Spatial dependence, trends, functional outliers and sparsity in compositional data analysis

Abstract

Many real world datasets are nowadays multivariate. The last decades have seen an explosion of the number of methods for multivariate data. Such data, usually considered to be real valued, and the corresponding statistical methods, are based on the Euclidean geometry. The underlying property of the Euclidean geometry is to measure similarity of two points as the length of the line between such. For settings in which data points consist of strictly positive variables, which are plentiful in the applied sciences, a different approach needs to be considered. Recently, a concept quickly gaining popularity, called Compositional Data Analysis (CoDA) has emerged. In CoDA one measures similarity of two multivariate strictly positive data points by comparing the log-ratios respectively. With this change of perspective comes the necessity to adapt classical statistical methods. Many such methods have been extended to this framework in recent years. In this thesis we look into applications of a combined approach between the log-ratio methodology, which is at the heart of CoDA, and Generalized Additive Models, to find important elements in a geochemical exploration setting. Furthermore we explore various extensions of CoDA such as change point detection of compositional time series, outlier detection of compositional functional data and very fruitful connections with signal processing on graphs.