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Data Science is the new and exciting interdisciplinary response that has emerged as a consequence of the staggering amounts of data generated in many new forms from digital images to audio to text. It is an interdisciplinary field involving Statistics, Computer Science and Mathematics. It involves the study of data, how they are collected, stored, accessed, visualised, modelled and ultimately used to inform decision making by turning data into intelligence. Despite this ’data revolution’ and the development of Data Science as a consequence, the aim of any data analysis is still the same, to make inference about unknown population parameters using sample statistics. One fundamental challenge in inference is the identification of outliers. Such oddities, or atypical observations, could be indicative of poor data management or biased sampling. In this situation the presence of such outliers are considered a negative aspect and efforts are needed to account for them (e.g. correct data entry errors) accordingly to avoid introducing bias in parameter estimation. On the other hand, finding an outlier may be the key focus of the exercise as an outlier may represent something new and novel. Many statistical methods have been developed to identify outlying data points and robust methods developed to account for outliers in statistical models. A central property of all such methods is that an observation is classified as an outlier or not (i.e. a binary decision); being able to quantify an observations ’outlyingness’ is clearly an attractive alternative. In this thesis, a novel method is presented for outlier detection in multivariate data based on the idea of a statistical depth function. The proposed approach enables outlier detection in multivariate data while taking into consideration the local geometry of the underlying probability distribution. 2021-09-14

We study statistical properties after a sudden episode of wind for water waves propagating in one direction. A wave with random initial conditions is propagated using a forced-damped higher order Nonlinear Schr\"odinger equation (NLS). During the wind episode, the wave action increases, the spectrum broadens, the spectral mean shifts up and the Benjamin-Feir index (BFI) and the kurtosis increase. Conversely, after the wind episode, the opposite occurs for each quantity. The kurtosis of the wave height distribution is considered the main parameter that can indicate whether rogue waves are likely to occur in a sea state, and the BFI is often mentioned as a means to predict the kurtosis. However, we find that while there is indeed a quadratic relation between these two, this relationship is dependent on the details of the forcing and damping. Instead, a simple and robust quadratic relation does exist between the kurtosis and the bandwidth. This could allow for a single-spectrum assessment of the likelihood of rogue waves in a given sea state. In addition, as the kurtosis depends strongly on the damping and forcing coefficients, by combining the bandwidth measurement with the damping coefficient, the evolution of the kurtosis after the wind episode can be predicted., Comment: 30 pages, 12 figures

Ewens sampling formula is the sampling distribution for a population assumed to follow a one parameter Poisson-Dirichlet distribution, where the parameter is fixed. In this project this assumption will be loosened and we will look at the parameter as a function of the sample size. This will result in sampling from a family of Poisson-Dirichlet distributions. Estimators for this new construction will be tested using two different simulation methods. Thesis Master of Science (MSc)