Flexible Detection of Groups in Data

Observations often come from a heterogeneous population which consists of different groups. However, the information from which group each observation stems is not observed. This occurs either due to difficulties in the measurement of the group indicator or because not a single characteristic could be identified that captures the grouping. In statistical modeling finite mixtures have been used for more than 100 years as a flexible model class to describe this kind of data and determine the group memberships of the given observations as well as the group sizes and a group-specific statistical model. The areas of application consist of astronomy, biology, economics, marketing and medicine. The usefulness of the application of finite mixture models often suffers from the fact that a-priori knowledge about certain characteristics of the grouping is available, but cannot be easily included in the model. This project aims at overcoming this drawback by offering a suitable approach for fitting a finite mixture model while also taking this additional information into account. Especially the possibility to include information on which observations are likely to be in the same group or should rather end up in different groups will be considered. A possible area of application for this newly developed approach is market segmentation. In market segmentation the aim is to partition the market into sub-markets. Segments are often defined to consist of consumers with similar behavior. However, the possibility to implement a successful marketing strategy is only ensured if these segments do not only differ in their behavior, but also with respect to socio-demographic characteristics. A combined approach taking all requirements on the segments directly into account will ease the statistical analysis and improve the finally derived solution. In addition the rigorous application of advanced mixtures of regression models will be investigated for two different problems: the validation of credit ratings systems using a latent variable approach and the simultaneous accounting for response style heterogeneity among respondents in a segmentation study when survey data is available.

In data often latent groups are suspected to be present, but the group memberships are unobserved. In this case statistical methods are required to unravel the latent structure and learn about the group specific characteristics. Finite mixture models constitute the state-of the-art technique to perform this task with a statistical model-based approach. In this project several extensions of the general model class of finite mixtures were considered. These extensions allow to suitably model different types of data in a range of applications and enlarge the toolbox of statistical methods in order to better capture available information in data. The theoretical statistical properties and estimation methods of these models were analyzed and the algorithms were implemented in the freely available open-source add-on package flexmix for the statistical software environment R. Within the Bayesian framework prior choices were investigated and developed which lead to sparse solutions.Applications included the modeling of HIV RNV levels over time using mixtures of linear mixed-effects models for censored data, of time-course gene expression levels over time using mixtures of linear additive models, of reading skill evaluations in children using mixtures of beta regression models and text corpora using topic models based on the latent Dirichlet allocation model and mixtures of von Mises-Fisher distributions. In addition sample size recommendations were developed for market segmentation applications in tourism.