and deserves further study. / Business Administration/Finance

Classical models for multivariate or spatial extremes are mainly based upon the asymptotically justified max-stable or generalized Pareto processes. These models are suitable when asymptotic dependence is present. However, recent environmental data applications suggest that asymptotic independence is equally important. Therefore, development of flexible subasymptotic models is in pressing need. This dissertation consists of four major contributions to subasymptotic modeling of multivariate and spatial extremes. Firstly, the dissertation proposes a new spatial copula model for extremes based on the multivariate generalized hyperbolic distribution. The extremal dependence of this distribution is revisited and a corrected theoretical description is provided. Secondly, the dissertation thoroughly investigates the extremal dependence of stochastic processes driven by exponential-tailed Lévy noise. It shows that the discrete approximation models, which are linear transformations of a random vector with independent components, bridge asymptotic independence and asymptotic dependence in a novel way, whilst the exact stochastic processes exhibit only asymptotic independence. Thirdly, the dissertation explores two different notions of optimal prediction for extremes, and compares the classical linear kriging predictor and the conditional mean predictor for certain non-Gaussian models. Finally, the dissertation proposes a multivariate skew-elliptical link model for correlated highly-imbalanced (extreme) binary responses, and shows that the regression coefficients have a closed-form unified skew-elliptical posterior with an elliptical prior.

My dissertation advances a solution to what I have called the problem of opposites in Heraclitus. The problem is this: Heraclitus often juxtaposes pairs of opposites, but the opposites he cites seem to be of many different kinds. How are we to explain this feature of the fragments? The default method of solution for interpreters has been to find a single thesis under which to subsume all the divergent examples of opposites. Some such theses are as follows: opposites are identical (Aristotle, Barnes), opposites are essentially connected (Kirk), opposites are transformationally equivalent (Graham), identical things can have opposite significances in different situations (Osborne). The main problem all these solutions face is that each is only able to make sense of some of the examples of opposition in Heraclitus, while ignoring or downplaying the significance of others. In order to solve this problem, I offer an interpretation on which Heraclitus was advancing multiple opposites theses, each of which contains interesting, philosophical content. The theses are as follows: The Transformation Thesis: the world contains opposing stuffs which transform into one another in such a way that they are transformationally equivalent, and therefore unified. The Dependence Thesis: objects are ontologically dependent for their existence (i.e. that they exist) and their identity (i.e. their ‘nature’ or φύσις) on opposing, yet essential properties which are necessarily inherent in them. The Value Thesis: it is possible for one and the same object to have opposing values (i.e. to be both objectively good and objectively bad). But why would Heraclitus promote multiple opposites theses? On my interpretation Heraclitus was responding to his Ionian predecessors who treated opposites as explanatory principles. Heraclitus seems to be saying that opposites are not explanatory principles since opposites themselves need to be explained. Hence the opposites are explananda, for Heraclitus, and the three theses are his explanantia. / Dissertation / Doctor of Philosophy (PhD) / In this dissertation I offer a new interpretation of an ancient Greek philosopher named Heraclitus who stands at the beginning of the timeline of Western philosophy (around 500BC). It has often been thought that Heraclitus had something interesting to say about opposites (e.g. hot and cold, up and down). Most scholars think that Heraclitus intended to say that opposites are connected; that is, hot is connected to cold since we cannot think of hot without its opposite, cold. I argue in this dissertation that this interpretation and other, alternative interpretations, fail to make good sense of what Heraclitus said about opposites. Rather, I argue that Heraclitus was treating opposites (e.g. hot and cold, up and down) as philosophical problems that need to be explained in order to be solved.

Med narrativ ges en mer begriplig och engagerande information av transparens än vad den traditionella numeriska transparensen bidrar med. Istället för att använda olika certifieringar, siffror och grafer som belyser transparensen består narrativ istället av berättelser och visualiseringar som öppnar upp till dialog med konsumenterna. Tidigare forskning säger att narrativ transparens är det som konsumenter eftersträvar, men hur gröna konsumenter skapar ett värde kring mottagandet av informationen är desto mindre uppmärksammat. Med det ökade gröna konsumtionssamhället ställer gröna konsumenter mer krav på modeföretagen. Därmed blir ökad transparens i värdekedjan en fördelaktig väg för modeföretagen att gå för att förbättra sin legitimitet gentemot intressenterna. Consumption Value Theory menar på att värdeskapande och tillfredsställande av ett visst behov är det som utgör grunden för konsumtion. Det blir därför viktigt för modeföretag att ha förståelse för konsumenters upplevda värde, för att kunna erhålla en fördel på marknaden. I denna studie följer därmed en undersökning om vilket värde narrativ transparensinformation har för gröna konsumenter i en modekontext. Med hjälp av semistrukturerade intervjuer faställdes en slutsats om att alla fem värden inom CVT-modellen går att kartlägga i samband med uppvisande av exempel på narrativ transparensinformation. En ytterligare slutsats som fastställdes var att narrativ bidrar till upplevelse av värde för gröna konsumenter genom att modeföretagen presenterar sin transparens på ett pålitligt, sammanhangsskapande, känslomässigt engagerande samt nyfikenhets- och kunskapsväckande sätt. Gröna konsumenters värden skapas dessutom av en association utifrån en specifik omständighet samt via social interaktion med andra individer eller grupper. En tydlig och tillräcklig narrativ transparensinformation motverkar att skepsis uppstår hos konsumenterna. Modeföretagen kan framföra sin transparens genom narrativ för att kunna nå ut till den gröna konsumenten på ett mer effektfullt sätt. / Narrative provides more comprehensible and engaging information of transparency than traditional numerical transparency. Instead of using different certifications, numbers and graphs that highlight transparency, narratives consist of stories and visualizations that open up for dialogue with the consumers. Previous research implies that narrative transparency is what consumers strive for, but how green consumers create value around this type of information has been less focused on. With the increased green consumer society, green consumers place more demands on fashion companies. Thus, increased transparency in the value chain becomes an advantageous way for fashion companies to improve their legitimacy in relation to their customers. Consumption Value Theory implies that the perception of value and the satisfaction of a certain need creates the basis for consumption. Therefore, it becomes important for fashion companies to have an understanding of consumers' perceived value, in order to obtain an advantage in the industry. This study follows an examination of the value of narrative transparency information for green consumers in a fashion context. With the help of semi-structured interviews, a conclusion was proposed that all five values ??within the CVT model can be located in relation to the presentation of narrative transparency information. Further, a conclusion was established that narratives contribute to the experience of value for green consumers, if the fashion companies present their transparency in a reliable, context-creating, emotionally engaging and curious and knowledge-inspiring way. The values ??of green consumers are also created by an association based on a specific circumstance and through social interaction with other individuals or groups. A clear and sufficient narrative transparency information counteracts that skepticism arises among consumers. Fashion companies can express their transparency through narrative in order to reach the green consumer in a more effective way.