Ensemble Copula Coupling as a Multivariate Discrete Copula Approach

In probability and statistics, copulas play important roles theoretically as well as to address a wide range of problems in various application areas. In this paper, we introduce the concept of multivariate discrete copulas, discuss their equivalence to stochastic arrays, and provide a multivariate discrete version of Sklar's theorem. These results provide the theoretical frame for the ensemble copula coupling approach proposed by Schefzik et al. (2013) for the multivariate statistical postprocessing of weather forecasts made by ensemble systems.Comment: references correcte

Greek Life : Greek-letter Student Societies in the United States Higher Education System on the Local and National Scale

This diploma thesis takes a look at what researchers sometimes label as a phenomenon that is exclusive to the United States of America, i.e. Greek life (DeSantis 6). The aim is to explain to the reader what Greek life or a Greek life organization is and what its features are besides being student groups that name themselves with Greek letters. There is much more to be elaborated about these groups. The two core elements of this paper are the explanation of the Greek life phenomenon on a US nationwide basis and the comparison of the information obtained about Greek life on a national basis to how this system operates and takes form on a local scale. The site for comparison on the local scale will be the University of New Orleans, located in Louisiana. For this paper, research has been conducted at the University of New Orleans in order to obtain information on how the Greek life system is to be described and evaluated at that institution. Comparing Greek life on the national and local scale has the benefit of providing the opportunity to take a detailed look at how aspects of a national phenomenon are put into practice on the local scale. Authentic source materials, firsthand experience, interviews and on-site research serve as the basis for this diploma thesis

Greek Life : Greek-letter Student Societies in the United States Higher Education System on the Local and National Scale

This diploma thesis takes a look at what researchers sometimes label as a phenomenon that is exclusive to the United States of America, i.e. Greek life (DeSantis 6). The aim is to explain to the reader what Greek life or a Greek life organization is and what its features are besides being student groups that name themselves with Greek letters. There is much more to be elaborated about these groups. The two core elements of this paper are the explanation of the Greek life phenomenon on a US nationwide basis and the comparison of the information obtained about Greek life on a national basis to how this system operates and takes form on a local scale. The site for comparison on the local scale will be the University of New Orleans, located in Louisiana. For this paper, research has been conducted at the University of New Orleans in order to obtain information on how the Greek life system is to be described and evaluated at that institution. Comparing Greek life on the national and local scale has the benefit of providing the opportunity to take a detailed look at how aspects of a national phenomenon are put into practice on the local scale. Authentic source materials, firsthand experience, interviews and on-site research serve as the basis for this diploma thesis

SimBPDD: Simulating differential distributions in Beta-Poisson models, in particular for single-cell RNA sequencing data

Beta-Poisson (BP) models employ Poisson distributions, where the corresponding rate parameter itself is a Beta-distributed random variable. They have been shown to appropriately mimic gene expression distributions in the context of single-cell ribonucleic acid sequencing (scRNA-seq), a breakthrough technology allowing to sequence information from individual biological cells and facilitating fundamental insights into numerous fields of biology. A prominent scRNA-seq data analysis task is to identify differences in gene expression distributions across two conditions. To validate new statistical approaches in this context, one typically has to rely on accurate simulations, as usually no ground truth for an assessment is available. We introduce several simulation procedures that allow to generate differential distributions (DDs) based on BP models. In particular, we describe how to create different types of DDs, mirroring various sources or origins of a difference, and different degrees of DDs, from a weak to a strong difference. The soundness of the simulation procedures is shown in a validation study in which theoretically expected model properties of the DD simulations are confirmed. The findings are in principle not restricted to the scRNA-seq context and may be generally applicable also to other application areas. The simulation approaches are implemented in the publicly available R package SimBPDD

Physically coherent probabilistic weather forecasts using multivariate discrete copula-based ensemble postprocessing methods

Being able to provide accurate forecasts of future quantities has always been a great human desire and is essential in numerous situations in daily life. Meanwhile, it has become routine to work with probabilistic forecasts in the form of full predictive distributions rather than with single deterministic point forecasts in many disciplines, with weather prediction acting as a key example. Nowadays, probabilistic weather forecasts are usually constructed from ensemble prediction systems, which consist of multiple runs of numerical weather prediction models differing in the initial conditions and/or the parameterized numerical representation of the atmosphere. The raw ensemble forecasts typically reveal biases and dispersion errors and thus call for statistical postprocessing to realize their full potential. Several ensemble postprocessing methods have been developed and are partly recapitulated in this thesis, yet many of them only apply to a single weather quantity at a single location and for a single prediction horizon. In many applications, however, there is a critical need to account for spatial, temporal and inter-variable dependencies. To address this, a tool called ensemble copula coupling (ECC) is introduced and examined. Essentially, ECC uses the empirical copula induced by the raw ensemble to aggregate samples from predictive distributions for each location, variable and look-ahead time separately, which are obtained via existing univariate postprocessing methods. The ECC ensemble inherits the multivariate rank dependence pattern from the raw ensemble, thereby capturing the flow dependence. Several variants and modifications of ECC are studied, and it is demonstrated that the ECC concept provides an overarching frame for existing techniques scattered in the literature. From a mathematical point of view, it is shown that ECC can be considered a copula approach by pointing out relationships to multivariate discrete copulas, which are introduced in this thesis and for which relevant mathematical properties are derived. A generalization of standard ECC is introduced, which aggregates samples from not necessarily univariate, but general predictive distributions obtained by low-dimensional postprocessing in an ECC-like manner. Finally, the SimSchaake approach, which combines the notion of similarity-based ensemble methods with that of the so-called Schaake shuffle, is presented as an alternative to ECC. In this technique, the dependence patterns are based on verifying observations rather than on raw ensemble forecasts as in ECC. The methods and concepts are illustrated and evaluated based on case studies, using real ensemble forecast data of the European Centre for Medium-Range Weather Forecasts. Essentially, the new multivariate approaches developed in this thesis reveal good predictive performances, thus contributing to improved probabilistic forecasts

SimBPDD: Simulating differential distributions in Beta-Poisson models, in particular for single-cell RNA sequencing data

Beta-Poisson (BP) models employ Poisson distributions, where the corresponding rate parameter itself is a Beta-distributed random variable. They have been shown to appropriately mimic gene expression distributions in the context of single-cell ribonucleic acid sequencing (scRNA-seq), a breakthrough technology allowing to sequence information from individual biological cells and facilitating fundamental insights into numerous fields of biology. A prominent scRNA-seq data analysis task is to identify differences in gene expression distributions across two conditions. To validate new statistical approaches in this context, one typically has to rely on accurate simulations, as usually no ground truth for an assessment is available. We introduce several simulation procedures that allow to generate differential distributions (DDs) based on BP models. In particular, we describe how to create different types of DDs, mirroring various sources or origins of a difference, and different degrees of DDs, from a weak to a strong difference. The soundness of the simulation procedures is shown in a validation study in which theoretically expected model properties of the DD simulations are confirmed. The findings are in principle not restricted to the scRNA-seq context and may be generally applicable also to other application areas. The simulation approaches are implemented in the publicly available R package SimBPDD