Doctoral Degrees (Mathematical Statistics and Actuarial Science)

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Browsing Doctoral Degrees (Mathematical Statistics and Actuarial Science) by Author "Chikobvu, Delson"

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    Continuous-time Markov modelling of the effects of treatment regimens on HIV/AIDS immunology and virology
    (University of the Free State, 2019) Shoko, Claris; Chikobvu, Delson
    As the Human immunodeficiency virus (HIV) enters the human body, its main target is the CD4+ cell, which it turns into a factory that produces millions of other HIV particles, thus compromising the immune system and resulting in opportunistic infections, for example tuberculosis (TB). Combination Anti-retroviral therapy (cART) has become the standard of care for patients with HIV infection and has led to the reduction in acquired immunodeficiency syndrome (AIDS) related morbidity and mortality, an increase in CD4+ cell counts and a decrease in viral load count to undetectable levels. In modelling HIV/AIDS progression in patients, researchers mostly deal with either viral load only or CD4+cellcountsonly, as they expect these two variables to be collinear. The purpose of this study is to fit a continuous-time Markov model that best describes mortality of HIV infected patients on cART by eventually including both CD4+ cell counts monitoring and viral load monitoring in a single model after treating for collinearity of these variables using the Principal Component approach. Acohortof320HIVinfectedpatientsoncARTfollowedupat a Wellness Clinic in Bela Bela, South Africa, is used in this thesis. These patients are administered with a triple therapy of two nucleoside reverse transcriptase inhibitor (NRTIs) and one non-nucleoside reverse transcriptase inhibitor (NNRTI). The thesis is divided into five sections. In the first section, a continuous-time homogeneous Markov model based on CD4+ cell count states is fitted. The model is used to analyse the effects of tuberculosis (TB) co-infection on the immunologic progression of HIV/AIDS patients on cART. TB co-infection was of interest because it is an opportunistic infection that takes advantage of the compromised immune system. Results from this section showed that once TB is diagnosed prior to treatment initiation and managed, mortality rates are reduced. However, if TB is diagnosed during the course of treatment, it increases the rates of immune deterioration in patients, leading to high rates of mortality. Therefore, this section proposes the need for routine TB screening before treatment initiation and a tevery stage of the follow up period, to avoid loss of lives. The goal of cART is not only to boost the immune system but also to suppress the viral load to undetectable levels. Thus, in the second section, a non-homogeneous continuous-time Markov model based on viral load states is fitted. This model helped in revealing possibilities of viral rebound among patient son cART. Although there were no significant gender differences on HIV/AIDS virology, the model explained the progression of patients better than the model based on CD4+ cell count fitted in the first section. In the third section, determinants of viral rebound are analysed. Viral rebound was notable mainly after patients had attained a viral load suppressed to the levels between 50 copies/mL and 10 000 copies/mL. The major attributes of viral rebound were non-adherence, lactic acid, resistance to treatment, and different combination therapy such as AZT-3TC-LPV/r and FTC-TDF-EFV. This section suggests the need to closely monitor HIV patients to ensure attainment of undetectable viral load (below 50 copies/mL) during the first six months of treatment uptake, as this reduces chances of viral rebound, leading to life gain by HIV/AIDS patients. The fourth section compares the use of viral load count and CD4+cell count in monitoring HIV/AIDS disease progression on patients receiving cART in order to establish the superiority of viral load over CD4+ cell count. This was done by fitting two separate models, one for CD4+ cell count states and the other one for viral load states. Comparison of the fitted models were based on percentage prevalence plots for the fitted model and for the observed data and likelihood ratio tests. The test confirmed that viral load monitoring is superior compared to CD4+cell count monitoring. Viral load monitoring is very good at detecting virologic failure, thereby avoiding unnecessary switches of treatment lines. However, this section suggests the use of both CD4+cellcount monitoring and viral load monitoring because CD4+ cell count monitoring helps in managing possibilities of the development of opportunistic infections. In the fifth section, continuous-time homogeneous Markov models are fitted, including both CD4+ cell count monitoring and viral load monitoring in one model. Since these variables are assumed to be collinear, principal component analysis was used to treat for the collinearity among these two variables. The models are fitted in such a way that when Markov states are based on CD4+ cell count, the principal component of viral load is included as a covariate, and when the Markov states are based on viral load, the principal component of CD4+cell count is included as a covariate. Results from the models show an improvement in the power of the continuous-time Markov model to explain and predict mortality when both CD4+cellcount and viral load routine monitoring are included in one model.
