Theses and Dissertations (Statistics)
http://hdl.handle.net/2263/32483
Fri, 26 Apr 2019 09:35:53 GMT2019-04-26T09:35:53ZProbabilistic SEM : an augmentation to classical Structural Equation Modelling
http://hdl.handle.net/2263/66521
Probabilistic SEM : an augmentation to classical Structural Equation Modelling
Structural equation modelling (SEM) is carried out with the aim of testing hypotheses
on the model of the researcher in a quantitative way, using the sampled data. Although
SEM has developed in many aspects over the past few decades, there are still numerous
advances which can make SEM an even more powerful technique. We propose representing
the nal theoretical SEM by a Bayesian Network (BN), which we would like to call a
Probabilistic Structural Equation Model (PSEM). With the PSEM, we can take things
a step further and conduct inference by explicitly entering evidence into the network and
performing di erent types of inferences. Because the direction of the inference is not an
issue, various scenarios can be simulated using the BN. The augmentation of SEM with
BN provides signi cant contributions to the eld. Firstly, structural learning can mine
data for additional causal information which is not necessarily clear when hypothesising
causality from theory. Secondly, the inference ability of the BN provides not only insight
as mentioned before, but acts as an interactive tool as the `what-if' analysis is dynamic.
Mini Dissertation (MCom)--University of Pretoria, 2018.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/2263/665212018-01-01T00:00:00ZOn the use of Lèvy processes in option pricing
http://hdl.handle.net/2263/65902
On the use of Lèvy processes in option pricing
In this dissertation, we t various nancial models to observed stock prices and we calculate the option prices under each of these models. All of the models considered are based on Lévy processes, which are processes with independent and identically distributed increments. The processes are popular in nance due to their exibility and their desirable mathematical properties. The models considered include the celebrated Black-Scholes model, under which the log-retuns are assumed to be driven by a Brownian motion. Two other classes of models are included in this study, both of which are generalizations of the Black-Scholes model. The rst class is the geometric Lévy process models, of which the Black-Scholes is a special case. Two speci c examples within this class are considered, the two models use the normal inverse Gaussian and Meixner processes to model log-returns. The second class of model considered generalizes the Black-Scholes while modeling the passing of time using an increasing stochastic process. The two speci c examples considered models time using a Pareto and a lognormal process. The aim of this dissertation is to explore the question of which model to use in a given nancial market. To this end, we t each of the models considered to observed log-returns. Following this step we calculate the prices of options available in this market. This is done in order to compare the prices calculated under the models to the prices observed in the market. In each case the Esscher transform is used in order to calculate the equivalent martingale measure used for the calculation of the option prices. Note that this is not the approach typically employed by nancial practitioners. In practice these models are often calibrated to the observed option prices, meaning that the parameters of the models are chosen so as to minimise some distance measure between the observed and calculated option prices. In this dissertation we depart from this methodology in order to determine if the models tted to the stock prices are capable of producing realistic option prices. When analysing the results obtained we use a two fold approach. The rst step is to determine which of the models considered provides the best t to the observed log-returns (this is done by comparing the integrated squared errors between the resulting densities and a kernel density estimate), and the second step is to compare the calculated and observed option prices (using the root mean square error calculated between the two sets of option prices). We conclude that, surprisingly, the model that ts the stock price data best often does not provide an adequate t to the option prices, and vice versa.
Dissertation (MSc)--University of Pretoria, 2018.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/2263/659022018-01-01T00:00:00ZA spatial variant of the Gaussian mixture of regressions model
http://hdl.handle.net/2263/65883
A spatial variant of the Gaussian mixture of regressions model
In this study the nite mixture of multivariate Gaussian distributions is discussed in detail including the derivation of maximum likelihood estimators, a discussion on identi ability of mixture components as well as a discussion on the singularities typically occurring during the estimation process. Examples demonstrate the application of the nite mixture of univariate and bivariate Gaussian distributions. The nite mixture of multivariate Gaussian regressions is discussed including the derivation of maximum likelihood estimators. An example is used to demonstrate the application of the mixture of regressions model. Two methods of calculating the coe cient of determination for measuring model performance are introduced. The application of nite mixtures of Gaussian distributions and regressions to image segmentation problems is examined. The traditional nite mixture models however, have a shortcoming in that commonality of location of observations (pixels) is not taken into account when clustering the data. In literature, this shortcoming is addressed by including a Markov random eld prior for the mixing probabilities and the present study discusses this theoretical development. The resulting nite spatial variant mixture of Gaussian regressions model is de ned and its application is demonstrated in a simulated example. It was found that the spatial variant mixture of Gaussian regressions delivered accurate spatial clustering results and simultaneously accurately estimated the component model parameters. This study contributes an application of the spatial variant mixture of Gaussian regressions model in the agricultural context: maize yields in the Free State are modelled as a function of precipitation, type of maize and season; GPS coordinates linked to the observations provide the location information. A simple linear regression and traditional mixture of Gaussian regressions model were tted for comparative purposes and the latter identi ed three distinct clusters without accounting for location information. It was found that the application of the spatial variant mixture of regressions model resulted in spatially distinct and informative clusters, especially with respect to the type of maize covariate. However, the estimated component regression models for this data set were quite similar. The investigated data set was not perfectly suited for the spatial variant mixture of regressions model application and possible solutions were proposed to improve the model results in future studies. A key learning from the present study is that the e ectiveness of the spatial variant mixture of regressions model is dependent on the clear and distinguishable spatial dependencies in the underlying data set when it is applied to map-type data.
Dissertation (MSc)--University of Pretoria, 2018.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/2263/658832018-01-01T00:00:00ZBayesian kernel density estimation
http://hdl.handle.net/2263/64692
Bayesian kernel density estimation
This dissertation investigates the performance of two-class classi cation credit scoring data
sets with low default ratios. The standard two-class parametric Gaussian and naive Bayes
(NB), as well as the non-parametric Parzen classi ers are extended, using Bayes' rule, to
include either a class imbalance or a Bernoulli prior. This is done with the aim of addressing
the low default probability problem. Furthermore, the performance of Parzen classi cation
with Silverman and Minimum Leave-one-out Entropy (MLE) Gaussian kernel bandwidth
estimation is also investigated. It is shown that the non-parametric Parzen classi ers yield
superior classi cation power.
However, there is a longing for these non-parametric classi ers to posses a predictive power,
such as exhibited by the odds ratio found in logistic regression (LR). The dissertation therefore
dedicates a section to, amongst other things, study the paper entitled \Model-Free Objective
Bayesian Prediction" (Bernardo 1999). Since this approach to Bayesian kernel density
estimation is only developed for the univariate and the uncorrelated multivariate case, the
section develops a theoretical multivariate approach to Bayesian kernel density estimation.
This approach is theoretically capable of handling both correlated as well as uncorrelated
features in data. This is done through the assumption of a multivariate Gaussian kernel
function and the use of an inverse Wishart prior.
Dissertation (MSc)--University of Pretoria, 2018.
Thu, 15 Feb 2018 00:00:00 GMThttp://hdl.handle.net/2263/646922018-02-15T00:00:00Z