Multiscale summaries of probability measures with applications to plant and microbiome data

Traditional descriptors such as the mean and the covariance matrix give useful global summaries of data and probability measures. Nevertheless, when distributions with more complex topological and geometrical behaviors arise, these methods fall short in accurately describing them. This dissertation explores and develops new methods that provide more informative summaries of complex probability measures using multiscale analogs of the Frechet function and the covariance tensor which encode variation of data with respect to any point in the domain. These multiscale methods are developed using kernel functions and diffusion distances and are helpful in obtaining more information on local-to-regional-to-global behavior of probability measures, unlike the traditional take which only gives global summaries. We applied the methods to the analysis of climatic data of the Fabaceae plant family (legumes) and to microbiome data related to the Clostridium difficile infection in the human gut. Our studies reveal patterns of climatological adaptation of various legume taxa and changes in the interactions of microbial communities in the presence of infection which are helpful in monitoring the resolution of the disease.