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    Modelling and analysing risk in precious metals
    (University of the Free State, 2018) Chinhamu, Knowledge; Chikobvu, Delson
    The prices of precious metals are volatile and financial market participants are interested in knowing the downside of holding precious metals in their portfolios. Risk management tools such as Value-at-Risk (VaR) are highly dependent on the underlying distributional assumption. Identifying a distribution that may best capture all the aspects of the given financial data can provide immense advantages to both investors and risk managers. In the analysis and modelling of financial returns, there are stylised facts that are observed. These include volatility clustering, heavy-tails, asymmetry, conditional heavy tails and long range dependence (long memory). In this study, we investigated the stylised facts of gold, platinum and silver returns. We thus propose models that are able to capture their empirical features. The models capture extreme tails of profit and loss distributions and improve the estimation of Value-at-Risk (VaR) of precious metal prices returns. Firstly, we evaluate the performance of existing heavy-tailed and flexible distributions in modelling extreme risk for precious metal returns. The heavy-tailed and flexible distributions used are: Generalised Hyperbolic Distributions (GHDs), Generalised Lambda Distribution (GLD), Stable Distribution (SD), Generalised Pareto Distribution (GPD), Generalised Extreme Value Distribution (GEVD), Pearson type-IV Distribution (PIVD), Symmetrical Student-t Distribution (STD) and Skewed Studentt Distribution (SSTD). Secondly, we couple ARMA-GARCH models and ARMAAPARCH models with heavy-tailed and flexible distributions. We fit the models to precious metal returns and evaluate their relative performance in estimating Valueat-Risk (VaR) using a number of conditional assumptions. The proposed models performed favourably when compared with the APARCH models with a Student-t distribution and the APARCH models with a skewed Student-t distribution usually used in the literature. This provides financial analysts with an alternative distributional scheme to be used in economic modelling. Thirdly, because all daily precious metal price returns exhibit volatility clustering, heavy tails, asymmetry and long range dependence, we fit the long-memory GARCH models under the GHDs, the GPD, the GEVD, the SD, the STD, the SSTD, the GLD and the PIVD assumptions to our price return data. The Anderson-Darling test is used to check for model adequacy. Kupiec likelihood ratio tests and Christoffersen conditional coverage tests are also used in this study to evaluate objectively whether VaR model is adequate. The backtesting results confirm that the long-memory GARCH-heavy-tailed models are adequate for improving risk management assessments and hedging strategies in the highly volatile metal markets. ARFIMA-HYGARCH, ARFIMA-FIGARCH and ARFIMA-FIAPARCH models with PIVD, Normal-Inverse Gaussian Distribution (NIGD), full GHD, FMKL GLD and Generalised Hyperbolic Student-t Distribution (GHStD) innovations are found to be suitable for VaR estimation of precious metals, thereby providing a good alternative candidate for modelling financial returns.
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    Modelling electricity demand in South Africa
    (University of the Free State, 2014-01) Sigauke, Caston; Chikobvu, Delson
    English: Peak electricity demand is an energy policy concern for all countries throughout the world, causing blackouts and increasing electricity tariffs for consumers. This calls for load curtailment strategies to either redistribute or reduce electricity demand during peak periods. This thesis attempts to address this problem by providing relevant information through a frequentist and Bayesian modelling framework for daily peak electricity demand using South African data. The thesis is divided into two parts. The first part deals with modelling of short term daily peak electricity demand. This is done through the investigation of important drivers of electricity demand using (i) piecewise linear regression models, (ii) a multivariate adaptive regression splines (MARS) modelling approach, (iii) a regression with seasonal autoregressive integrated moving average (Reg-SARIMA) model (iv) a Reg-SARIMA model with generalized autoregressive conditional heteroskedastic errors (Reg-SARIMA-GARCH). The second part of the thesis explores the use of extreme value theory in modelling winter peaks, extreme daily positive changes in hourly peak electricity demand and same day of the week increases in peak electricity demand. This is done through fitting the generalized Pareto, generalized single Pareto and the generalized extreme value distributions. One of the major contributions of this thesis is quantification of the amount of electricity which should be shifted to off peak hours. This is achieved through accurate assessment of the level and frequency of future extreme load forecasts. This modelling approach provides a policy framework for load curtailment and determination of the number of critical peak days for power utility companies. This has not been done for electricity demand in the context of South Africa to the best of our knowledge. The thesis further extends the autoregressive moving average-exponential generalized autoregressive conditional heteroskedasticity model to an autoregressive moving average exponential generalized autoregressive conditional heteroskedasticity-generalized single Pareto distribution. The benefit of this hybrid model is in risk modelling of under and over demand predictions of peak electricity demand. Some of the key findings of this thesis are (i) peak electricity demand is influenced by the tails of probability distributions as well as by means or averages, (ii) electricity demand in South Africa rises significantly for average temperature values below 180C and rises slightly for average temperature values above 220C and (iii) modelling under and over demand electricity forecasts provides a basis for risk assessment and quantification of such risk associated with forecasting uncertainty including demand variability.